3.234 \(\int \frac{1}{(a-b \sin ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=319 \[ -\frac{b \tan (c+d x) \left (\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}+\frac{(33 a-13 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}+\frac{\left (-50 \sqrt{a} \sqrt{b}+32 a+21 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{11/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (50 \sqrt{a} \sqrt{b}+32 a+21 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{11/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{b^2 \tan (c+d x) \left (4 (a+b) \tan ^2(c+d x)+3 a+b\right )}{8 a d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

[Out]

((32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt
[a] - Sqrt[b])^(5/2)*d) + ((32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^
(1/4)])/(64*a^(11/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b^2*Tan[c + d*x]*(3*a + b + 4*(a + b)*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) - (b*Tan[c + d*x]*((17*a^2 - 40*a*b + 7*b^
2)/(a - b)^3 + ((33*a - 13*b)*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 = 0.651839, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3209, 1205, 1678, 1166, 205} \[ -\frac{b \tan (c+d x) \left (\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}+\frac{(33 a-13 b) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^2 d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}+\frac{\left (-50 \sqrt{a} \sqrt{b}+32 a+21 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{11/4} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\left (50 \sqrt{a} \sqrt{b}+32 a+21 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{11/4} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}-\frac{b^2 \tan (c+d x) \left (4 (a+b) \tan ^2(c+d x)+3 a+b\right )}{8 a d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Sin[c + d*x]^4)^(-3),x]

[Out]

((32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(11/4)*(Sqrt
[a] - Sqrt[b])^(5/2)*d) + ((32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^
(1/4)])/(64*a^(11/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b^2*Tan[c + d*x]*(3*a + b + 4*(a + b)*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) - (b*Tan[c + d*x]*((17*a^2 - 40*a*b + 7*b^
2)/(a - b)^3 + ((33*a - 13*b)*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 3209

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

Rule 1205

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coeff[Polynom
ialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(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)*ExpandToS
um[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(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]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*
a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]

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{1}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^5}{\left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{b^2 \tan (c+d x) \left (3 a+b+4 (a+b) \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 b \left (8 a^3-24 a^2 b+27 a b^2-7 b^3\right )}{(a-b)^3}-\frac{8 a b \left (6 a^3-18 a^2 b+15 a b^2-5 b^3\right ) x^2}{(a-b)^3}-\frac{16 a^2 (3 a-5 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^2 \tan (c+d x) \left (3 a+b+4 (a+b) \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{b \tan (c+d x) \left (\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}+\frac{(33 a-13 b) \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{4 a^2 b^2 \left (32 a^2-47 a b+21 b^2\right )}{(a-b)^2}+\frac{4 a^2 b^2 \left (32 a^2-33 a b+13 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^2 \tan (c+d x) \left (3 a+b+4 (a+b) \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{b \tan (c+d x) \left (\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}+\frac{(33 a-13 b) \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 (32 a-50 \sqrt{a} \sqrt{b}+21 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^{5/2} \left (\sqrt{a}-\sqrt{b}\right )^2 d}+\frac{\left (\left (\sqrt{a}-\sqrt{b}\right ) \left (32 a+50 \sqrt{a} \sqrt{b}+21 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^{5/2} \left (\sqrt{a}+\sqrt{b}\right )^2 d}\\ &=\frac{\left (32 a-50 \sqrt{a} \sqrt{b}+21 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{11/4} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} d}+\frac{\left (32 a+50 \sqrt{a} \sqrt{b}+21 b\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{11/4} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} d}-\frac{b^2 \tan (c+d x) \left (3 a+b+4 (a+b) \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{b \tan (c+d x) \left (\frac{17 a^2-40 a b+7 b^2}{(a-b)^3}+\frac{(33 a-13 b) \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 = 3.12397, size = 333, normalized size = 1.04 \[ \frac{\frac{64 a^{3/2} b (a-b) (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{\left (50 \sqrt{a} \sqrt{b}+32 a+21 b\right ) \left (\sqrt{a}-\sqrt{b}\right )^2 \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{8 \sqrt{a} b \sin (2 (c+d x)) ((6 a-3 b) \cos (2 (c+d x))-19 a+10 b)}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{\left (\sqrt{a}+\sqrt{b}\right )^2 \left (-50 \sqrt{a} \sqrt{b}+32 a+21 b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{64 a^{5/2} d (a-b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((Sqrt[a] - Sqrt[b])^2*(32*a + 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a +
Sqrt[a]*Sqrt[b]]])/Sqrt[a + Sqrt[a]*Sqrt[b]] - ((Sqrt[a] + Sqrt[b])^2*(32*a - 50*Sqrt[a]*Sqrt[b] + 21*b)*ArcTa
nh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (8*Sqrt[a]*b*(
-19*a + 10*b + (6*a - 3*b)*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c
+ d*x)]) + (64*a^(3/2)*(a - b)*b*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(-8*a + 3*b - 4*b*Cos[2*(c + d*x)]
+ b*Cos[4*(c + d*x)])^2)/(64*a^(5/2)*(a - b)^2*d)

________________________________________________________________________________________

Maple [B]  time = 0.148, size = 1803, normalized size = 5.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a-b*sin(d*x+c)^4)^3,x)

[Out]

23/32/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))-23/32/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))-67/32/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)^3-83/32/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)^5+13/32/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+43/32/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+19/16/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^3-19/16
/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^3-7/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/a^2*b^3/(a^2-2*a*b+b^2)*tan(d*
x+c)^5-33/32/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+1/2/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))-21/64/d/a^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))*b^4+21/64/d/a^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))*b^4+1/2/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))+23/32/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))+23/32/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))-65/64/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))-65/64/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))-101/64/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))+101
/64/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))-13/64/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^3-13/64/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+33/16/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^2-17/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a^2-2*a*b+b^2
)*tan(d*x+c)*b+11/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b^2/a/(a^2-2*a*b+b^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(1/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

1/8*(4*(120*a^2*b^3 - 77*a*b^4 + 14*b^5)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((7*a*b^4 - 4*b^5)*sin(14*d*x + 1
4*c) - (32*a^2*b^3 + 2*a*b^4 - 7*b^5)*sin(12*d*x + 12*c) - (16*a^2*b^3 - 3*a*b^4 - 28*b^5)*sin(10*d*x + 10*c)
+ 3*(256*a^3*b^2 - 320*a^2*b^3 + 166*a*b^4 - 35*b^5)*sin(8*d*x + 8*c) + (784*a^2*b^3 - 723*a*b^4 + 140*b^5)*si
n(6*d*x + 6*c) - (160*a^2*b^3 - 266*a*b^4 + 91*b^5)*sin(4*d*x + 4*c) - (55*a*b^4 - 28*b^5)*sin(2*d*x + 2*c))*c
os(16*d*x + 16*c) + 2*(2*(120*a^2*b^3 - 77*a*b^4 + 14*b^5)*sin(12*d*x + 12*c) - 8*(48*a^2*b^3 - 55*a*b^4 + 28*
b^5)*sin(10*d*x + 10*c) - (3968*a^3*b^2 - 5024*a^2*b^3 + 2621*a*b^4 - 560*b^5)*sin(8*d*x + 8*c) - 16*(224*a^2*
b^3 - 209*a*b^4 + 42*b^5)*sin(6*d*x + 6*c) + 2*(376*a^2*b^3 - 613*a*b^4 + 210*b^5)*sin(4*d*x + 4*c) + 8*(31*a*
b^4 - 16*b^5)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 2*(2*(1152*a^3*b^2 - 520*a^2*b^3 - 455*a*b^4 + 294*b^5)*s
in(10*d*x + 10*c) - (8192*a^4*b - 23296*a^3*b^2 + 21376*a^2*b^3 - 9394*a*b^4 + 1715*b^5)*sin(8*d*x + 8*c) - 2*
(5248*a^3*b^2 - 10888*a^2*b^3 + 6433*a*b^4 - 1078*b^5)*sin(6*d*x + 6*c) + 4*(512*a^3*b^2 - 1520*a^2*b^3 + 1330
*a*b^4 - 343*b^5)*sin(4*d*x + 4*c) + 2*(376*a^2*b^3 - 613*a*b^4 + 210*b^5)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c
) + 2*((51200*a^4*b - 84864*a^3*b^2 + 56016*a^2*b^3 - 18081*a*b^4 + 1960*b^5)*sin(8*d*x + 8*c) + 8*(6400*a^3*b
^2 - 8608*a^2*b^3 + 3437*a*b^4 - 392*b^5)*sin(6*d*x + 6*c) - 2*(5248*a^3*b^2 - 10888*a^2*b^3 + 6433*a*b^4 - 10
78*b^5)*sin(4*d*x + 4*c) - 16*(224*a^2*b^3 - 209*a*b^4 + 42*b^5)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) + 2*((51
200*a^4*b - 84864*a^3*b^2 + 56016*a^2*b^3 - 18081*a*b^4 + 1960*b^5)*sin(6*d*x + 6*c) - (8192*a^4*b - 23296*a^3
*b^2 + 21376*a^2*b^3 - 9394*a*b^4 + 1715*b^5)*sin(4*d*x + 4*c) - (3968*a^3*b^2 - 5024*a^2*b^3 + 2621*a*b^4 - 5
60*b^5)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) + 4*((1152*a^3*b^2 - 520*a^2*b^3 - 455*a*b^4 + 294*b^5)*sin(4*d*x +
4*c) - 4*(48*a^2*b^3 - 55*a*b^4 + 28*b^5)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) - 8*((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 + 75
3*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 - 67
648*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 + 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 + 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 + 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) + 1
6*(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*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)*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 - 514
1*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 - 381
3*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*(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 - 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 - 3
9*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))*integrate(1/4*(4*(7*a*b^2 - 4*b^3)*cos(6*d*x + 6*c)^2 - 4*(256*a^3
- 416*a^2*b + 256*a*b^2 - 51*b^3)*cos(4*d*x + 4*c)^2 + 4*(7*a*b^2 - 4*b^3)*cos(2*d*x + 2*c)^2 + 4*(7*a*b^2 -
4*b^3)*sin(6*d*x + 6*c)^2 - 4*(256*a^3 - 416*a^2*b + 256*a*b^2 - 51*b^3)*sin(4*d*x + 4*c)^2 - 2*(72*a^2*b - 10
7*a*b^2 + 56*b^3)*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*(7*a*b^2 - 4*b^3)*sin(2*d*x + 2*c)^2 - ((7*a*b^2 - 4*b
^3)*cos(6*d*x + 6*c) - 2*(32*a^2*b - 40*a*b^2 + 17*b^3)*cos(4*d*x + 4*c) + (7*a*b^2 - 4*b^3)*cos(2*d*x + 2*c))
*cos(8*d*x + 8*c) - (7*a*b^2 - 4*b^3 + 2*(72*a^2*b - 107*a*b^2 + 56*b^3)*cos(4*d*x + 4*c) - 8*(7*a*b^2 - 4*b^3
)*cos(2*d*x + 2*c))*cos(6*d*x + 6*c) + 2*(32*a^2*b - 40*a*b^2 + 17*b^3 - (72*a^2*b - 107*a*b^2 + 56*b^3)*cos(2
*d*x + 2*c))*cos(4*d*x + 4*c) - (7*a*b^2 - 4*b^3)*cos(2*d*x + 2*c) - ((7*a*b^2 - 4*b^3)*sin(6*d*x + 6*c) - 2*(
32*a^2*b - 40*a*b^2 + 17*b^3)*sin(4*d*x + 4*c) + (7*a*b^2 - 4*b^3)*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) - 2*((72
*a^2*b - 107*a*b^2 + 56*b^3)*sin(4*d*x + 4*c) - 4*(7*a*b^2 - 4*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 + 1
6*(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) - (6*a*b^4 - 3*b^5 + (7*a*b^4 - 4*b^5)*cos(14*d*x + 14*c) - (32*a^2*b^3 + 2*a*b^4 - 7*b^5)*cos(12*d*x + 12*
c) - (16*a^2*b^3 - 3*a*b^4 - 28*b^5)*cos(10*d*x + 10*c) + 3*(256*a^3*b^2 - 320*a^2*b^3 + 166*a*b^4 - 35*b^5)*c
os(8*d*x + 8*c) + (784*a^2*b^3 - 723*a*b^4 + 140*b^5)*cos(6*d*x + 6*c) - (160*a^2*b^3 - 266*a*b^4 + 91*b^5)*co
s(4*d*x + 4*c) - (55*a*b^4 - 28*b^5)*cos(2*d*x + 2*c))*sin(16*d*x + 16*c) + (55*a*b^4 - 28*b^5 - 4*(120*a^2*b^
3 - 77*a*b^4 + 14*b^5)*cos(12*d*x + 12*c) + 16*(48*a^2*b^3 - 55*a*b^4 + 28*b^5)*cos(10*d*x + 10*c) + 2*(3968*a
^3*b^2 - 5024*a^2*b^3 + 2621*a*b^4 - 560*b^5)*cos(8*d*x + 8*c) + 32*(224*a^2*b^3 - 209*a*b^4 + 42*b^5)*cos(6*d
*x + 6*c) - 4*(376*a^2*b^3 - 613*a*b^4 + 210*b^5)*cos(4*d*x + 4*c) - 16*(31*a*b^4 - 16*b^5)*cos(2*d*x + 2*c))*
sin(14*d*x + 14*c) + (160*a^2*b^3 - 266*a*b^4 + 91*b^5 - 4*(1152*a^3*b^2 - 520*a^2*b^3 - 455*a*b^4 + 294*b^5)*
cos(10*d*x + 10*c) + 2*(8192*a^4*b - 23296*a^3*b^2 + 21376*a^2*b^3 - 9394*a*b^4 + 1715*b^5)*cos(8*d*x + 8*c) +
4*(5248*a^3*b^2 - 10888*a^2*b^3 + 6433*a*b^4 - 1078*b^5)*cos(6*d*x + 6*c) - 8*(512*a^3*b^2 - 1520*a^2*b^3 + 1
330*a*b^4 - 343*b^5)*cos(4*d*x + 4*c) - 4*(376*a^2*b^3 - 613*a*b^4 + 210*b^5)*cos(2*d*x + 2*c))*sin(12*d*x + 1
2*c) - (784*a^2*b^3 - 723*a*b^4 + 140*b^5 + 2*(51200*a^4*b - 84864*a^3*b^2 + 56016*a^2*b^3 - 18081*a*b^4 + 196
0*b^5)*cos(8*d*x + 8*c) + 16*(6400*a^3*b^2 - 8608*a^2*b^3 + 3437*a*b^4 - 392*b^5)*cos(6*d*x + 6*c) - 4*(5248*a
^3*b^2 - 10888*a^2*b^3 + 6433*a*b^4 - 1078*b^5)*cos(4*d*x + 4*c) - 32*(224*a^2*b^3 - 209*a*b^4 + 42*b^5)*cos(2
*d*x + 2*c))*sin(10*d*x + 10*c) - (768*a^3*b^2 - 960*a^2*b^3 + 498*a*b^4 - 105*b^5 + 2*(51200*a^4*b - 84864*a^
3*b^2 + 56016*a^2*b^3 - 18081*a*b^4 + 1960*b^5)*cos(6*d*x + 6*c) - 2*(8192*a^4*b - 23296*a^3*b^2 + 21376*a^2*b
^3 - 9394*a*b^4 + 1715*b^5)*cos(4*d*x + 4*c) - 2*(3968*a^3*b^2 - 5024*a^2*b^3 + 2621*a*b^4 - 560*b^5)*cos(2*d*
x + 2*c))*sin(8*d*x + 8*c) + (16*a^2*b^3 - 3*a*b^4 - 28*b^5 - 4*(1152*a^3*b^2 - 520*a^2*b^3 - 455*a*b^4 + 294*
b^5)*cos(4*d*x + 4*c) + 16*(48*a^2*b^3 - 55*a*b^4 + 28*b^5)*cos(2*d*x + 2*c))*sin(6*d*x + 6*c) + (32*a^2*b^3 +
2*a*b^4 - 7*b^5 - 4*(120*a^2*b^3 - 77*a*b^4 + 14*b^5)*cos(2*d*x + 2*c))*sin(4*d*x + 4*c) - (7*a*b^4 - 4*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 + 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 + 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 + 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 - 2
3*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 + 2
2*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 + 3
37*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)*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 - 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 + 2
2*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*(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 - 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*si
n(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 = 24.4428, size = 15823, normalized size = 49.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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(-(1024*a^4 - 1916*a^3*b + 1501*a
^2*b^2 - 570*a*b^3 + 105*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686
400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 916
2574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 25
2*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10
*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(491520*a^6*b - 1742720*a^5*b^2 + 2747904*a^4*b^3 - 2435
877*a^3*b^4 + 5106989/4*a^2*b^5 - 750141/2*a*b^6 + 194481/4*b^7 - 1/4*(1966080*a^6*b - 6970880*a^5*b^2 + 10991
616*a^4*b^3 - 9743508*a^3*b^4 + 5106989*a^2*b^5 - 1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 + 1/2*((32*a^16 -
193*a^15*b + 498*a^14*b^2 - 715*a^13*b^3 + 620*a^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^7)*d^3*sqrt((
3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 +
9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4
- 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d*x + c)*sin(d
*x + c) + (88320*a^9*b - 319040*a^8*b^2 + 510294*a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 71421*a^4*b^6 + 9
261*a^3*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 -
(a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 +
39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 +
194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120
*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b
^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 1121
4*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2*cos(d*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005
*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b - 17817600*a^
7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*
a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b
^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*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(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 - (a^10 - 5*a^9*b +
10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 -
51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21
- 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^1
3*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)
)*log(491520*a^6*b - 1742720*a^5*b^2 + 2747904*a^4*b^3 - 2435877*a^3*b^4 + 5106989/4*a^2*b^5 - 750141/2*a*b^6
+ 194481/4*b^7 - 1/4*(1966080*a^6*b - 6970880*a^5*b^2 + 10991616*a^4*b^3 - 9743508*a^3*b^4 + 5106989*a^2*b^5 -
1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 - 1/2*((32*a^16 - 193*a^15*b + 498*a^14*b^2 - 715*a^13*b^3 + 620*a
^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^7)*d^3*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b
^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((
a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45
*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (88320*a^9*b - 319040*a^8*b^2 + 510294*
a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 71421*a^4*b^6 + 9261*a^3*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(
1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 - (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6
*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^
4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 12
0*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10
)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*a^13 - 6276*a^
12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2*cos(d
*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*
a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 443
35881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*
