3.233 \(\int \frac{\sin ^2(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=347 \[ -\frac{\tan (c+d x) \left (\frac{5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}\right )}{32 a^2 d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac{b \tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 a d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}+\frac{\left (-14 \sqrt{a} \sqrt{b}+12 a+5 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \sqrt{b} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}-\frac{\left (14 \sqrt{a} \sqrt{b}+12 a+5 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \sqrt{b} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}} \]

[Out]

((12*a - 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a
] - Sqrt[b])^(5/2)*Sqrt[b]*d) - ((12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x
])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b +
b^2)*Tan[c + d*x]^2))/(8*a*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((
2*a*(5*a^2 - 9*a*b - 4*b^2))/(a - b)^3 + (5*(2*a^2 + 3*a*b - b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a + 2
*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

________________________________________________________________________________________

Rubi [A]  time = 0.724284, antiderivative size = 347, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3217, 1333, 1678, 1166, 205} \[ -\frac{\tan (c+d x) \left (\frac{5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac{2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}\right )}{32 a^2 d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac{b \tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 a d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}+\frac{\left (-14 \sqrt{a} \sqrt{b}+12 a+5 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \sqrt{b} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}-\frac{\left (14 \sqrt{a} \sqrt{b}+12 a+5 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \sqrt{b} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((12*a - 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a
] - Sqrt[b])^(5/2)*Sqrt[b]*d) - ((12*a + 14*Sqrt[a]*Sqrt[b] + 5*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x
])/a^(1/4)])/(64*a^(9/4)*(Sqrt[a] + Sqrt[b])^(5/2)*Sqrt[b]*d) - (b*Tan[c + d*x]*(a*(a + 3*b) + (a^2 + 6*a*b +
b^2)*Tan[c + d*x]^2))/(8*a*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4)^2) - (Tan[c + d*x]*((
2*a*(5*a^2 - 9*a*b - 4*b^2))/(a - b)^3 + (5*(2*a^2 + 3*a*b - b^2)*Tan[c + d*x]^2)/(a - b)^2))/(32*a^2*d*(a + 2
*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

Rule 3217

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rule 1333

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coe
ff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d +
e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*
f - 2*a*g)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)
^(p + 1)*Simp[ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x
] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)*x^2, x], x], x], x]] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && IGtQ[m/2, 0]

Rule 1678

Int[(Pq_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Pq, a +
b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[Pq, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2
+ c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*
a*(p + 1)*(b^2 - 4*a*c)), Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuot
ient[Pq, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4*p + 7)*(b*d - 2*a*e)*x^2,
x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && Expon[Pq, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (1+x^2\right )^4}{\left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{2 a^2 b^2 (a+3 b)}{(a-b)^3}-\frac{2 a b \left (8 a^3-29 a^2 b+18 a b^2-5 b^3\right ) x^2}{(a-b)^3}-\frac{32 a^2 (a-2 b) b x^4}{(a-b)^2}-\frac{16 a^2 b x^6}{a-b}}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{16 a^2 b d}\\ &=-\frac{b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\tan (c+d x) \left (\frac{2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac{5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{\frac{8 a^3 (5 a-2 b) b^2}{(a-b)^2}+\frac{4 a^2 b^2 \left (22 a^2-15 a b+5 b^2\right ) x^2}{(a-b)^2}}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{128 a^4 b^2 d}\\ &=-\frac{b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\tan (c+d x) \left (\frac{2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac{5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac{\left (\left (\sqrt{a}+\sqrt{b}\right ) \left (12 a-14 \sqrt{a} \sqrt{b}+5 b\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 a^2 \left (\sqrt{a}-\sqrt{b}\right )^2 \sqrt{b} d}-\frac{\left (\left (\sqrt{a}-\sqrt{b}\right ) \left (12 a+14 \sqrt{a} \sqrt{b}+5 b\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 a^2 \left (\sqrt{a}+\sqrt{b}\right )^2 \sqrt{b} d}\\ &=\frac{\left (12 a-14 \sqrt{a} \sqrt{b}+5 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} \sqrt{b} d}-\frac{\left (12 a+14 \sqrt{a} \sqrt{b}+5 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{9/4} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} \sqrt{b} d}-\frac{b \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac{\tan (c+d x) \left (\frac{2 a \left (5 a^2-9 a b-4 b^2\right )}{(a-b)^3}+\frac{5 \left (2 a^2+3 a b-b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 6.43048, size = 457, normalized size = 1.32 \[ \frac{\left (11 a^{3/2} b^{3/2}-12 a^{5/2} \sqrt{b}+10 a^2 b-4 a b^2-5 \sqrt{a} b^{5/2}\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a} \sqrt{b}+b\right ) \tan (c+d x)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{64 a^{5/2} b d \sqrt{\sqrt{a} \sqrt{b}+a} (a-b)^2}+\frac{24 a^2 \sin (2 (c+d x))+22 a b \sin (2 (c+d x))-11 a b \sin (4 (c+d x))-10 b^2 \sin (2 (c+d x))+5 b^2 \sin (4 (c+d x))}{32 a^2 d (a-b)^2 (-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)}-\frac{\left (-11 a^{3/2} b^{3/2}+12 a^{5/2} \sqrt{b}+10 a^2 b-4 a b^2+5 \sqrt{a} b^{5/2}\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a} \sqrt{b}-b\right ) \tan (c+d x)}{\sqrt{b} \sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{64 a^{5/2} b d \sqrt{\sqrt{a} \sqrt{b}-a} (a-b)^2}+\frac{-4 a \sin (2 (c+d x))-2 b \sin (2 (c+d x))+b \sin (4 (c+d x))}{a d (a-b) (-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

((-12*a^(5/2)*Sqrt[b] + 10*a^2*b + 11*a^(3/2)*b^(3/2) - 4*a*b^2 - 5*Sqrt[a]*b^(5/2))*ArcTan[((Sqrt[a]*Sqrt[b]
+ b)*Tan[c + d*x])/(Sqrt[a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(64*a^(5/2)*Sqrt[a + Sqrt[a]*Sqrt[b]]*(a - b)^2*b*d)
- ((12*a^(5/2)*Sqrt[b] + 10*a^2*b - 11*a^(3/2)*b^(3/2) - 4*a*b^2 + 5*Sqrt[a]*b^(5/2))*ArcTanh[((Sqrt[a]*Sqrt[b
] - b)*Tan[c + d*x])/(Sqrt[-a + Sqrt[a]*Sqrt[b]]*Sqrt[b])])/(64*a^(5/2)*Sqrt[-a + Sqrt[a]*Sqrt[b]]*(a - b)^2*b
*d) + (-4*a*Sin[2*(c + d*x)] - 2*b*Sin[2*(c + d*x)] + b*Sin[4*(c + d*x)])/(a*(a - b)*d*(-8*a + 3*b - 4*b*Cos[2
*(c + d*x)] + b*Cos[4*(c + d*x)])^2) + (24*a^2*Sin[2*(c + d*x)] + 22*a*b*Sin[2*(c + d*x)] - 10*b^2*Sin[2*(c +
d*x)] - 11*a*b*Sin[4*(c + d*x)] + 5*b^2*Sin[4*(c + d*x)])/(32*a^2*(a - b)^2*d*(-8*a + 3*b - 4*b*Cos[2*(c + d*x
)] + b*Cos[4*(c + d*x)]))

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Maple [B]  time = 0.145, size = 1906, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x)

[Out]

-13/64/d*b/(a^2-2*a*b+b^2)*a/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(
1/2)+a)*(a-b))^(1/2))+13/64/d*b/(a^2-2*a*b+b^2)*a/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+
b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-3/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b/(a^
2-2*a*b+b^2)*tan(d*x+c)^3-3/8/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b/(a^2-2*a*b+b^2)*tan(d*x
+c)^5-15/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*a/(a^2-2*a*b+b^2)*tan(d*x+c)^3-5/16/d/(tan(
d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*a/(a^2-2*a*b+b^2)*tan(d*x+c)+5/32/d/(tan(d*x+c)^4*a-tan(d*x+c)
^4*b+2*a*tan(d*x+c)^2+a)^2/a^2/(a-b)*tan(d*x+c)^7*b^2+9/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a
)^2/a/(a^2-2*a*b+b^2)*tan(d*x+c)^3*b^2-1/64/d/a/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2
)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*b^3+1/64/d/a/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b
)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*b^3-5/16/d/(tan(d*x+c)^4*a-ta
n(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a-b)*tan(d*x+c)^7-15/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+
a)^2/(a-b)/a*b*tan(d*x+c)^7+11/32/d/(a^2-2*a*b+b^2)*a/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x
+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-15/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a^2-2*a*b+b^2
)*tan(d*x+c)^5*a+11/32/d/(a^2-2*a*b+b^2)*a/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*
b)^(1/2)-a)*(a-b))^(1/2))+5/16/d/a*b^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*
x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+5/16/d/a*b^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((
-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-37/64/d*b/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*
arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-37/64/d*b/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))
^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+3/16/d/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*
b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*a^2-3/16/d/(a^2-2*a*b+b^2)/(a*
b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*a^2+1/32
/d*b^2/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a
)*(a-b))^(1/2))-1/32/d*b^2/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(
d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-5/64/d/a^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a
-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*b^3-5/64/d/a^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/
2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*b^3+9/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(
d*x+c)^2+a)^2/a/(a^2-2*a*b+b^2)*tan(d*x+c)^5*b^2+1/8/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a
^2-2*a*b+b^2)*tan(d*x+c)*b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/16*(4*(96*a^3*b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((12*a^2*b^3 - 11*a*
b^4 + 5*b^5)*sin(14*d*x + 14*c) - (104*a^2*b^3 - 85*a*b^4 + 35*b^5)*sin(12*d*x + 12*c) - (320*a^3*b^2 - 652*a^
2*b^3 + 407*a*b^4 - 105*b^5)*sin(10*d*x + 10*c) + (1408*a^3*b^2 - 1696*a^2*b^3 + 865*a*b^4 - 175*b^5)*sin(8*d*
x + 8*c) + (320*a^3*b^2 + 756*a^2*b^3 - 849*a*b^4 + 175*b^5)*sin(6*d*x + 6*c) - (248*a^2*b^3 - 383*a*b^4 + 105
*b^5)*sin(4*d*x + 4*c) - (12*a^2*b^3 + 77*a*b^4 - 35*b^5)*sin(2*d*x + 2*c))*cos(16*d*x + 16*c) + 2*(2*(96*a^3*
b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*sin(12*d*x + 12*c) + 8*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4 - 35*b^5)*s
in(10*d*x + 10*c) - 3*(512*a^4*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*sin(8*d*x + 8*c) - 16*(1
28*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*sin(6*d*x + 6*c) + 2*(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4 + 17
5*b^5)*sin(4*d*x + 4*c) + 24*(4*a^2*b^3 + 11*a*b^4 - 5*b^5)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 2*(2*(2560*
a^4*b - 4128*a^3*b^2 + 3644*a^2*b^3 - 1379*a*b^4 + 245*b^5)*sin(10*d*x + 10*c) - (9216*a^4*b - 25984*a^3*b^2 +
21304*a^2*b^3 - 8575*a*b^4 + 1225*b^5)*sin(8*d*x + 8*c) - 2*(2560*a^4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*a
*b^4 - 735*b^5)*sin(6*d*x + 6*c) + 4*(576*a^3*b^2 - 1696*a^2*b^3 + 1323*a*b^4 - 245*b^5)*sin(4*d*x + 4*c) + 2*
(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4 + 175*b^5)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) + 2*((40960*a^5 - 24064*
a^4*b - 22080*a^3*b^2 + 27516*a^2*b^3 - 11095*a*b^4 + 1225*b^5)*sin(8*d*x + 8*c) + 8*(5120*a^4*b - 1408*a^3*b^
2 - 3900*a^2*b^3 + 2107*a*b^4 - 245*b^5)*sin(6*d*x + 6*c) - 2*(2560*a^4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*
a*b^4 - 735*b^5)*sin(4*d*x + 4*c) - 16*(128*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*sin(2*d*x + 2*c))*cos(
10*d*x + 10*c) + 2*((40960*a^5 - 24064*a^4*b - 22080*a^3*b^2 + 27516*a^2*b^3 - 11095*a*b^4 + 1225*b^5)*sin(6*d
*x + 6*c) - (9216*a^4*b - 25984*a^3*b^2 + 21304*a^2*b^3 - 8575*a*b^4 + 1225*b^5)*sin(4*d*x + 4*c) - 3*(512*a^4
*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 4*((2560*a^4*b -
4128*a^3*b^2 + 3644*a^2*b^3 - 1379*a*b^4 + 245*b^5)*sin(4*d*x + 4*c) + 4*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4
- 35*b^5)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 16*((a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(16*d*x + 16*c)^2 + 6
4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^
3*b^5 + 49*a^2*b^6)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^
2*b^6)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^8 - 57344*a^7*b + 83712*a^6*b^2 - 67648*a^5*b^3 + 32841*a^4*b^4 - 9
170*a^3*b^5 + 1225*a^2*b^6)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 +
49*a^2*b^6)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*c
os(4*d*x + 4*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c)^2 + (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*
d*sin(16*d*x + 16*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^6*b^2 - 240*a^5*
b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4
*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^8 - 57344*a^7*b + 83712*a^6*b^2 - 67648*a
^5*b^3 + 32841*a^4*b^4 - 9170*a^3*b^5 + 1225*a^2*b^6)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 + 7
53*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210
*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x
+ 4*c)*sin(2*d*x + 2*c) + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x + 2*c)^2 - 16*(a^4*b^4 - 2*a^3*b^5 +
a^2*b^6)*d*cos(2*d*x + 2*c) + (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d - 2*(8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(
14*d*x + 14*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^5*b^3 - 39
*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3
*b^5 + 35*a^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c
) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)
*d*cos(2*d*x + 2*c) - (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^5*b^3 - 23*a^4*b^4 +
22*a^3*b^5 - 7*a^2*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d
*x + 10*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16
*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 -
7*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c) - (a^4*b^4 - 2*a^3*b^5 +
a^2*b^6)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*c
os(10*d*x + 10*c) + 2*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d
*cos(8*d*x + 8*c) + 8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6*c)
- 4*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(8*a^5*b^3 - 23
*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*c
os(12*d*x + 12*c) + 16*(2*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^
6)*d*cos(8*d*x + 8*c) + 8*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6
*c) - 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(16*a^5*b^
3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^
6)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*
a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*
a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*cos(2*d
*x + 2*c) + (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(
128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(16*a^5*b^3 - 39*a^
4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) - (16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d)*cos
(6*d*x + 6*c) + 8*(8*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) - (8*a^5*b^3 - 23*a^
4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*cos(4*d*x + 4*c) - 4*(4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(14*d*x + 14*c
) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30
*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b
^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^5*b^
3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c) + 4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x +
2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(12*d*x + 12*c) - 4*(1
6*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b
^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin
(6*d*x + 6*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c) + 4*(a^4*b^4 - 2*a^3*b^
5 + a^2*b^6)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3
*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c) + (1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*
b^5 - 245*a^2*b^6)*d*sin(8*d*x + 8*c) + 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)
*d*sin(6*d*x + 6*c) - 2*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c)
- 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(12*d*x + 12*c) + 32*((2048*a^7*
b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*sin(8*d*x + 8*c) + 4*(256*a^6*b
^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c) - 2*(128*a^6*b^2 - 424*a^5*b^3 +
513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*
b^6)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 +
1722*a^3*b^5 - 245*a^2*b^6)*d*sin(6*d*x + 6*c) - (1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b^4 + 1
442*a^3*b^5 - 245*a^2*b^6)*d*sin(4*d*x + 4*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*
a^2*b^6)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 4
9*a^2*b^6)*d*sin(4*d*x + 4*c) + 2*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(6
*d*x + 6*c))*integrate(-1/8*(4*(12*a^2*b - 11*a*b^2 + 5*b^3)*cos(6*d*x + 6*c)^2 - 4*(256*a^3 - 248*a^2*b + 97*
a*b^2 - 15*b^3)*cos(4*d*x + 4*c)^2 + 4*(12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c)^2 + 4*(12*a^2*b - 11*a*b
^2 + 5*b^3)*sin(6*d*x + 6*c)^2 - 4*(256*a^3 - 248*a^2*b + 97*a*b^2 - 15*b^3)*sin(4*d*x + 4*c)^2 + 2*(96*a^3 -
252*a^2*b + 149*a*b^2 - 35*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(12*a^2*b - 11*a*b^2 + 5*b^3)*sin(2*d*x
+ 2*c)^2 - ((12*a^2*b - 11*a*b^2 + 5*b^3)*cos(6*d*x + 6*c) - 2*(32*a^2*b - 19*a*b^2 + 5*b^3)*cos(4*d*x + 4*c)
+ (12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - (12*a^2*b - 11*a*b^2 + 5*b^3 - 2*(96*a^3
- 252*a^2*b + 149*a*b^2 - 35*b^3)*cos(4*d*x + 4*c) - 8*(12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c))*cos(6*d
*x + 6*c) + 2*(32*a^2*b - 19*a*b^2 + 5*b^3 + (96*a^3 - 252*a^2*b + 149*a*b^2 - 35*b^3)*cos(2*d*x + 2*c))*cos(4
*d*x + 4*c) - (12*a^2*b - 11*a*b^2 + 5*b^3)*cos(2*d*x + 2*c) - ((12*a^2*b - 11*a*b^2 + 5*b^3)*sin(6*d*x + 6*c)
- 2*(32*a^2*b - 19*a*b^2 + 5*b^3)*sin(4*d*x + 4*c) + (12*a^2*b - 11*a*b^2 + 5*b^3)*sin(2*d*x + 2*c))*sin(8*d*
x + 8*c) + 2*((96*a^3 - 252*a^2*b + 149*a*b^2 - 35*b^3)*sin(4*d*x + 4*c) + 4*(12*a^2*b - 11*a*b^2 + 5*b^3)*sin
(2*d*x + 2*c))*sin(6*d*x + 6*c))/(a^4*b^2 - 2*a^3*b^3 + a^2*b^4 + (a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(8*d*x +
8*c)^2 + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(6*d*x + 6*c)^2 + 4*(64*a^6 - 176*a^5*b + 169*a^4*b^2 - 66*a^3*
b^3 + 9*a^2*b^4)*cos(4*d*x + 4*c)^2 + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(2*d*x + 2*c)^2 + (a^4*b^2 - 2*a^3
*b^3 + a^2*b^4)*sin(8*d*x + 8*c)^2 + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(6*d*x + 6*c)^2 + 4*(64*a^6 - 176*a
^5*b + 169*a^4*b^2 - 66*a^3*b^3 + 9*a^2*b^4)*sin(4*d*x + 4*c)^2 + 16*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^
2*b^4)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(2*d*x + 2*c)^2 + 2*(a^4*b^2
- 2*a^3*b^3 + a^2*b^4 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(6*d*x + 6*c) - 2*(8*a^5*b - 19*a^4*b^2 + 14*a^3*
b^3 - 3*a^2*b^4)*cos(4*d*x + 4*c) - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(
a^4*b^2 - 2*a^3*b^3 + a^2*b^4 - 2*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4)*cos(4*d*x + 4*c) - 4*(a^4*b^
2 - 2*a^3*b^3 + a^2*b^4)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4
- 4*(8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^4*b^2 - 2*a^3*b
^3 + a^2*b^4)*cos(2*d*x + 2*c) - 4*(2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(6*d*x + 6*c) + (8*a^5*b - 19*a^4*b^2
+ 14*a^3*b^3 - 3*a^2*b^4)*sin(4*d*x + 4*c) + 2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*sin(2*d*x + 2*c))*sin(8*d*x +
8*c) + 16*((8*a^5*b - 19*a^4*b^2 + 14*a^3*b^3 - 3*a^2*b^4)*sin(4*d*x + 4*c) + 2*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4
)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - (11*a*b^4 - 5*b^5 + (12*a^2*b^3 - 11*a*b^4 + 5*b^5)*cos(14*d*x + 1
4*c) - (104*a^2*b^3 - 85*a*b^4 + 35*b^5)*cos(12*d*x + 12*c) - (320*a^3*b^2 - 652*a^2*b^3 + 407*a*b^4 - 105*b^5
)*cos(10*d*x + 10*c) + (1408*a^3*b^2 - 1696*a^2*b^3 + 865*a*b^4 - 175*b^5)*cos(8*d*x + 8*c) + (320*a^3*b^2 + 7
56*a^2*b^3 - 849*a*b^4 + 175*b^5)*cos(6*d*x + 6*c) - (248*a^2*b^3 - 383*a*b^4 + 105*b^5)*cos(4*d*x + 4*c) - (1
2*a^2*b^3 + 77*a*b^4 - 35*b^5)*cos(2*d*x + 2*c))*sin(16*d*x + 16*c) + (12*a^2*b^3 + 77*a*b^4 - 35*b^5 - 4*(96*
a^3*b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*cos(12*d*x + 12*c) - 16*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4 - 35*b
^5)*cos(10*d*x + 10*c) + 6*(512*a^4*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*cos(8*d*x + 8*c) +
32*(128*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*cos(6*d*x + 6*c) - 4*(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4
+ 175*b^5)*cos(4*d*x + 4*c) - 48*(4*a^2*b^3 + 11*a*b^4 - 5*b^5)*cos(2*d*x + 2*c))*sin(14*d*x + 14*c) + (248*a
^2*b^3 - 383*a*b^4 + 105*b^5 - 4*(2560*a^4*b - 4128*a^3*b^2 + 3644*a^2*b^3 - 1379*a*b^4 + 245*b^5)*cos(10*d*x
+ 10*c) + 2*(9216*a^4*b - 25984*a^3*b^2 + 21304*a^2*b^3 - 8575*a*b^4 + 1225*b^5)*cos(8*d*x + 8*c) + 4*(2560*a^
4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*a*b^4 - 735*b^5)*cos(6*d*x + 6*c) - 8*(576*a^3*b^2 - 1696*a^2*b^3 + 13
23*a*b^4 - 245*b^5)*cos(4*d*x + 4*c) - 4*(96*a^3*b^2 + 324*a^2*b^3 - 649*a*b^4 + 175*b^5)*cos(2*d*x + 2*c))*si
n(12*d*x + 12*c) - (320*a^3*b^2 + 756*a^2*b^3 - 849*a*b^4 + 175*b^5 + 2*(40960*a^5 - 24064*a^4*b - 22080*a^3*b
^2 + 27516*a^2*b^3 - 11095*a*b^4 + 1225*b^5)*cos(8*d*x + 8*c) + 16*(5120*a^4*b - 1408*a^3*b^2 - 3900*a^2*b^3 +
2107*a*b^4 - 245*b^5)*cos(6*d*x + 6*c) - 4*(2560*a^4*b + 480*a^3*b^2 - 7908*a^2*b^3 + 5033*a*b^4 - 735*b^5)*c
os(4*d*x + 4*c) - 32*(128*a^3*b^2 + 124*a^2*b^3 - 173*a*b^4 + 35*b^5)*cos(2*d*x + 2*c))*sin(10*d*x + 10*c) - (
1408*a^3*b^2 - 1696*a^2*b^3 + 865*a*b^4 - 175*b^5 + 2*(40960*a^5 - 24064*a^4*b - 22080*a^3*b^2 + 27516*a^2*b^3
- 11095*a*b^4 + 1225*b^5)*cos(6*d*x + 6*c) - 2*(9216*a^4*b - 25984*a^3*b^2 + 21304*a^2*b^3 - 8575*a*b^4 + 122
5*b^5)*cos(4*d*x + 4*c) - 6*(512*a^4*b + 1024*a^3*b^2 - 1556*a^2*b^3 + 865*a*b^4 - 175*b^5)*cos(2*d*x + 2*c))*
sin(8*d*x + 8*c) + (320*a^3*b^2 - 652*a^2*b^3 + 407*a*b^4 - 105*b^5 - 4*(2560*a^4*b - 4128*a^3*b^2 + 3644*a^2*
b^3 - 1379*a*b^4 + 245*b^5)*cos(4*d*x + 4*c) - 16*(64*a^3*b^2 - 196*a^2*b^3 + 125*a*b^4 - 35*b^5)*cos(2*d*x +
2*c))*sin(6*d*x + 6*c) + (104*a^2*b^3 - 85*a*b^4 + 35*b^5 - 4*(96*a^3*b^2 + 36*a^2*b^3 - 53*a*b^4 + 35*b^5)*co
s(2*d*x + 2*c))*sin(4*d*x + 4*c) - (12*a^2*b^3 - 11*a*b^4 + 5*b^5)*sin(2*d*x + 2*c))/((a^4*b^4 - 2*a^3*b^5 + a
^2*b^6)*d*cos(16*d*x + 16*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(14*d*x + 14*c)^2 + 16*(64*a^6*b^2 -
240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^6*b^2 - 736*a^5*b^3 +
753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^8 - 57344*a^7*b + 83712*a^6*b^2 -
67648*a^5*b^3 + 32841*a^4*b^4 - 9170*a^3*b^5 + 1225*a^2*b^6)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^6*b^2 - 736*a^5
*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b
^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*cos(4*d*x + 4*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c)^2
+ (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(16*d*x + 16*c)^2 + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(14*d*x + 1
4*c)^2 + 16*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(12*d*x + 12*c)^2 + 64*(2
56*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^8 - 573
44*a^7*b + 83712*a^6*b^2 - 67648*a^5*b^3 + 32841*a^4*b^4 - 9170*a^3*b^5 + 1225*a^2*b^6)*d*sin(8*d*x + 8*c)^2 +
64*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^6*b^2
- 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c)^2 + 64*(8*a^5*b^3 - 23*a^4*b^4 + 2
2*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x +
2*c)^2 - 16*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c) + (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d - 2*(8*(a^4
*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(14*d*x + 14*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(1
2*d*x + 12*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^6*b^2 - 3
52*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*
b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(4*d*x + 4*c) +
8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*x + 2*c) - (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d)*cos(16*d*x + 16*c)
+ 16*(4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(12*d*x + 12*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 +
30*a^3*b^5 - 7*a^2*b^6)*d*cos(10*d*x + 10*c) - 2*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a
^2*b^6)*d*cos(8*d*x + 8*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(6*d*x + 6*c) + 4*(8*a^
5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(4*d*x + 4*c) + 8*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*cos(2*d*
x + 2*c) - (a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d)*cos(14*d*x + 14*c) - 8*(8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b
^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(10*d*x + 10*c) + 2*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 - 3813*a^4*b
^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*cos(8*d*x + 8*c) + 8*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5
+ 49*a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^5 + 49*a^2*b^6)*d*co
s(4*d*x + 4*c) - 8*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (8*a^5*b^3 - 23*a^4*
b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a
^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*cos(8*d*x + 8*c) + 8*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3
*b^5 + 49*a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)
*d*cos(4*d*x + 4*c) - 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) + (16*a^5*b^3 -
39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d)*cos(10*d*x + 10*c) + 4*(8*(2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 -
5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*cos(6*d*x + 6*c) - 4*(1024*a^7*b - 3712*a^6*b^2 + 5304*a^5*b^3 -
3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*cos(4*d*x + 4*c) - 8*(128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 1
66*a^3*b^5 + 35*a^2*b^6)*d*cos(2*d*x + 2*c) + (128*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*
b^6)*d)*cos(8*d*x + 8*c) - 16*(4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*cos(4*
d*x + 4*c) + 8*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*cos(2*d*x + 2*c) - (16*a^5*b^3 - 39*a^4*b^
4 + 30*a^3*b^5 - 7*a^2*b^6)*d)*cos(6*d*x + 6*c) + 8*(8*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*cos
(2*d*x + 2*c) - (8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d)*cos(4*d*x + 4*c) - 4*(4*(a^4*b^4 - 2*a^3*
b^5 + a^2*b^6)*d*sin(14*d*x + 14*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(12*d*x + 12*c)
- 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128*a^6*b^2 - 352*a^5*b^3 + 35
5*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6
)*d*sin(6*d*x + 6*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*d*x + 4*c) + 4*(a^4*b^4 - 2
*a^3*b^5 + a^2*b^6)*d*sin(2*d*x + 2*c))*sin(16*d*x + 16*c) + 32*(2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^
2*b^6)*d*sin(12*d*x + 12*c) - 4*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(10*d*x + 10*c) - (128
*a^6*b^2 - 352*a^5*b^3 + 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(8*d*x + 8*c) - 4*(16*a^5*b^3 - 39*a^4*b
^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(6*d*x + 6*c) + 2*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(4*
d*x + 4*c) + 4*(a^4*b^4 - 2*a^3*b^5 + a^2*b^6)*d*sin(2*d*x + 2*c))*sin(14*d*x + 14*c) - 16*(4*(128*a^6*b^2 - 4
24*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(10*d*x + 10*c) + (1024*a^7*b - 3712*a^6*b^2 + 5304*
a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*sin(8*d*x + 8*c) + 4*(128*a^6*b^2 - 424*a^5*b^3 + 513*a
^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c) - 2*(64*a^6*b^2 - 240*a^5*b^3 + 337*a^4*b^4 - 210*a^3*b^
5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) - 4*(8*a^5*b^3 - 23*a^4*b^4 + 22*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*s
in(12*d*x + 12*c) + 32*((2048*a^7*b - 6528*a^6*b^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)
*d*sin(8*d*x + 8*c) + 4*(256*a^6*b^2 - 736*a^5*b^3 + 753*a^4*b^4 - 322*a^3*b^5 + 49*a^2*b^6)*d*sin(6*d*x + 6*c
) - 2*(128*a^6*b^2 - 424*a^5*b^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) - 4*(16*a^5*b^3
- 39*a^4*b^4 + 30*a^3*b^5 - 7*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 16*(2*(2048*a^7*b - 6528*a^6*b
^2 + 8144*a^5*b^3 - 5141*a^4*b^4 + 1722*a^3*b^5 - 245*a^2*b^6)*d*sin(6*d*x + 6*c) - (1024*a^7*b - 3712*a^6*b^2
+ 5304*a^5*b^3 - 3813*a^4*b^4 + 1442*a^3*b^5 - 245*a^2*b^6)*d*sin(4*d*x + 4*c) - 2*(128*a^6*b^2 - 352*a^5*b^3
+ 355*a^4*b^4 - 166*a^3*b^5 + 35*a^2*b^6)*d*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 64*((128*a^6*b^2 - 424*a^5*b
^3 + 513*a^4*b^4 - 266*a^3*b^5 + 49*a^2*b^6)*d*sin(4*d*x + 4*c) + 2*(16*a^5*b^3 - 39*a^4*b^4 + 30*a^3*b^5 - 7*
a^2*b^6)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

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Fricas [B]  time = 24.3843, size = 15023, normalized size = 43.29 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/256*(((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6
- 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4
)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*
b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456
*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700
*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13
*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*
b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(13824*a^6 - 24576*a^5*b + 24084*a^4*b^2 - 14455*a^3*b^3 + 22509/4*a^2*b^4
- 2625/2*a*b^5 + 625/4*b^6 - 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5
250*a*b^5 + 625*b^6)*cos(d*x + c)^2 + 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*
b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*
b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17
*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 +
a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^
4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3
- 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a
^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^
8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12
*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5
- a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5
+ 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*
a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 4379
52*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 +
45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10
*b^10 + a^9*b^11)*d^4))) - ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^
4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 +
3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 +
76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^
6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 3
5406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*
a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 1
0*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(13824*a^6 - 24576*a^5*b + 24084*a^4*b^2 - 14455*a^3*b^
3 + 22509/4*a^2*b^4 - 2625/2*a*b^5 + 625/4*b^6 - 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3
+ 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 - 1/2*((22*a^14*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a
^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^
6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10
*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b
^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) + (4608*a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a
^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*
a^2*b^2 + 94*a*b^3 - 15*b^4 - (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((14
7456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 -
6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*
a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*
a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8
*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2
325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8 - 368640*a^7*b + 49
8432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19
*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45
*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))) + ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8 - 4*(a^4*b^2 -
2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*x + c)^4 + 4*(a
^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b^3 + a^2*b^4)*d
)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 +
5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 -
117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 +
210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^
9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(-13824*a^6 + 24576*a^5*b - 24084*a^
4*b^2 + 14455*a^3*b^3 - 22509/4*a^2*b^4 + 2625/2*a*b^5 - 625/4*b^6 + 1/4*(55296*a^6 - 98304*a^5*b + 96336*a^4*
b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 + 1/2*((22*a^14*b - 125*a^13*b^2 +
300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqrt((147456*a^8 - 3686
40*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 62
5*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*
a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) - (4608*a^9 - 6144*a^8*b + 5
052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(144*a
^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a
^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^
5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 -
252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^
2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b + 1845*a^10*b^2 - 232
5*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (144*a^12 - 796*a^11*b
+ 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2)*sqrt((147456*a^8
- 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b
^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7
- 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))) - ((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x +
c)^8 - 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*c
os(d*x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*
a^3*b^3 + a^2*b^4)*d)*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^
7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^
3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b
^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a
^9*b^11)*d^4)))/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2))*log(-13824*a^6 + 24
576*a^5*b - 24084*a^4*b^2 + 14455*a^3*b^3 - 22509/4*a^2*b^4 + 2625/2*a*b^5 - 625/4*b^6 + 1/4*(55296*a^6 - 9830
4*a^5*b + 96336*a^4*b^2 - 57820*a^3*b^3 + 22509*a^2*b^4 - 5250*a*b^5 + 625*b^6)*cos(d*x + c)^2 - 1/2*((22*a^14
*b - 125*a^13*b^2 + 300*a^12*b^3 - 395*a^11*b^4 + 310*a^10*b^5 - 147*a^9*b^6 + 40*a^8*b^7 - 5*a^7*b^8)*d^3*sqr
t((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 35406*a^2*b
^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^14*b^6 +
210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))*cos(d*x + c)*sin(d*x + c) - (4608*
a^9 - 6144*a^8*b + 5052*a^7*b^2 - 2437*a^6*b^3 + 783*a^5*b^4 - 159*a^4*b^5 + 25*a^3*b^6)*d*cos(d*x + c)*sin(d*
x + c))*sqrt(-(144*a^4 + 76*a^3*b - 155*a^2*b^2 + 94*a*b^3 - 15*b^4 + (a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6
*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4
*b^4 - 117532*a^3*b^5 + 35406*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*
b^4 + 210*a^15*b^5 - 252*a^14*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))
)/((a^9*b - 5*a^8*b^2 + 10*a^7*b^3 - 10*a^6*b^4 + 5*a^5*b^5 - a^4*b^6)*d^2)) - 1/4*(2*(144*a^12 - 796*a^11*b +
1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d^2*cos(d*x + c)^2 - (1
44*a^12 - 796*a^11*b + 1845*a^10*b^2 - 2325*a^9*b^3 + 1730*a^8*b^4 - 774*a^7*b^5 + 201*a^6*b^6 - 25*a^5*b^7)*d
^2)*sqrt((147456*a^8 - 368640*a^7*b + 498432*a^6*b^2 - 437952*a^5*b^3 + 269641*a^4*b^4 - 117532*a^3*b^5 + 3540
6*a^2*b^6 - 6700*a*b^7 + 625*b^8)/((a^19*b - 10*a^18*b^2 + 45*a^17*b^3 - 120*a^16*b^4 + 210*a^15*b^5 - 252*a^1
4*b^6 + 210*a^13*b^7 - 120*a^12*b^8 + 45*a^11*b^9 - 10*a^10*b^10 + a^9*b^11)*d^4))) + 8*((11*a*b^2 - 5*b^3)*co
s(d*x + c)^7 - 3*(2*a^2*b + 11*a*b^2 - 5*b^3)*cos(d*x + c)^5 - 3*(a^2*b - 14*a*b^2 + 5*b^3)*cos(d*x + c)^3 + 5
*(2*a^3 + a^2*b - 4*a*b^2 + b^3)*cos(d*x + c))*sin(d*x + c))/((a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^8
- 4*(a^4*b^2 - 2*a^3*b^3 + a^2*b^4)*d*cos(d*x + c)^6 - 2*(a^5*b - 5*a^4*b^2 + 7*a^3*b^3 - 3*a^2*b^4)*d*cos(d*
x + c)^4 + 4*(a^5*b - 3*a^4*b^2 + 3*a^3*b^3 - a^2*b^4)*d*cos(d*x + c)^2 + (a^6 - 4*a^5*b + 6*a^4*b^2 - 4*a^3*b
^3 + a^2*b^4)*d)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**2/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError