3.3.30 \(\int \frac {\sin ^8(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [230]

Optimal. Leaf size=319 \[ -\frac {\left (2 \sqrt {a}-5 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/2} d}+\frac {\left (2 \sqrt {a}+5 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/2} d}-\frac {(a+5 b) \tan (c+d x)}{32 a (a-b)^2 b d}+\frac {\tan ^3(c+d x)}{32 a (a-b) b d}+\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

[Out]

-1/64*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)-5*b^(1/2))/a^(3/4)/b^(3/2)/d/(a^(1/2)-b^(1
/2))^(5/2)+1/64*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*(2*a^(1/2)+5*b^(1/2))/a^(3/4)/b^(3/2)/d/(a^
(1/2)+b^(1/2))^(5/2)-1/32*(a+5*b)*tan(d*x+c)/a/(a-b)^2/b/d+1/32*tan(d*x+c)^3/a/(a-b)/b/d+1/8*tan(d*x+c)^9/a/d/
(a+2*a*tan(d*x+c)^2+(a-b)*tan(d*x+c)^4)^2-1/32*sec(d*x+c)^2*tan(d*x+c)^5/a/b/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d
*x+c)^4)

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Rubi [A]
time = 0.35, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {3296, 1289, 12, 1134, 1293, 1180, 211} \begin {gather*} -\frac {\left (2 \sqrt {a}-5 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}+\frac {\left (2 \sqrt {a}+5 \sqrt {b}\right ) \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}+\frac {\tan ^3(c+d x)}{32 a b d (a-b)}+\frac {\tan ^9(c+d x)}{8 a d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}-\frac {(a+5 b) \tan (c+d x)}{32 a b d (a-b)^2}-\frac {\tan ^5(c+d x) \sec ^2(c+d x)}{32 a b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

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

Rule 1134

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-d^3)*(d*x)^(m - 3)*(2*a
+ b*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*(p + 1)*(b^2 - 4*a*c))), x] + Dist[d^4/(2*(p + 1)*(b^2 - 4*a*c)), Int
[(d*x)^(m - 4)*(2*a*(m - 3) + b*(m + 4*p + 3)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 3] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1180

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 1289

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[f*(
f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)*((b*d - 2*a*e - (b*e - 2*c*d)*x^2)/(2*(p + 1)*(b^2 - 4*a*c))), x] - D
ist[f^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)*Simp[(m - 1)*(b*d - 2*a*e) -
(4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[
p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1293

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e*f*
(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m -
2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[
{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (Inte
gerQ[p] || IntegerQ[m])

Rule 3296

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]

Rubi steps

\begin {align*} \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^8 \left (1+x^2\right )}{\left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}+\frac {\text {Subst}\left (\int -\frac {2 b x^8}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{16 a b d}\\ &=\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\text {Subst}\left (\int \frac {x^8}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{8 a d}\\ &=\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {x^4 \left (10 a+6 a x^2\right )}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{64 a^2 b d}\\ &=\frac {\tan ^3(c+d x)}{32 a (a-b) b d}+\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {x^2 \left (18 a^2+6 a (a+5 b) x^2\right )}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{192 a^2 (a-b) b d}\\ &=-\frac {(a+5 b) \tan (c+d x)}{32 a (a-b)^2 b d}+\frac {\tan ^3(c+d x)}{32 a (a-b) b d}+\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {6 a^2 (a+5 b)-6 a^2 (a-13 b) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{192 a^2 (a-b)^2 b d}\\ &=-\frac {(a+5 b) \tan (c+d x)}{32 a (a-b)^2 b d}+\frac {\tan ^3(c+d x)}{32 a (a-b) b d}+\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\left (2 a+3 \sqrt {a} \sqrt {b}-5 b\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )^2 b^{3/2} d}-\frac {\left (\left (2 \sqrt {a}-5 \sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right )^3\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 \sqrt {a} (a-b)^2 b^{3/2} d}\\ &=-\frac {\left (2 \sqrt {a}-5 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} b^{3/2} d}+\frac {\left (2 \sqrt {a}+5 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} b^{3/2} d}-\frac {(a+5 b) \tan (c+d x)}{32 a (a-b)^2 b d}+\frac {\tan ^3(c+d x)}{32 a (a-b) b d}+\frac {\tan ^9(c+d x)}{8 a d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\sec ^2(c+d x) \tan ^5(c+d x)}{32 a b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [A]
time = 2.95, size = 331, normalized size = 1.04 \begin {gather*} \frac {\frac {\left (2 a^{3/2} \sqrt {b}+a b-8 \sqrt {a} b^{3/2}+5 b^2\right ) \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {\left (2 \sqrt {a}-5 \sqrt {b}\right ) \left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {b} \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {8 b (5 a-14 b+(-2 a+5 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))}+\frac {64 a (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-b)^2 b^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

(((2*a^(3/2)*Sqrt[b] + a*b - 8*Sqrt[a]*b^(3/2) + 5*b^2)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqr
t[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]]) + ((2*Sqrt[a] - 5*Sqrt[b])*(Sqrt[a] + Sqrt[b])^2*Sqrt[b]*A
rcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) +
(8*b*(5*a - 14*b + (-2*a + 5*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*(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 - b)^2*b^2*d)

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Maple [A]
time = 1.16, size = 374, normalized size = 1.17

method	result	size
derivativedivides	\(\frac {\frac {-\frac {\left (a +19 b \right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 \left (a -b \right ) b}-\frac {3 \left (a^{2}+10 a b -3 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (a +7 b \right ) a \left (\tan ^{3}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (a +5 b \right ) \tan \left (d x +c \right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )^{2}}+\frac {\left (a -b \right ) \left (\frac {\left (-a \sqrt {a b}+13 \sqrt {a b}\, b -2 a^{2}+9 a b +5 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (-a \sqrt {a b}+13 \sqrt {a b}\, b +2 a^{2}-9 a b -5 b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(374\)
default	\(\frac {\frac {-\frac {\left (a +19 b \right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 \left (a -b \right ) b}-\frac {3 \left (a^{2}+10 a b -3 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 \left (a +7 b \right ) a \left (\tan ^{3}\left (d x +c \right )\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}-\frac {a \left (a +5 b \right ) \tan \left (d x +c \right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}}{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )^{2}}+\frac {\left (a -b \right ) \left (\frac {\left (-a \sqrt {a b}+13 \sqrt {a b}\, b -2 a^{2}+9 a b +5 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (-a \sqrt {a b}+13 \sqrt {a b}\, b +2 a^{2}-9 a b -5 b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{32 b \left (a^{2}-2 a b +b^{2}\right )}}{d}\)	\(374\)
risch	\(\text {Expression too large to display}\)	\(1589\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/d*((-1/32*(a+19*b)/(a-b)/b*tan(d*x+c)^7-3/32*(a^2+10*a*b-3*b^2)/b/(a^2-2*a*b+b^2)*tan(d*x+c)^5-3/32*(a+7*b)*
a/b/(a^2-2*a*b+b^2)*tan(d*x+c)^3-1/32*a*(a+5*b)/b/(a^2-2*a*b+b^2)*tan(d*x+c))/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2
*a*tan(d*x+c)^2+a)^2+1/32/b/(a^2-2*a*b+b^2)*(a-b)*(1/2*(-a*(a*b)^(1/2)+13*(a*b)^(1/2)*b-2*a^2+9*a*b+5*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))+1/2*(-a*(a
*b)^(1/2)+13*(a*b)^(1/2)*b+2*a^2-9*a*b-5*b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*t
an(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/8*(4*(72*a^2*b^2 - 155*a*b^3 + 26*b^4)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + ((a*b^3 - 4*b^4)*sin(14*d*x + 14
*c) - (32*a^2*b^2 - 58*a*b^3 - b^4)*sin(12*d*x + 12*c) + 3*(48*a^2*b^2 - 73*a*b^3 + 20*b^4)*sin(10*d*x + 10*c)
+ (256*a^3*b - 832*a^2*b^2 + 550*a*b^3 - 175*b^4)*sin(8*d*x + 8*c) + (112*a^2*b^2 - 533*a*b^3 + 220*b^4)*sin(
6*d*x + 6*c) - (32*a^2*b^2 - 158*a*b^3 + 141*b^4)*sin(4*d*x + 4*c) - (17*a*b^3 - 44*b^4)*sin(2*d*x + 2*c))*cos
(16*d*x + 16*c) + 2*(2*(72*a^2*b^2 - 155*a*b^3 + 26*b^4)*sin(12*d*x + 12*c) - 8*(80*a^2*b^2 - 145*a*b^3 + 44*b
^4)*sin(10*d*x + 10*c) - 3*(384*a^3*b - 1312*a^2*b^2 + 873*a*b^3 - 280*b^4)*sin(8*d*x + 8*c) - 16*(32*a^2*b^2
- 151*a*b^3 + 62*b^4)*sin(6*d*x + 6*c) + 2*(72*a^2*b^2 - 355*a*b^3 + 310*b^4)*sin(4*d*x + 4*c) + 24*(3*a*b^3 -
8*b^4)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) - 2*(2*(128*a^3*b - 456*a^2*b^2 + 1233*a*b^3 - 434*b^4)*sin(10*d*
x + 10*c) - (6400*a^3*b - 13888*a^2*b^2 + 8566*a*b^3 - 2485*b^4)*sin(8*d*x + 8*c) - 2*(128*a^3*b + 2744*a^2*b^
2 - 4711*a*b^3 + 1554*b^4)*sin(6*d*x + 6*c) + 4*(400*a^2*b^2 - 918*a*b^3 + 497*b^4)*sin(4*d*x + 4*c) - 2*(72*a
^2*b^2 - 355*a*b^3 + 310*b^4)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c) - 2*((2048*a^4 + 18560*a^3*b - 24752*a^2*b^
2 + 13175*a*b^3 - 2800*b^4)*sin(8*d*x + 8*c) + 8*(256*a^3*b + 2400*a^2*b^2 - 2379*a*b^3 + 560*b^4)*sin(6*d*x +
6*c) - 2*(128*a^3*b + 2744*a^2*b^2 - 4711*a*b^3 + 1554*b^4)*sin(4*d*x + 4*c) + 16*(32*a^2*b^2 - 151*a*b^3 + 6
2*b^4)*sin(2*d*x + 2*c))*cos(10*d*x + 10*c) - 2*((2048*a^4 + 18560*a^3*b - 24752*a^2*b^2 + 13175*a*b^3 - 2800*
b^4)*sin(6*d*x + 6*c) - (6400*a^3*b - 13888*a^2*b^2 + 8566*a*b^3 - 2485*b^4)*sin(4*d*x + 4*c) + 3*(384*a^3*b -
1312*a^2*b^2 + 873*a*b^3 - 280*b^4)*sin(2*d*x + 2*c))*cos(8*d*x + 8*c) - 4*((128*a^3*b - 456*a^2*b^2 + 1233*a
*b^3 - 434*b^4)*sin(4*d*x + 4*c) + 4*(80*a^2*b^2 - 145*a*b^3 + 44*b^4)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) - 8*
((a^2*b^5 - 2*a*b^6 + b^7)*d*cos(16*d*x + 16*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(14*d*x + 14*c)^2 + 16*(
64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*cos(12*d*x + 12*c)^2 + 64*(256*a^4*b^3 - 736*a^
3*b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a^6*b - 57344*a^5*b^2 + 83712*a^4*
b^3 - 67648*a^3*b^4 + 32841*a^2*b^5 - 9170*a*b^6 + 1225*b^7)*d*cos(8*d*x + 8*c)^2 + 64*(256*a^4*b^3 - 736*a^3*
b^4 + 753*a^2*b^5 - 322*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 21
0*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c)^2 + (a^2*b^5 - 2*a*b^
6 + b^7)*d*sin(16*d*x + 16*c)^2 + 64*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(14*d*x + 14*c)^2 + 16*(64*a^4*b^3 - 240*a
^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*d*sin(12*d*x + 12*c)^2 + 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^
5 - 322*a*b^6 + 49*b^7)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^6*b - 57344*a^5*b^2 + 83712*a^4*b^3 - 67648*a^3*b^
4 + 32841*a^2*b^5 - 9170*a*b^6 + 1225*b^7)*d*sin(8*d*x + 8*c)^2 + 64*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^5
- 322*a*b^6 + 49*b^7)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5 - 210*a*b^6 + 49*b^7)*
d*sin(4*d*x + 4*c)^2 + 64*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 64
*(a^2*b^5 - 2*a*b^6 + b^7)*d*sin(2*d*x + 2*c)^2 - 16*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) + (a^2*b^5 -
2*a*b^6 + b^7)*d - 2*(8*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(14*d*x + 14*c) + 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6
- 7*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(10*d*x + 10*c) - 2*(128*
a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30
*a*b^6 - 7*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(4*d*x + 4*c) + 8*(a^2
*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) - (a^2*b^5 - 2*a*b^6 + b^7)*d)*cos(16*d*x + 16*c) + 16*(4*(8*a^3*b^4
- 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(12*d*x + 12*c) - 8*(16*a^3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(1
0*d*x + 10*c) - 2*(128*a^4*b^3 - 352*a^3*b^4 + 355*a^2*b^5 - 166*a*b^6 + 35*b^7)*d*cos(8*d*x + 8*c) - 8*(16*a^
3*b^4 - 39*a^2*b^5 + 30*a*b^6 - 7*b^7)*d*cos(6*d*x + 6*c) + 4*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*co
s(4*d*x + 4*c) + 8*(a^2*b^5 - 2*a*b^6 + b^7)*d*cos(2*d*x + 2*c) - (a^2*b^5 - 2*a*b^6 + b^7)*d)*cos(14*d*x + 14
*c) - 8*(8*(128*a^4*b^3 - 424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*cos(10*d*x + 10*c) + 2*(1024*a^5*b
^2 - 3712*a^4*b^3 + 5304*a^3*b^4 - 3813*a^2*b^5 + 1442*a*b^6 - 245*b^7)*d*cos(8*d*x + 8*c) + 8*(128*a^4*b^3 -
424*a^3*b^4 + 513*a^2*b^5 - 266*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c) - 4*(64*a^4*b^3 - 240*a^3*b^4 + 337*a^2*b^5
- 210*a*b^6 + 49*b^7)*d*cos(4*d*x + 4*c) - 8*(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d*cos(2*d*x + 2*c) +
(8*a^3*b^4 - 23*a^2*b^5 + 22*a*b^6 - 7*b^7)*d)*cos(12*d*x + 12*c) + 16*(2*(2048*a^5*b^2 - 6528*a^4*b^3 + 8144
*a^3*b^4 - 5141*a^2*b^5 + 1722*a*b^6 - 245*b^7)*d*cos(8*d*x + 8*c) + 8*(256*a^4*b^3 - 736*a^3*b^4 + 753*a^2*b^
5 - 322*a*b^6 + 49*b^7)*d*cos(6*d*x + 6*c) - 4*...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 5219 vs. \(2 (263) = 526\).
time = 1.47, size = 5219, normalized size = 16.36 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/256*(((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^
2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 +
(a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a
^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*
b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a
^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3
*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(35*a^3 - 1491/4*a^2*b + 1875/2*a*b^2 + 625/4*b^3 - 1/4*(140*a^3 - 1491*a^2
*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 + 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^7*b^5 - 140*a^6*b^6 + 135*a^5*b^
7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b
^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^
5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (70*a^5*b - 623*a^4*b^2 + 1161*a^3*b^3 + 99
5*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a
^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*
b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a
^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3
*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 - 129
*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 - 290*a^5*b^4 + 270*a^4*b^5 -
129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3
- 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*
b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) - ((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b
^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3
*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^5)*d)*sqrt(((a^6*b^3 - 5*a^5*b^4
+ 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3
+ 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 1
20*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 -
5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(35*a^3 - 1491/4*a^2*b + 1875/2*a*b^2 + 625
/4*b^3 - 1/4*(140*a^3 - 1491*a^2*b + 3750*a*b^2 + 625*b^3)*cos(d*x + c)^2 - 1/2*((a^9*b^3 - 18*a^8*b^4 + 75*a^
7*b^5 - 140*a^6*b^6 + 135*a^5*b^7 - 66*a^4*b^8 + 13*a^3*b^9)*d^3*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2
+ 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 21
0*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (70*a^5*b -
623*a^4*b^2 + 1161*a^3*b^3 + 995*a^2*b^4 + 125*a*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^6*b^3 - 5*a^5*b^4
+ 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 + 7700*a*b^3
+ 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*a^7*b^9 - 1
20*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) - 4*a^3 + 35*a^2*b - 70*a*b^2 - 105*b^3)/((a^6*b^3 -
5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2)) + 1/4*(2*(4*a^8*b - 45*a^7*b^2 + 165*a^6*b^3 -
290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2*cos(d*x + c)^2 - (4*a^8*b - 45*a^7*b^2 + 165*a^6*b^
3 - 290*a^5*b^4 + 270*a^4*b^5 - 129*a^3*b^6 + 25*a^2*b^7)*d^2)*sqrt((1225*a^4 - 10780*a^3*b + 21966*a^2*b^2 +
7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^7 - 252*a^8*b^8 + 210*
a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4))) + ((a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x +
c)^8 - 4*(a^2*b^3 - 2*a*b^4 + b^5)*d*cos(d*x + c)^6 - 2*(a^3*b^2 - 5*a^2*b^3 + 7*a*b^4 - 3*b^5)*d*cos(d*x + c
)^4 + 4*(a^3*b^2 - 3*a^2*b^3 + 3*a*b^4 - b^5)*d*cos(d*x + c)^2 + (a^4*b - 4*a^3*b^2 + 6*a^2*b^3 - 4*a*b^4 + b^
5)*d)*sqrt(-((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2*sqrt((1225*a^4 - 10780*a^
3*b + 21966*a^2*b^2 + 7700*a*b^3 + 625*b^4)/((a^13*b^3 - 10*a^12*b^4 + 45*a^11*b^5 - 120*a^10*b^6 + 210*a^9*b^
7 - 252*a^8*b^8 + 210*a^7*b^9 - 120*a^6*b^10 + 45*a^5*b^11 - 10*a^4*b^12 + a^3*b^13)*d^4)) + 4*a^3 - 35*a^2*b
+ 70*a*b^2 + 105*b^3)/((a^6*b^3 - 5*a^5*b^4 + 10*a^4*b^5 - 10*a^3*b^6 + 5*a^2*b^7 - a*b^8)*d^2))*log(-35*a^3 +
1491/4*a^2*b - 1875/2*a*b^2 - 625/4*b^3 + 1/4*...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**8/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1989 vs. \(2 (263) = 526\).
time = 1.32, size = 1989, normalized size = 6.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/64*(((3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 45*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b
)*a^2*b + 77*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 13*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt
(a*b)*b^3)*(a^2*b - 2*a*b^2 + b^3)^2*abs(-a + b) - (3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^6*b - 49*sqrt(a^2
- a*b + sqrt(a*b)*(a - b))*a^4*b^3 + 112*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^4 - 87*sqrt(a^2 - a*b + sqr
t(a*b)*(a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))
*b^7)*abs(a^2*b - 2*a*b^2 + b^3)*abs(-a + b) - (6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b - 63*sqr
t(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^2 + 229*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^3
- 367*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 + 233*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*
b)*a^4*b^5 + 27*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^6 - 89*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))
*sqrt(a*b)*a^2*b^7 + 19*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a
- b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3*b - 2*a^2*b^2
+ a*b^3 + sqrt((a^3*b - 2*a^2*b^2 + a*b^3)^2 - (a^3*b - 2*a^2*b^2 + a*b^3)*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)
))/(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4))))/((3*a^10*b^2 - 27*a^9*b^3 + 104*a^8*b^4 - 224*a^7*b^5 + 294*a^6*b^6
- 238*a^5*b^7 + 112*a^4*b^8 - 24*a^3*b^9 - a^2*b^10 + a*b^11)*abs(a^2*b - 2*a*b^2 + b^3)) + ((3*sqrt(a^2 - a*b
- sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 45*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b + 77*sqrt(a^2 - a
*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 13*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^3)*(a^2*b - 2*a*b
^2 + b^3)^2*abs(-a + b) - (3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b - 49*sqrt(a^2 - a*b - sqrt(a*b)*(a - b)
)*a^4*b^3 + 112*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^4 - 87*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^2*b^5 +
16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^7)*abs(a^2*b - 2*a*b^2
+ b^3)*abs(-a + b) - (6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b - 63*sqrt(a^2 - a*b - sqrt(a*b)*(
a - b))*sqrt(a*b)*a^7*b^2 + 229*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^6*b^3 - 367*sqrt(a^2 - a*b - s
qrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 + 233*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^5 + 27*sqrt(a^2
- a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^6 - 89*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 19*s
qrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(
-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c)/sqrt((a^3*b - 2*a^2*b^2 + a*b^3 - sqrt((a^3*b - 2
*a^2*b^2 + a*b^3)^2 - (a^3*b - 2*a^2*b^2 + a*b^3)*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)))/(a^3*b - 3*a^2*b^2 + 3
*a*b^3 - b^4))))/((3*a^10*b^2 - 27*a^9*b^3 + 104*a^8*b^4 - 224*a^7*b^5 + 294*a^6*b^6 - 238*a^5*b^7 + 112*a^4*b
^8 - 24*a^3*b^9 - a^2*b^10 + a*b^11)*abs(a^2*b - 2*a*b^2 + b^3)) + 2*(a^2*tan(d*x + c)^7 + 18*a*b*tan(d*x + c)
^7 - 19*b^2*tan(d*x + c)^7 + 3*a^2*tan(d*x + c)^5 + 30*a*b*tan(d*x + c)^5 - 9*b^2*tan(d*x + c)^5 + 3*a^2*tan(d
*x + c)^3 + 21*a*b*tan(d*x + c)^3 + a^2*tan(d*x + c) + 5*a*b*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*x + c)
^4 + 2*a*tan(d*x + c)^2 + a)^2*(a^2*b - 2*a*b^2 + b^3)))/d

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Mupad [B]
time = 19.65, size = 2500, normalized size = 7.84 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^8/(a - b*sin(c + d*x)^4)^3,x)

[Out]

(atan(((((229376*a^2*b^6 - 81920*a*b^7 - 196608*a^3*b^5 + 32768*a^4*b^4 + 16384*a^5*b^3)/(32768*(3*a*b^5 - b^6
- 3*a^2*b^4 + a^3*b^3)) - (tan(c + d*x)*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*
a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^
6*b^8 + 5*a^7*b^7 - a^8*b^6)))^(1/2)*(16384*a^2*b^8 - 81920*a^3*b^7 + 163840*a^4*b^6 - 163840*a^5*b^5 + 81920*
a^6*b^4 - 16384*a^7*b^3))/(256*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*
b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5
*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))^(1/2) + (tan(c + d*x)*(259*a*b^3 - 35*a^3*b + 4*a
^4 + 25*b^4 + 35*a^2*b^2))/(256*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3
*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 -
5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))^(1/2)*1i - (((229376*a^2*b^6 - 81920*a*b^7 - 196
608*a^3*b^5 + 32768*a^4*b^4 + 16384*a^5*b^3)/(32768*(3*a*b^5 - b^6 - 3*a^2*b^4 + a^3*b^3)) + (tan(c + d*x)*((2
5*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^3 + 154*a*b*(
a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))^(1/2)*(16384*
a^2*b^8 - 81920*a^3*b^7 + 163840*a^4*b^6 - 163840*a^5*b^5 + 81920*a^6*b^4 - 16384*a^7*b^3))/(256*(3*a*b^4 - b^
5 - 3*a^2*b^3 + a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a
^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*
b^7 - a^8*b^6)))^(1/2) - (tan(c + d*x)*(259*a*b^3 - 35*a^3*b + 4*a^4 + 25*b^4 + 35*a^2*b^2))/(256*(3*a*b^4 - b
^5 - 3*a^2*b^3 + a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*
a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7
*b^7 - a^8*b^6)))^(1/2)*1i)/((4*a^2 - 77*a*b + 325*b^2)/(16384*(3*a*b^5 - b^6 - 3*a^2*b^4 + a^3*b^3)) + (((229
376*a^2*b^6 - 81920*a*b^7 - 196608*a^3*b^5 + 32768*a^4*b^4 + 16384*a^5*b^3)/(32768*(3*a*b^5 - b^6 - 3*a^2*b^4
+ a^3*b^3)) - (tan(c + d*x)*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*
a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7
*b^7 - a^8*b^6)))^(1/2)*(16384*a^2*b^8 - 81920*a^3*b^7 + 163840*a^4*b^6 - 163840*a^5*b^5 + 81920*a^6*b^4 - 163
84*a^7*b^3))/(256*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) +
105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 1
0*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))^(1/2) + (tan(c + d*x)*(259*a*b^3 - 35*a^3*b + 4*a^4 + 25*b^4 +
35*a^2*b^2))/(256*(3*a*b^4 - b^5 - 3*a^2*b^3 + a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) +
105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 +
10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))^(1/2) + (((229376*a^2*b^6 - 81920*a*b^7 - 196608*a^3*b^5 + 32
768*a^4*b^4 + 16384*a^5*b^3)/(32768*(3*a*b^5 - b^6 - 3*a^2*b^4 + a^3*b^3)) + (tan(c + d*x)*((25*b^2*(a^3*b^9)^
(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^3 + 154*a*b*(a^3*b^9)^(1/2))/
(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))^(1/2)*(16384*a^2*b^8 - 81920*
a^3*b^7 + 163840*a^4*b^6 - 163840*a^5*b^5 + 81920*a^6*b^4 - 16384*a^7*b^3))/(256*(3*a*b^4 - b^5 - 3*a^2*b^3 +
a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b
^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))
^(1/2) - (tan(c + d*x)*(259*a*b^3 - 35*a^3*b + 4*a^4 + 25*b^4 + 35*a^2*b^2))/(256*(3*a*b^4 - b^5 - 3*a^2*b^3 +
a^3*b^2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*
b^3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6))
)^(1/2)))*((25*b^2*(a^3*b^9)^(1/2) - 35*a^2*(a^3*b^9)^(1/2) + 105*a^2*b^6 + 70*a^3*b^5 - 35*a^4*b^4 + 4*a^5*b^
3 + 154*a*b*(a^3*b^9)^(1/2))/(16384*(a^3*b^11 - 5*a^4*b^10 + 10*a^5*b^9 - 10*a^6*b^8 + 5*a^7*b^7 - a^8*b^6)))^
(1/2)*2i)/d - ((tan(c + d*x)^7*(a + 19*b))/(32*(a*b - b^2)) + (3*tan(c + d*x)^3*(7*a*b + a^2))/(32*(a^2*b - 2*
a*b^2 + b^3)) + (a*tan(c + d*x)*(a + 5*b))/(32*(a^2*b - 2*a*b^2 + b^3)) + (3*tan(c + d*x)^5*(10*a*b + a^2 - 3*
b^2))/(32*(a - b)*(a*b - b^2)))/(d*(tan(c + d*x)^8*(a^2 - 2*a*b + b^2) + a^2 - tan(c + d*x)^4*(2*a*b - 6*a^2)
- tan(c + d*x)^6*(4*a*b - 4*a^2) + 4*a^2*tan(c + d*x)^2)) + (atan(((((229376*a^2*b^6 - 81920*a*b^7 - 196608*a^
3*b^5 + 32768*a^4*b^4 + 16384*a^5*b^3)/(32768*(...

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