3.8 \(\int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx\)

Optimal. Leaf size=91 \[ \frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}-\frac {b \sin (x) \cos (x)}{2 \left (a^2+b^2\right )}-\frac {a^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2} \]

[Out]

a^2*b*x/(a^2+b^2)^2+1/2*b*x/(a^2+b^2)-a^3*ln(a*cos(x)+b*sin(x))/(a^2+b^2)^2-1/2*b*cos(x)*sin(x)/(a^2+b^2)-1/2*
a*sin(x)^2/(a^2+b^2)

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Rubi [A]  time = 0.11, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3099, 3097, 3133, 2635, 8} \[ \frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}-\frac {b \sin (x) \cos (x)}{2 \left (a^2+b^2\right )}-\frac {a^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[x]^3/(a*Cos[x] + b*Sin[x]),x]

[Out]

(a^2*b*x)/(a^2 + b^2)^2 + (b*x)/(2*(a^2 + b^2)) - (a^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^2 - (b*Cos[x]*Sin
[x])/(2*(a^2 + b^2)) - (a*Sin[x]^2)/(2*(a^2 + b^2))

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 3097

Int[sin[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(b*x)/(a^2 + b^2), x] - Dist[a/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c +
d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3099

Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(a*Sin[c + d*x]^(m - 1))/(d*(a^2 + b^2)*(m - 1)), x] + (Dist[a^2/(a^2 + b^2), Int[Sin[c + d*x]^(m - 2)/
(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Dist[b/(a^2 + b^2), Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a,
b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]

Rule 3133

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rubi steps

\begin {align*} \int \frac {\sin ^3(x)}{a \cos (x)+b \sin (x)} \, dx &=-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}+\frac {a^2 \int \frac {\sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac {b \int \sin ^2(x) \, dx}{a^2+b^2}\\ &=\frac {a^2 b x}{\left (a^2+b^2\right )^2}-\frac {b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {b \int 1 \, dx}{2 \left (a^2+b^2\right )}\\ &=\frac {a^2 b x}{\left (a^2+b^2\right )^2}+\frac {b x}{2 \left (a^2+b^2\right )}-\frac {a^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^2}-\frac {b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )}-\frac {a \sin ^2(x)}{2 \left (a^2+b^2\right )}\\ \end {align*}

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Mathematica [C]  time = 0.20, size = 94, normalized size = 1.03 \[ \frac {-2 a^3 \log \left ((a \cos (x)+b \sin (x))^2\right )-4 i a^3 x+4 i a^3 \tan ^{-1}(\tan (x))+a \left (a^2+b^2\right ) \cos (2 x)+6 a^2 b x-a^2 b \sin (2 x)+2 b^3 x-b^3 \sin (2 x)}{4 \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[x]^3/(a*Cos[x] + b*Sin[x]),x]

[Out]

((-4*I)*a^3*x + 6*a^2*b*x + 2*b^3*x + (4*I)*a^3*ArcTan[Tan[x]] + a*(a^2 + b^2)*Cos[2*x] - 2*a^3*Log[(a*Cos[x]
+ b*Sin[x])^2] - a^2*b*Sin[2*x] - b^3*Sin[2*x])/(4*(a^2 + b^2)^2)

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fricas [A]  time = 0.48, size = 93, normalized size = 1.02 \[ -\frac {a^{3} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) - {\left (a^{3} + a b^{2}\right )} \cos \relax (x)^{2} + {\left (a^{2} b + b^{3}\right )} \cos \relax (x) \sin \relax (x) - {\left (3 \, a^{2} b + b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="fricas")

[Out]

-1/2*(a^3*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - (a^3 + a*b^2)*cos(x)^2 + (a^2*b + b^3)*cos(x
)*sin(x) - (3*a^2*b + b^3)*x)/(a^4 + 2*a^2*b^2 + b^4)

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giac [A]  time = 0.21, size = 148, normalized size = 1.63 \[ -\frac {a^{3} b \log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} + \frac {a^{3} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (3 \, a^{2} b + b^{3}\right )} x}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a^{3} \tan \relax (x)^{2} + a^{2} b \tan \relax (x) + b^{3} \tan \relax (x) - a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\tan \relax (x)^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="giac")

[Out]

-a^3*b*log(abs(b*tan(x) + a))/(a^4*b + 2*a^2*b^3 + b^5) + 1/2*a^3*log(tan(x)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) +
1/2*(3*a^2*b + b^3)*x/(a^4 + 2*a^2*b^2 + b^4) - 1/2*(a^3*tan(x)^2 + a^2*b*tan(x) + b^3*tan(x) - a*b^2)/((a^4 +
2*a^2*b^2 + b^4)*(tan(x)^2 + 1))

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maple [B]  time = 0.47, size = 173, normalized size = 1.90 \[ -\frac {a^{3} \ln \left (a +b \tan \relax (x )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {\tan \relax (x ) a^{2} b}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}-\frac {\tan \relax (x ) b^{3}}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {a^{3}}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {b^{2} a}{2 \left (a^{2}+b^{2}\right )^{2} \left (1+\tan ^{2}\relax (x )\right )}+\frac {a^{3} \ln \left (1+\tan ^{2}\relax (x )\right )}{2 \left (a^{2}+b^{2}\right )^{2}}+\frac {3 \arctan \left (\tan \relax (x )\right ) a^{2} b}{2 \left (a^{2}+b^{2}\right )^{2}}+\frac {\arctan \left (\tan \relax (x )\right ) b^{3}}{2 \left (a^{2}+b^{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(a*cos(x)+b*sin(x)),x)

[Out]

-a^3/(a^2+b^2)^2*ln(a+b*tan(x))-1/2/(a^2+b^2)^2/(1+tan(x)^2)*tan(x)*a^2*b-1/2/(a^2+b^2)^2/(1+tan(x)^2)*tan(x)*
b^3+1/2/(a^2+b^2)^2/(1+tan(x)^2)*a^3+1/2/(a^2+b^2)^2/(1+tan(x)^2)*b^2*a+1/2/(a^2+b^2)^2*a^3*ln(1+tan(x)^2)+3/2
/(a^2+b^2)^2*arctan(tan(x))*a^2*b+1/2/(a^2+b^2)^2*arctan(tan(x))*b^3

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maxima [B]  time = 0.43, size = 209, normalized size = 2.30 \[ -\frac {a^{3} \log \left (-a - \frac {2 \, b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a^{3} \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (3 \, a^{2} b + b^{3}\right )} \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {\frac {b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {2 \, a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {b \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}}}{a^{2} + b^{2} + \frac {2 \, {\left (a^{2} + b^{2}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {{\left (a^{2} + b^{2}\right )} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)^3/(a*cos(x)+b*sin(x)),x, algorithm="maxima")

[Out]

-a^3*log(-a - 2*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/(a^4 + 2*a^2*b^2 + b^4) + a^3*log(sin(x)^2/
(cos(x) + 1)^2 + 1)/(a^4 + 2*a^2*b^2 + b^4) + (3*a^2*b + b^3)*arctan(sin(x)/(cos(x) + 1))/(a^4 + 2*a^2*b^2 + b
^4) - (b*sin(x)/(cos(x) + 1) + 2*a*sin(x)^2/(cos(x) + 1)^2 - b*sin(x)^3/(cos(x) + 1)^3)/(a^2 + b^2 + 2*(a^2 +
b^2)*sin(x)^2/(cos(x) + 1)^2 + (a^2 + b^2)*sin(x)^4/(cos(x) + 1)^4)

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mupad [B]  time = 7.50, size = 3512, normalized size = 38.59 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(x)^3/(a*cos(x) + b*sin(x)),x)

[Out]

(4*a^3*log(1/(cos(x) + 1)))/(4*a^4 + 4*b^4 + 8*a^2*b^2) - ((b*tan(x/2))/(a^2 + b^2) + (2*a*tan(x/2)^2)/(a^2 +
b^2) - (b*tan(x/2)^3)/(a^2 + b^2))/(2*tan(x/2)^2 + tan(x/2)^4 + 1) - (a^3*log(a + 2*b*tan(x/2) - a*tan(x/2)^2)
)/(a^4 + b^4 + 2*a^2*b^2) - (b*atan((tan(x/2)*((((4*a^3*((b*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4
- 8*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a^3*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4
+ 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(3*a^2 + b^2))/(2*(a^4 + b^4
+ 2*a^2*b^2)) + (16*a^3*b*(3*a^2 + b^2)*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*
a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2
*b^2) + (b*(3*a^2 + b^2)*((8*(2*a*b^8 + 13*a^3*b^6 + 32*a^5*b^4 + 21*a^7*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*
b^2) + (4*a^3*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4 - 8*a^8*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b
^2) + (32*a^3*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a
^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2)))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (b^3*(3*a^2 +
b^2)^3*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((a^4 + b^4 + 2*a^2*b^2)^3*(a^6 + b^6
+ 3*a^2*b^4 + 3*a^4*b^2)))*(16*a^8 + b^8 + 5*a^2*b^6 - 13*a^4*b^4 - 73*a^6*b^2))/(16*a^8 + b^8 + 7*a^2*b^6 +
15*a^4*b^4 + 25*a^6*b^2)^2 + (2*a*b*(b^6 - 28*a^6 + 10*a^2*b^4 + 17*a^4*b^2)*((8*(8*a^8 + 2*a^4*b^4 + 9*a^6*b^
2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*a^3*((8*(2*a*b^8 + 13*a^3*b^6 + 32*a^5*b^4 + 21*a^7*b^2))/(a^6 +
b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (4*a^3*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4 - 8*a^8*b^2))/(a^6 + b
^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a^3*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4
+ 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2)))/(4*a^4 + 4*b^4 + 8*
a^2*b^2) - (b*(3*a^2 + b^2)*((b*((8*(4*a^2*b^8 - 8*a^10 + 16*a^4*b^6 + 12*a^6*b^4 - 8*a^8*b^2))/(a^6 + b^6 + 3
*a^2*b^4 + 3*a^4*b^2) + (32*a^3*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b
^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(3*a^2 + b^2))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (16*a^3*b*(
3*a^2 + b^2)*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^
4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (8*a^3*b^2*(3*a^2 +
b^2)^2*(12*a*b^10 + 48*a^3*b^8 + 72*a^5*b^6 + 48*a^7*b^4 + 12*a^9*b^2))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^
4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(16*a^8 + b^8 + 7*a^2*b^6 + 15*a^4*b^4 + 25*a^6*b^2)^2
)*(a^10 + b^10 + 5*a^2*b^8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2))/(4*a*b^3 + 12*a^3*b) - (((4*a^3*((b*(3*a^2
+ b^2)*((8*(2*a*b^9 - 10*a^9*b + 8*a^3*b^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^3*(12*a^
10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4
+ 3*a^4*b^2))))/(2*(a^4 + b^4 + 2*a^2*b^2)) - (16*a^3*b*(3*a^2 + b^2)*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 +
72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*
b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2) - (b*((8*(a^2*b^7 + 2*a^4*b^5 + a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b
^2) - (4*a^3*((8*(2*a*b^9 - 10*a^9*b + 8*a^3*b^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^3*
(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a
^2*b^4 + 3*a^4*b^2))))/(4*a^4 + 4*b^4 + 8*a^2*b^2))*(3*a^2 + b^2))/(2*(a^4 + b^4 + 2*a^2*b^2)) + (b^3*(3*a^2 +
b^2)^3*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((a^4 + b^4 + 2*a^2*b^2)^3*(a^6 + b^6
+ 3*a^2*b^4 + 3*a^4*b^2)))*(16*a^8 + b^8 + 5*a^2*b^6 - 13*a^4*b^4 - 73*a^6*b^2)*(a^10 + b^10 + 5*a^2*b^8 + 10
*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2))/((4*a*b^3 + 12*a^3*b)*(16*a^8 + b^8 + 7*a^2*b^6 + 15*a^4*b^4 + 25*a^6*b^2)
^2) + (2*a*b*(b^6 - 28*a^6 + 10*a^2*b^4 + 17*a^4*b^2)*((8*(2*a^7*b + a^5*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*
b^2) + (4*a^3*((8*(a^2*b^7 + 2*a^4*b^5 + a^6*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (4*a^3*((8*(2*a*b^9 -
10*a^9*b + 8*a^3*b^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^3*(12*a^10*b + 12*a^2*b^9 + 4
8*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(4*a
^4 + 4*b^4 + 8*a^2*b^2)))/(4*a^4 + 4*b^4 + 8*a^2*b^2) + (b*((b*(3*a^2 + b^2)*((8*(2*a*b^9 - 10*a^9*b + 8*a^3*b
^7 - 16*a^7*b^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (32*a^3*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*
b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^4 + 8*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(a^4 + b^4 + 2*a^2*b
^2)) - (16*a^3*b*(3*a^2 + b^2)*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^4 + 4*b^
4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(3*a^2 + b^2))/(2*(a^4 + b^4 + 2*
a^2*b^2)) - (8*a^3*b^2*(3*a^2 + b^2)^2*(12*a^10*b + 12*a^2*b^9 + 48*a^4*b^7 + 72*a^6*b^5 + 48*a^8*b^3))/((4*a^
4 + 4*b^4 + 8*a^2*b^2)*(a^4 + b^4 + 2*a^2*b^2)^2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)))*(a^10 + b^10 + 5*a^2*b^
8 + 10*a^4*b^6 + 10*a^6*b^4 + 5*a^8*b^2))/((4*a*b^3 + 12*a^3*b)*(16*a^8 + b^8 + 7*a^2*b^6 + 15*a^4*b^4 + 25*a^
6*b^2)^2))*(3*a^2 + b^2))/(a^4 + b^4 + 2*a^2*b^2)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(x)**3/(a*cos(x)+b*sin(x)),x)

[Out]

Timed out

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