b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a
^11*b^10)*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(-(1024*a^4 - 1916*a^
3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^
2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a
^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a
^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 -
5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2))*log(-491520*a^6*b + 1742720*a^5*b^2 - 2747904*a
^4*b^3 + 2435877*a^3*b^4 - 5106989/4*a^2*b^5 + 750141/2*a*b^6 - 194481/4*b^7 + 1/4*(1966080*a^6*b - 6970880*a^
5*b^2 + 10991616*a^4*b^3 - 9743508*a^3*b^4 + 5106989*a^2*b^5 - 1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 + 1/
2*((32*a^16 - 193*a^15*b + 498*a^14*b^2 - 715*a^13*b^3 + 620*a^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^
7)*d^3*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065
628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 +
210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d
*x + c)*sin(d*x + c) - (88320*a^9*b - 319040*a^8*b^2 + 510294*a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 7142
1*a^4*b^6 + 9261*a^3*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3
+ 105*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 178176
00*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 198
0972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a
^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*
b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a
^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2*cos(d*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^1
1*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b
- 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b
^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5
+ 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*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(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 + (a^10
- 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458
560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 19448
1*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14
*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 -
a^5*b^5)*d^2))*log(-491520*a^6*b + 1742720*a^5*b^2 - 2747904*a^4*b^3 + 2435877*a^3*b^4 - 5106989/4*a^2*b^5 + 7
50141/2*a*b^6 - 194481/4*b^7 + 1/4*(1966080*a^6*b - 6970880*a^5*b^2 + 10991616*a^4*b^3 - 9743508*a^3*b^4 + 510
6989*a^2*b^5 - 1500282*a*b^6 + 194481*b^7)*cos(d*x + c)^2 - 1/2*((32*a^16 - 193*a^15*b + 498*a^14*b^2 - 715*a^
13*b^3 + 620*a^12*b^4 - 327*a^11*b^5 + 98*a^10*b^6 - 13*a^9*b^7)*d^3*sqrt((3686400*a^8*b - 17817600*a^7*b^2 +
39458560*a^6*b^3 - 51952960*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 +
194481*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120
*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (88320*a^9*b - 319040*a^8
*b^2 + 510294*a^7*b^3 - 457551*a^6*b^4 + 241865*a^5*b^5 - 71421*a^4*b^6 + 9261*a^3*b^7)*d*cos(d*x + c)*sin(d*x
+ c))*sqrt(-(1024*a^4 - 1916*a^3*b + 1501*a^2*b^2 - 570*a*b^3 + 105*b^4 + (a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a
^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960*a^5*b^4
+ 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^20*b + 45
*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b
^9 + a^11*b^10)*d^4)))/((a^10 - 5*a^9*b + 10*a^8*b^2 - 10*a^7*b^3 + 5*a^6*b^4 - a^5*b^5)*d^2)) - 1/4*(2*(1024*
a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a^8*b^5 + 3361*a^7*b^6 - 441*a^6*
b^7)*d^2*cos(d*x + c)^2 - (1024*a^13 - 6276*a^12*b + 16461*a^11*b^2 - 24005*a^10*b^3 + 21090*a^9*b^4 - 11214*a
^8*b^5 + 3361*a^7*b^6 - 441*a^6*b^7)*d^2)*sqrt((3686400*a^8*b - 17817600*a^7*b^2 + 39458560*a^6*b^3 - 51952960
*a^5*b^4 + 44335881*a^4*b^5 - 25065628*a^3*b^6 + 9162574*a^2*b^7 - 1980972*a*b^8 + 194481*b^9)/((a^21 - 10*a^2
0*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 1
0*a^12*b^9 + a^11*b^10)*d^4))) - 8*(6*(2*a*b^2 - b^3)*cos(d*x + c)^7 - (49*a*b^2 - 25*b^3)*cos(d*x + c)^5 - 8*
(2*a^2*b - 9*a*b^2 + 4*b^3)*cos(d*x + c)^3 + (33*a^2*b - 46*a*b^2 + 13*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(1/(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(1/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError