3.222 \(\int \frac{1}{(a-b \sin ^4(c+d x))^2} \, dx\)

Optimal. Leaf size=210 \[ \frac{\left (4 \sqrt{a}-3 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}+\frac{\left (4 \sqrt{a}+3 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{b \tan (c+d x) \left (2 \tan ^2(c+d x)+1\right )}{4 a d (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \]

[Out]

((4*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] - Sqrt[b]
)^(3/2)*d) + ((4*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt
[a] + Sqrt[b])^(3/2)*d) - (b*Tan[c + d*x]*(1 + 2*Tan[c + d*x]^2))/(4*a*(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 = 0.260398, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3209, 1205, 1166, 205} \[ \frac{\left (4 \sqrt{a}-3 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}+\frac{\left (4 \sqrt{a}+3 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{b \tan (c+d x) \left (2 \tan ^2(c+d x)+1\right )}{4 a d (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*Sin[c + d*x]^4)^(-2),x]

[Out]

((4*Sqrt[a] - 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt[a] - Sqrt[b]
)^(3/2)*d) + ((4*Sqrt[a] + 3*Sqrt[b])*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*a^(7/4)*(Sqrt
[a] + Sqrt[b])^(3/2)*d) - (b*Tan[c + d*x]*(1 + 2*Tan[c + d*x]^2))/(4*a*(a - b)*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 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 )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{\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 (1+2 \tan ^2(c+d x)\right )}{4 a (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{-\frac{2 a (4 a-3 b) b}{a-b}-\frac{4 a (2 a-b) b x^2}{a-b}}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{8 a^2 b d}\\ &=-\frac{b \tan (c+d x) \left (1+2 \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac{\left (4 a-\sqrt{a} \sqrt{b}-3 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^{3/2} \left (\sqrt{a}+\sqrt{b}\right ) d}+\frac{\left (4 a+\sqrt{a} \sqrt{b}-3 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^{3/2} \left (\sqrt{a}-\sqrt{b}\right ) d}\\ &=\frac{\left (4 \sqrt{a}-3 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt{a}-\sqrt{b}\right )^{3/2} d}+\frac{\left (4 \sqrt{a}+3 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{7/4} \left (\sqrt{a}+\sqrt{b}\right )^{3/2} d}-\frac{b \tan (c+d x) \left (1+2 \tan ^2(c+d x)\right )}{4 a (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 = 2.82413, size = 230, normalized size = 1.1 \[ \frac{\frac{\left (-\sqrt{a} \sqrt{b}+4 a-3 b\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{2 \sqrt{a} b (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b}-\frac{\left (\sqrt{a} \sqrt{b}+4 a-3 b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{8 a^{3/2} d (a-b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((4*a - Sqrt[a]*Sqrt[b] - 3*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqrt[b]]])/Sqrt[a +
Sqrt[a]*Sqrt[b]] - ((4*a + Sqrt[a]*Sqrt[b] - 3*b)*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a
]*Sqrt[b]]])/Sqrt[-a + Sqrt[a]*Sqrt[b]] + (2*Sqrt[a]*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)]))/(8*a^(3/2)*(a - b)*d)

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Maple [B]  time = 0.129, size = 618, normalized size = 2.9 \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)^2,x)

[Out]

-1/2/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/(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)+1/2/d/(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))-1/4/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/(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-3/8/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+1/2/d/(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))-1/4/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/(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+3/8/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

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Maxima [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)^2,x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 8.20175, size = 7862, normalized size = 37.44 \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)^2,x, algorithm="fricas")

[Out]

-1/32*(((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqr
t(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b
^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b
+ 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(96*a^3*b - 170*a^2*b^2 + 405/4*a*b^3 - 81/4*b^4 - 1/
4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 + 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7
*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15
*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (120*a^5*b - 217
*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)
*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*
a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^
3*b^3)*d^2)) + 1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 -
57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^
4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) - ((a^2*b
- a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(-((a^6 - 3*a
^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6
*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6
- 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2))*log(96*a^3*b - 170*a^2*b^2 + 405/4*a*b^3 - 81/4*b^4 - 1/4*(384*a^3*b -
680*a^2*b^2 + 405*a*b^3 - 81*b^4)*cos(d*x + c)^2 - 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4
)*d^3*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20
*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) - (120*a^5*b - 217*a^4*b^2 + 132
*a^3*b^3 - 27*a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576
*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*
a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) + 16*a^2 - 15*a*b + 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) +
1/4*(2*(16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75
*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((
a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) + ((a^2*b - a*b^2)*d*co
s(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^
2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^
11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*
a^4*b^2 - a^3*b^3)*d^2))*log(-96*a^3*b + 170*a^2*b^2 - 405/4*a*b^3 + 81/4*b^4 + 1/4*(384*a^3*b - 680*a^2*b^2 +
405*a*b^3 - 81*b^4)*cos(d*x + c)^2 + 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((57
6*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15
*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*
a^2*b^4)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a
^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8
*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) + 1/4*(2*(16*a^8
- 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a
^5*b^3 + 9*a^4*b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b
+ 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) - ((a^2*b - a*b^2)*d*cos(d*x + c)^4 -
2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*d)*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^
2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^1
0*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b
^3)*d^2))*log(-96*a^3*b + 170*a^2*b^2 - 405/4*a*b^3 + 81/4*b^4 + 1/4*(384*a^3*b - 680*a^2*b^2 + 405*a*b^3 - 81
*b^4)*cos(d*x + c)^2 - 1/2*(2*(2*a^10 - 7*a^9*b + 9*a^8*b^2 - 5*a^7*b^3 + a^6*b^4)*d^3*sqrt((576*a^4*b - 1392*
a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^
8*b^5 + a^7*b^6)*d^4))*cos(d*x + c)*sin(d*x + c) + (120*a^5*b - 217*a^4*b^2 + 132*a^3*b^3 - 27*a^2*b^4)*d*cos(
d*x + c)*sin(d*x + c))*sqrt(((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a
^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2 - 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)
*d^4)) - 16*a^2 + 15*a*b - 3*b^2)/((a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)*d^2)) + 1/4*(2*(16*a^8 - 57*a^7*b + 7
5*a^6*b^2 - 43*a^5*b^3 + 9*a^4*b^4)*d^2*cos(d*x + c)^2 - (16*a^8 - 57*a^7*b + 75*a^6*b^2 - 43*a^5*b^3 + 9*a^4*
b^4)*d^2)*sqrt((576*a^4*b - 1392*a^3*b^2 + 1273*a^2*b^3 - 522*a*b^4 + 81*b^5)/((a^13 - 6*a^12*b + 15*a^11*b^2
- 20*a^10*b^3 + 15*a^9*b^4 - 6*a^8*b^5 + a^7*b^6)*d^4))) + 8*(b*cos(d*x + c)^3 - 2*b*cos(d*x + c))*sin(d*x + c
))/((a^2*b - a*b^2)*d*cos(d*x + c)^4 - 2*(a^2*b - a*b^2)*d*cos(d*x + c)^2 - (a^3 - 2*a^2*b + a*b^2)*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)**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(1/(a-b*sin(d*x+c)^4)^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError