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

Optimal. Leaf size=236 \[ \frac{\sqrt{b} \left (6 \sqrt{a}-5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}-\frac{\sqrt{b} \left (6 \sqrt{a}+5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{b \tan (c+d x) \left ((a+b) \tan ^2(c+d x)+a\right )}{4 a^2 d (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac{\cot (c+d x)}{a^2 d} \]

[Out]

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

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Rubi [A]  time = 0.534179, antiderivative size = 236, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {3217, 1334, 1664, 1166, 205} \[ \frac{\sqrt{b} \left (6 \sqrt{a}-5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}-\frac{\sqrt{b} \left (6 \sqrt{a}+5 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{b \tan (c+d x) \left ((a+b) \tan ^2(c+d x)+a\right )}{4 a^2 d (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac{\cot (c+d x)}{a^2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*Sqrt[a] - 5*Sqrt[b])*Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(9/4)*(Sqrt[a] -
Sqrt[b])^(3/2)*d) - ((6*Sqrt[a] + 5*Sqrt[b])*Sqrt[b]*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/
(8*a^(9/4)*(Sqrt[a] + Sqrt[b])^(3/2)*d) - Cot[c + d*x]/(a^2*d) - (b*Tan[c + d*x]*(a + (a + b)*Tan[c + d*x]^2))
/(4*a^2*(a - b)*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 1334

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[x^m*(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])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x],
x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]

Rule 1664

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

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{\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{x^2 \left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{b \tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-8 a b-\frac{2 a (8 a-7 b) b x^2}{a-b}-\frac{2 b \left (4 a^2-a b-b^2\right ) x^4}{a-b}}{x^2 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{8 a^2 b d}\\ &=-\frac{b \tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{8 b}{x^2}+\frac{2 b^2 \left (-a-(7 a-5 b) x^2\right )}{(a-b) \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{8 a^2 b d}\\ &=-\frac{\cot (c+d x)}{a^2 d}-\frac{b \tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac{b \operatorname{Subst}\left (\int \frac{-a+(-7 a+5 b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{4 a^2 (a-b) d}\\ &=-\frac{\cot (c+d x)}{a^2 d}-\frac{b \tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac{\left (\left (7 a+\frac{2 \sqrt{a} (3 a-2 b)}{\sqrt{b}}-5 b\right ) b\right ) \operatorname{Subst}\left (\int \frac{1}{a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{8 a^2 (a-b) d}-\frac{\left (b \left (-7 a+\frac{2 \sqrt{a} (3 a-2 b)}{\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 )}{8 a^2 (a-b) d}\\ &=\frac{\left (6 \sqrt{a}-5 \sqrt{b}\right ) \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt{a}-\sqrt{b}\right )^{3/2} d}-\frac{\left (6 \sqrt{a}+5 \sqrt{b}\right ) \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{9/4} \left (\sqrt{a}+\sqrt{b}\right )^{3/2} d}-\frac{\cot (c+d x)}{a^2 d}-\frac{b \tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 2.15961, size = 274, normalized size = 1.16 \[ \frac{-\frac{\left (6 a \sqrt{b}+5 \sqrt{a} b\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\left (\sqrt{a}+\sqrt{b}\right ) \sqrt{\sqrt{a} \sqrt{b}+a}}-\frac{4 \sqrt{a} b \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(a-b) (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}-\frac{\left (6 a \sqrt{b}-5 \sqrt{a} b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\left (\sqrt{a}-\sqrt{b}\right ) \sqrt{\sqrt{a} \sqrt{b}-a}}-8 \sqrt{a} \cot (c+d x)}{8 a^{5/2} d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(((6*a*Sqrt[b] + 5*Sqrt[a]*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a
] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]])) - ((6*a*Sqrt[b] - 5*Sqrt[a]*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d
*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) - 8*Sqrt[a]*Cot[c + d*x] -
(4*Sqrt[a]*b*(2*a + b - b*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*C
os[4*(c + d*x)])))/(8*a^(5/2)*d)

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Maple [B]  time = 0.154, size = 708, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)/(a-b)/a*tan(d*x+c)^3*b-1/4/d*b^2/a^2/(tan(d*x+c)^4*a
-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)/(a-b)*tan(d*x+c)^3-1/4/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a
)*b/a/(a-b)*tan(d*x+c)+7/8/d/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))*b-5/8/d*b^2/a^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))+3/4/d/(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-1/2/d/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))*b^2+7/8/d/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))*b-5/8/d*b^2/a^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))-3/4/d/(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+1/2/d/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))*b^2-1/d/a^2/tan(d*x+c)

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

[Out]

1/2*(2*(48*a^2*b - 5*a*b^2 - 25*b^3)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((6*a*b^2 - 5*b^3)*sin(8*d*x + 8*c) -
2*(13*a*b^2 - 10*b^3)*sin(6*d*x + 6*c) - 2*(32*a^2*b - 47*a*b^2 + 15*b^3)*sin(4*d*x + 4*c) - 2*(7*a*b^2 - 10*
b^3)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + (2*(48*a^2*b - 5*a*b^2 - 25*b^3)*sin(6*d*x + 6*c) + 2*(112*a^2*b -
165*a*b^2 + 50*b^3)*sin(4*d*x + 4*c) + 5*(8*a*b^2 - 15*b^3)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 2*(2*(256*a^
3 - 432*a^2*b + 210*a*b^2 - 25*b^3)*sin(4*d*x + 4*c) + (112*a^2*b - 165*a*b^2 + 50*b^3)*sin(2*d*x + 2*c))*cos(
6*d*x + 6*c) + 2*((a^3*b^2 - a^2*b^3)*d*cos(10*d*x + 10*c)^2 + 25*(a^3*b^2 - a^2*b^3)*d*cos(8*d*x + 8*c)^2 + 4
*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^5 - 144*a^4*b + 105*a^3*b^2 -
25*a^2*b^3)*d*cos(4*d*x + 4*c)^2 + 25*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c)^2 + (a^3*b^2 - a^2*b^3)*d*sin(10*
d*x + 10*c)^2 + 25*(a^3*b^2 - a^2*b^3)*d*sin(8*d*x + 8*c)^2 + 4*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3
)*d*sin(6*d*x + 6*c)^2 + 4*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*sin(4*d*x + 4*c)^2 + 20*(8*a^4*b
- 13*a^3*b^2 + 5*a^2*b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*(a^3*b^2 - a^2*b^3)*d*sin(2*d*x + 2*c)^2 -
10*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d - 2*(5*(a^3*b^2 - a^2*b^3)*d*cos(8*d*x + 8*c
) + 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(6*d*x + 6*c) - 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(4*d*x
+ 4*c) - 5*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d)*cos(10*d*x + 10*c) + 10*(2*(8*a^4*
b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(6*d*x + 6*c) - 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(4*d*x + 4*c) - 5*(
a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d)*cos(8*d*x + 8*c) - 4*(2*(64*a^5 - 144*a^4*b + 1
05*a^3*b^2 - 25*a^2*b^3)*d*cos(4*d*x + 4*c) + 5*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(2*d*x + 2*c) - (8*a^4
*b - 13*a^3*b^2 + 5*a^2*b^3)*d)*cos(6*d*x + 6*c) + 4*(5*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(2*d*x + 2*c)
- (8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d)*cos(4*d*x + 4*c) - 2*(5*(a^3*b^2 - a^2*b^3)*d*sin(8*d*x + 8*c) + 2*(8*
a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*sin(6*d*x + 6*c) - 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*sin(4*d*x + 4*c) -
5*(a^3*b^2 - a^2*b^3)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 10*(2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*sin
(6*d*x + 6*c) - 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*sin(4*d*x + 4*c) - 5*(a^3*b^2 - a^2*b^3)*d*sin(2*d*x +
2*c))*sin(8*d*x + 8*c) - 4*(2*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*sin(4*d*x + 4*c) + 5*(8*a^4*b
- 13*a^3*b^2 + 5*a^2*b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))*integrate(-(4*(6*a*b^2 - 5*b^3)*cos(6*d*x + 6*
c)^2 - 4*(64*a^2*b - 64*a*b^2 + 15*b^3)*cos(4*d*x + 4*c)^2 + 4*(6*a*b^2 - 5*b^3)*cos(2*d*x + 2*c)^2 + 4*(6*a*b
^2 - 5*b^3)*sin(6*d*x + 6*c)^2 - 4*(64*a^2*b - 64*a*b^2 + 15*b^3)*sin(4*d*x + 4*c)^2 + 2*(48*a^2*b - 90*a*b^2
+ 35*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(6*a*b^2 - 5*b^3)*sin(2*d*x + 2*c)^2 - ((6*a*b^2 - 5*b^3)*cos(
6*d*x + 6*c) - 2*(8*a*b^2 - 5*b^3)*cos(4*d*x + 4*c) + (6*a*b^2 - 5*b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - (
6*a*b^2 - 5*b^3 - 2*(48*a^2*b - 90*a*b^2 + 35*b^3)*cos(4*d*x + 4*c) - 8*(6*a*b^2 - 5*b^3)*cos(2*d*x + 2*c))*co
s(6*d*x + 6*c) + 2*(8*a*b^2 - 5*b^3 + (48*a^2*b - 90*a*b^2 + 35*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - (6*a
*b^2 - 5*b^3)*cos(2*d*x + 2*c) - ((6*a*b^2 - 5*b^3)*sin(6*d*x + 6*c) - 2*(8*a*b^2 - 5*b^3)*sin(4*d*x + 4*c) +
(6*a*b^2 - 5*b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*((48*a^2*b - 90*a*b^2 + 35*b^3)*sin(4*d*x + 4*c) + 4*
(6*a*b^2 - 5*b^3)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))/(a^3*b^2 - a^2*b^3 + (a^3*b^2 - a^2*b^3)*cos(8*d*x + 8*c
)^2 + 16*(a^3*b^2 - a^2*b^3)*cos(6*d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b + 57*a^3*b^2 - 9*a^2*b^3)*cos(4*d*x +
4*c)^2 + 16*(a^3*b^2 - a^2*b^3)*cos(2*d*x + 2*c)^2 + (a^3*b^2 - a^2*b^3)*sin(8*d*x + 8*c)^2 + 16*(a^3*b^2 - a^
2*b^3)*sin(6*d*x + 6*c)^2 + 4*(64*a^5 - 112*a^4*b + 57*a^3*b^2 - 9*a^2*b^3)*sin(4*d*x + 4*c)^2 + 16*(8*a^4*b -
11*a^3*b^2 + 3*a^2*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*(a^3*b^2 - a^2*b^3)*sin(2*d*x + 2*c)^2 + 2*(a^
3*b^2 - a^2*b^3 - 4*(a^3*b^2 - a^2*b^3)*cos(6*d*x + 6*c) - 2*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*cos(4*d*x + 4*
c) - 4*(a^3*b^2 - a^2*b^3)*cos(2*d*x + 2*c))*cos(8*d*x + 8*c) - 8*(a^3*b^2 - a^2*b^3 - 2*(8*a^4*b - 11*a^3*b^2
+ 3*a^2*b^3)*cos(4*d*x + 4*c) - 4*(a^3*b^2 - a^2*b^3)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) - 4*(8*a^4*b - 11*a^
3*b^2 + 3*a^2*b^3 - 4*(8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 8*(a^3*b^2 - a^2
*b^3)*cos(2*d*x + 2*c) - 4*(2*(a^3*b^2 - a^2*b^3)*sin(6*d*x + 6*c) + (8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)*sin(4*
d*x + 4*c) + 2*(a^3*b^2 - a^2*b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*((8*a^4*b - 11*a^3*b^2 + 3*a^2*b^3)
*sin(4*d*x + 4*c) + 2*(a^3*b^2 - a^2*b^3)*sin(2*d*x + 2*c))*sin(6*d*x + 6*c)), x) - (4*a*b^2 - 5*b^3 + (6*a*b^
2 - 5*b^3)*cos(8*d*x + 8*c) - 2*(13*a*b^2 - 10*b^3)*cos(6*d*x + 6*c) - 2*(32*a^2*b - 47*a*b^2 + 15*b^3)*cos(4*
d*x + 4*c) - 2*(7*a*b^2 - 10*b^3)*cos(2*d*x + 2*c))*sin(10*d*x + 10*c) + (14*a*b^2 - 20*b^3 - 2*(48*a^2*b - 5*
a*b^2 - 25*b^3)*cos(6*d*x + 6*c) - 2*(112*a^2*b - 165*a*b^2 + 50*b^3)*cos(4*d*x + 4*c) - 5*(8*a*b^2 - 15*b^3)*
cos(2*d*x + 2*c))*sin(8*d*x + 8*c) + 2*(32*a^2*b - 47*a*b^2 + 15*b^3 - 2*(256*a^3 - 432*a^2*b + 210*a*b^2 - 25
*b^3)*cos(4*d*x + 4*c) - (112*a^2*b - 165*a*b^2 + 50*b^3)*cos(2*d*x + 2*c))*sin(6*d*x + 6*c) + 2*(13*a*b^2 - 1
0*b^3 - (48*a^2*b - 5*a*b^2 - 25*b^3)*cos(2*d*x + 2*c))*sin(4*d*x + 4*c) - (6*a*b^2 - 5*b^3)*sin(2*d*x + 2*c))
/((a^3*b^2 - a^2*b^3)*d*cos(10*d*x + 10*c)^2 + 25*(a^3*b^2 - a^2*b^3)*d*cos(8*d*x + 8*c)^2 + 4*(64*a^5 - 144*a
^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*cos(6*d*x + 6*c)^2 + 4*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*co
s(4*d*x + 4*c)^2 + 25*(a^3*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c)^2 + (a^3*b^2 - a^2*b^3)*d*sin(10*d*x + 10*c)^2 +
25*(a^3*b^2 - a^2*b^3)*d*sin(8*d*x + 8*c)^2 + 4*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*sin(6*d*x +
6*c)^2 + 4*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*sin(4*d*x + 4*c)^2 + 20*(8*a^4*b - 13*a^3*b^2 + 5
*a^2*b^3)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 25*(a^3*b^2 - a^2*b^3)*d*sin(2*d*x + 2*c)^2 - 10*(a^3*b^2 - a^
2*b^3)*d*cos(2*d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d - 2*(5*(a^3*b^2 - a^2*b^3)*d*cos(8*d*x + 8*c) + 2*(8*a^4*b -
13*a^3*b^2 + 5*a^2*b^3)*d*cos(6*d*x + 6*c) - 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(4*d*x + 4*c) - 5*(a^3
*b^2 - a^2*b^3)*d*cos(2*d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d)*cos(10*d*x + 10*c) + 10*(2*(8*a^4*b - 13*a^3*b^2 +
5*a^2*b^3)*d*cos(6*d*x + 6*c) - 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(4*d*x + 4*c) - 5*(a^3*b^2 - a^2*b^
3)*d*cos(2*d*x + 2*c) + (a^3*b^2 - a^2*b^3)*d)*cos(8*d*x + 8*c) - 4*(2*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*
a^2*b^3)*d*cos(4*d*x + 4*c) + 5*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(2*d*x + 2*c) - (8*a^4*b - 13*a^3*b^2
+ 5*a^2*b^3)*d)*cos(6*d*x + 6*c) + 4*(5*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*cos(2*d*x + 2*c) - (8*a^4*b - 13*
a^3*b^2 + 5*a^2*b^3)*d)*cos(4*d*x + 4*c) - 2*(5*(a^3*b^2 - a^2*b^3)*d*sin(8*d*x + 8*c) + 2*(8*a^4*b - 13*a^3*b
^2 + 5*a^2*b^3)*d*sin(6*d*x + 6*c) - 2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*sin(4*d*x + 4*c) - 5*(a^3*b^2 - a^
2*b^3)*d*sin(2*d*x + 2*c))*sin(10*d*x + 10*c) + 10*(2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*sin(6*d*x + 6*c) -
2*(8*a^4*b - 13*a^3*b^2 + 5*a^2*b^3)*d*sin(4*d*x + 4*c) - 5*(a^3*b^2 - a^2*b^3)*d*sin(2*d*x + 2*c))*sin(8*d*x
+ 8*c) - 4*(2*(64*a^5 - 144*a^4*b + 105*a^3*b^2 - 25*a^2*b^3)*d*sin(4*d*x + 4*c) + 5*(8*a^4*b - 13*a^3*b^2 + 5
*a^2*b^3)*d*sin(2*d*x + 2*c))*sin(6*d*x + 6*c))

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Fricas [B]  time = 10.5434, size = 8411, normalized size = 35.64 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(8*(4*a*b - 5*b^2)*cos(d*x + c)^5 - 8*(7*a*b - 10*b^2)*cos(d*x + c)^3 - ((a^3*b - a^2*b^2)*d*cos(d*x + c
)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 -
a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a
^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^
6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(432*a^3*b^2 - 921*a^2*b^3 + 2625/4*a*b^4 - 625/4*b^5 - 1/4*(1728*a^3*b^2
- 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cos(d*x + c)^2 + 1/2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5
*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*
a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(144*a^6*b -
303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b
^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b
^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^6*b +
3*a^5*b^2 - a^4*b^3)*d^2)) + 1/4*(2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*cos(d*x
+ c)^2 - (36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((2304*a^4*b^3 - 6624*a^3*b^4
+ 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b
^5 + a^9*b^6)*d^4)))*sin(d*x + c) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2
- (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3
*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^
10*b^5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(432*a
^3*b^2 - 921*a^2*b^3 + 2625/4*a*b^4 - 625/4*b^5 - 1/4*(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cos
(d*x + c)^2 - 1/2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^
3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a
^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d*
cos(d*x + c)*sin(d*x + c))*sqrt(-((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4
+ 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^
5 + a^9*b^6)*d^4)) + 36*a^2*b - 47*a*b^2 + 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2)) + 1/4*(2*(36*a
^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*cos(d*x + c)^2 - (36*a^9 - 133*a^8*b + 183*a^7*b^
2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((
a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)))*sin(d*x + c) - ((a^3*
b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7
- 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)
/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b
^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2))*log(-432*a^3*b^2 + 921*a^2*b^3 - 2625/4*a*b^4 + 625/
4*b^5 + 1/4*(1728*a^3*b^2 - 3684*a^2*b^3 + 2625*a*b^4 - 625*b^5)*cos(d*x + c)^2 + 1/2*((7*a^11 - 26*a^10*b + 3
6*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^
7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(
d*x + c) + 2*(144*a^6*b - 303*a^5*b^2 + 213*a^4*b^3 - 50*a^3*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^7 - 3*
a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^
15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^2 -
15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2)) + 1/4*(2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3
+ 25*a^5*b^4)*d^2*cos(d*x + c)^2 - (36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((23
04*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3
+ 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)))*sin(d*x + c) + ((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b -
a^2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt(
(2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*
b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 -
a^4*b^3)*d^2))*log(-432*a^3*b^2 + 921*a^2*b^3 - 2625/4*a*b^4 + 625/4*b^5 + 1/4*(1728*a^3*b^2 - 3684*a^2*b^3 +
2625*a*b^4 - 625*b^5)*cos(d*x + c)^2 - 1/2*((7*a^11 - 26*a^10*b + 36*a^9*b^2 - 22*a^8*b^3 + 5*a^7*b^4)*d^3*sqr
t((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^1
2*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + 2*(144*a^6*b - 303*a^5*b^2 + 213
*a^4*b^3 - 50*a^3*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b^3)*d^2*sqrt((2304
*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 - 3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 +
15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)) - 36*a^2*b + 47*a*b^2 - 15*b^3)/((a^7 - 3*a^6*b + 3*a^5*b^2 - a^4*b
^3)*d^2)) + 1/4*(2*(36*a^9 - 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2*cos(d*x + c)^2 - (36*a^9
- 133*a^8*b + 183*a^7*b^2 - 111*a^6*b^3 + 25*a^5*b^4)*d^2)*sqrt((2304*a^4*b^3 - 6624*a^3*b^4 + 7161*a^2*b^5 -
3450*a*b^6 + 625*b^7)/((a^15 - 6*a^14*b + 15*a^13*b^2 - 20*a^12*b^3 + 15*a^11*b^4 - 6*a^10*b^5 + a^9*b^6)*d^4)
))*sin(d*x + c) - 8*(4*a^2 - 7*a*b + 5*b^2)*cos(d*x + c))/(((a^3*b - a^2*b^2)*d*cos(d*x + c)^4 - 2*(a^3*b - a^
2*b^2)*d*cos(d*x + c)^2 - (a^4 - 2*a^3*b + a^2*b^2)*d)*sin(d*x + c))

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

[Out]

Exception raised: NotImplementedError