\(\int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx\) [22]

Optimal result

Integrand size = 16, antiderivative size = 98 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {b \left (3 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {a}{2 \left (a^2+b^2\right ) (b+a \cot (x))^2}+\frac {2 a b}{\left (a^2+b^2\right )^2 (b+a \cot (x))}+\frac {a \left (a^2-3 b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3164, 3564, 3610, 3612, 3611} \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {b x \left (3 a^2-b^2\right )}{\left (a^2+b^2\right )^3}+\frac {2 a b}{\left (a^2+b^2\right )^2 (a \cot (x)+b)}+\frac {a}{2 \left (a^2+b^2\right ) (a \cot (x)+b)^2}+\frac {a \left (a^2-3 b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

[In]

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

[Out]

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

Rule 3164

Int[sin[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symb
ol] :> Int[(b + a*Cot[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b
^2, 0]

Rule 3564

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((a + b*Tan[c + d*x])^(n + 1)/(d*(n + 1)*
(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3610

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b
*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

Rule 3611

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c/(b*f))
*Log[RemoveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.16 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {b \left (-3 a^2+b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac {a \left (a^2-3 b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac {a^3}{2 (a-i b)^2 (a+i b)^2 (a \cos (x)+b \sin (x))^2}+\frac {3 a b \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37

method	result	size
default	\(\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (a +b \tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a^{2} \left (a^{2}+3 b^{2}\right )}{\left (a^{2}+b^{2}\right )^{2} b^{2} \left (a +b \tan \left (x \right )\right )}+\frac {a^{3}}{2 b^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (x \right )\right )^{2}}+\frac {\frac {\left (-a^{3}+3 a \,b^{2}\right ) \ln \left (1+\tan \left (x \right )^{2}\right )}{2}+\left (-3 a^{2} b +b^{3}\right ) \arctan \left (\tan \left (x \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}\)	\(134\)
parallelrisch	\(\frac {2 a \left (a^{2}-3 b^{2}\right ) \left (\left (a^{2}-b^{2}\right ) \cos \left (2 x \right )+2 b a \sin \left (2 x \right )+a^{2}+b^{2}\right ) \ln \left (\frac {-a \cos \left (x \right )-b \sin \left (x \right )}{\cos \left (x \right )+1}\right )-2 a \left (a^{2}-3 b^{2}\right ) \left (\left (a^{2}-b^{2}\right ) \cos \left (2 x \right )+2 b a \sin \left (2 x \right )+a^{2}+b^{2}\right ) \ln \left (\frac {1}{\cos \left (x \right )+1}\right )+\left (-6 a^{4} b x +8 a^{2} b^{3} x -2 b^{5} x -a^{5}-6 a^{3} b^{2}-5 a \,b^{4}\right ) \cos \left (2 x \right )+4 b a \left (-3 x \,a^{2} b +x \,b^{3}+a^{3}+a \,b^{2}\right ) \sin \left (2 x \right )+\left (a^{2}+b^{2}\right ) \left (-6 x \,a^{2} b +2 x \,b^{3}+a^{3}+5 a \,b^{2}\right )}{2 \left (a^{2}+b^{2}\right )^{3} \left (\left (a^{2}-b^{2}\right ) \cos \left (2 x \right )+2 b a \sin \left (2 x \right )+a^{2}+b^{2}\right )}\)	\(259\)
risch	\(-\frac {i x}{3 i b \,a^{2}-i b^{3}-a^{3}+3 a \,b^{2}}-\frac {2 i a^{3} x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {6 i a x \,b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 a^{2} \left (2 i a b \,{\mathrm e}^{2 i x}+a^{2} {\mathrm e}^{2 i x}+3 b^{2} {\mathrm e}^{2 i x}+3 i b a -3 b^{2}\right )}{\left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right )^{2} \left (i b +a \right )^{2} \left (-i b +a \right )^{3}}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 a \ln \left ({\mathrm e}^{2 i x}-\frac {i b +a}{i b -a}\right ) b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\)	\(293\)
norman	\(\frac {\frac {\left (2 a^{5}+10 a^{3} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{2}}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\left (2 a^{5}+10 a^{3} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{8}}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (-3 a^{5}-15 a^{3} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{4}}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {2 \left (-3 a^{5}-15 a^{3} b^{2}\right ) \tan \left (\frac {x}{2}\right )^{6}}{a^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {4 a^{2} b \tan \left (\frac {x}{2}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {8 a^{2} b \tan \left (\frac {x}{2}\right )^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {8 a^{2} b \tan \left (\frac {x}{2}\right )^{7}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 a^{2} b \tan \left (\frac {x}{2}\right )^{9}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 a \,b^{2} \left (3 a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {8 a \,b^{2} \left (3 a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{3}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {8 a \,b^{2} \left (3 a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{7}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {4 a \,b^{2} \left (3 a^{2}-b^{2}\right ) x \tan \left (\frac {x}{2}\right )^{9}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 \left (a^{2}-6 b^{2}\right ) \left (3 a^{2}-b^{2}\right ) b x \tan \left (\frac {x}{2}\right )^{4}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {2 \left (a^{2}-6 b^{2}\right ) \left (3 a^{2}-b^{2}\right ) b x \tan \left (\frac {x}{2}\right )^{6}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {\left (a^{2}+4 b^{2}\right ) \left (3 a^{2}-b^{2}\right ) b x \tan \left (\frac {x}{2}\right )^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {\left (a^{2}+4 b^{2}\right ) \left (3 a^{2}-b^{2}\right ) b x \tan \left (\frac {x}{2}\right )^{8}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {\left (3 a^{2}-b^{2}\right ) a^{2} b x \tan \left (\frac {x}{2}\right )^{10}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )^{2}}+\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {a \left (a^{2}-3 b^{2}\right ) \ln \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\)	\(923\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (96) = 192\).

Time = 0.27 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.88 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\frac {a^{5} + 7 \, a^{3} b^{2} - 2 \, {\left (6 \, a^{3} b^{2} + {\left (3 \, a^{4} b - 4 \, a^{2} b^{3} + b^{5}\right )} x\right )} \cos \left (x\right )^{2} + 2 \, {\left (3 \, a^{4} b - 3 \, a^{2} b^{3} - 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} x\right )} \cos \left (x\right ) \sin \left (x\right ) - 2 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} x + {\left (a^{3} b^{2} - 3 \, a b^{4} + {\left (a^{5} - 4 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \cos \left (x\right ) \sin \left (x\right )\right )} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}\right )}{2 \, {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8} + {\left (a^{8} + 2 \, a^{6} b^{2} - 2 \, a^{2} b^{6} - b^{8}\right )} \cos \left (x\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \cos \left (x\right ) \sin \left (x\right )\right )}} \]

[In]

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

[Out]

1/2*(a^5 + 7*a^3*b^2 - 2*(6*a^3*b^2 + (3*a^4*b - 4*a^2*b^3 + b^5)*x)*cos(x)^2 + 2*(3*a^4*b - 3*a^2*b^3 - 2*(3*
a^3*b^2 - a*b^4)*x)*cos(x)*sin(x) - 2*(3*a^2*b^3 - b^5)*x + (a^3*b^2 - 3*a*b^4 + (a^5 - 4*a^3*b^2 + 3*a*b^4)*c
os(x)^2 + 2*(a^4*b - 3*a^2*b^3)*cos(x)*sin(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2))/(a^6*b^2
+ 3*a^4*b^4 + 3*a^2*b^6 + b^8 + (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*cos(x)^2 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b
^5 + a*b^7)*cos(x)*sin(x))

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\text {Exception raised: AttributeError} \]

[In]

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

[Out]

Exception raised: AttributeError >> 'NoneType' object has no attribute 'primitive'

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (96) = 192\).

Time = 0.29 (sec) , antiderivative size = 359, normalized size of antiderivative = 3.66 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {2 \, {\left (3 \, a^{2} b - b^{3}\right )} \arctan \left (\frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (-a - \frac {2 \, b \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {a \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\frac {\sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (\frac {2 \, a^{2} b \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, a^{2} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {{\left (a^{3} + 5 \, a b^{2}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}}\right )}}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} + \frac {4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} - \frac {2 \, {\left (a^{6} - 3 \, a^{2} b^{4} - 2 \, b^{6}\right )} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} - \frac {4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} \]

[In]

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

[Out]

-2*(3*a^2*b - b^3)*arctan(sin(x)/(cos(x) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (a^3 - 3*a*b^2)*log(-a -
2*b*sin(x)/(cos(x) + 1) + a*sin(x)^2/(cos(x) + 1)^2)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^3 - 3*a*b^2)*log
(sin(x)^2/(cos(x) + 1)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 2*(2*a^2*b*sin(x)/(cos(x) + 1) - 2*a^2*b*s
in(x)^3/(cos(x) + 1)^3 + (a^3 + 5*a*b^2)*sin(x)^2/(cos(x) + 1)^2)/(a^6 + 2*a^4*b^2 + a^2*b^4 + 4*(a^5*b + 2*a^
3*b^3 + a*b^5)*sin(x)/(cos(x) + 1) - 2*(a^6 - 3*a^2*b^4 - 2*b^6)*sin(x)^2/(cos(x) + 1)^2 - 4*(a^5*b + 2*a^3*b^
3 + a*b^5)*sin(x)^3/(cos(x) + 1)^3 + (a^6 + 2*a^4*b^2 + a^2*b^4)*sin(x)^4/(cos(x) + 1)^4)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (96) = 192\).

Time = 0.28 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.47 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=-\frac {{\left (3 \, a^{2} b - b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (x\right )^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{3} b - 3 \, a b^{3}\right )} \log \left ({\left | b \tan \left (x\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {3 \, a^{3} b^{4} \tan \left (x\right )^{2} - 9 \, a b^{6} \tan \left (x\right )^{2} + 2 \, a^{6} b \tan \left (x\right ) + 14 \, a^{4} b^{3} \tan \left (x\right ) - 12 \, a^{2} b^{5} \tan \left (x\right ) + a^{7} + 9 \, a^{5} b^{2} - 4 \, a^{3} b^{4}}{2 \, {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (x\right ) + a\right )}^{2}} \]

[In]

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

[Out]

-(3*a^2*b - b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 1/2*(a^3 - 3*a*b^2)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6) + (a^3*b - 3*a*b^3)*log(abs(b*tan(x) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - 1/2*(3
*a^3*b^4*tan(x)^2 - 9*a*b^6*tan(x)^2 + 2*a^6*b*tan(x) + 14*a^4*b^3*tan(x) - 12*a^2*b^5*tan(x) + a^7 + 9*a^5*b^
2 - 4*a^3*b^4)/((a^6*b^2 + 3*a^4*b^4 + 3*a^2*b^6 + b^8)*(b*tan(x) + a)^2)

Mupad [B] (verification not implemented)

Time = 30.10 (sec) , antiderivative size = 5324, normalized size of antiderivative = 54.33 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((2*tan(x/2)^2*(5*a*b^2 + a^3))/(a^4 + b^4 + 2*a^2*b^2) - (4*a^2*b*tan(x/2)^3)/(a^4 + b^4 + 2*a^2*b^2) + (4*a^
2*b*tan(x/2))/(a^4 + b^4 + 2*a^2*b^2))/(a^2 - tan(x/2)^2*(2*a^2 - 4*b^2) + a^2*tan(x/2)^4 + 4*a*b*tan(x/2) - 4
*a*b*tan(x/2)^3) - (log(a + 2*b*tan(x/2) - a*tan(x/2)^2)*(3*a*b^2 - a^3))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)
+ (log(1/(cos(x) + 1))*(6*a*b^2 - 2*a^3))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (2*b*atan((tan(x/2)*(((((6
*a*b^2 - 2*a^3)*((b*((32*(a^2*b^12 - 2*a^14 + 15*a^4*b^10 + 48*a^6*b^8 + 62*a^8*b^6 + 33*a^10*b^4 + 3*a^12*b^2
))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - (16*(6*a*b^2 - 2*a^3)*(3*a
*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^
6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2
)))*(3*a^2 - b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*b*(6*a*b^2 - 2*a^3)*(3*a^2 - b^2)*(3*a*b^16 + 21*
a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3
*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(
a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (b*((32*(2*a*b^10 - 24*a^3*b^8 - 36*a^5*b^6 + 8*a^7*b^4 + 18*a^9*b^2))/(
a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - ((6*a*b^2 - 2*a^3)*((32*(a^2*b
^12 - 2*a^14 + 15*a^4*b^10 + 48*a^6*b^8 + 62*a^8*b^6 + 33*a^10*b^4 + 3*a^12*b^2))/(a^12 + b^12 + 6*a^2*b^10 +
15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - (16*(6*a*b^2 - 2*a^3)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^
12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2
)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4
+ 3*a^4*b^2)))*(3*a^2 - b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*b^3*(3*a^2 - b^2)^3*(3*a*b^16 + 21*a^3
*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^
2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(4*a^8 +
b^8 - 67*a^2*b^6 + 155*a^4*b^4 - 61*a^6*b^2))/(4*a^8 + b^8 + 31*a^2*b^6 + 15*a^4*b^4 - 11*a^6*b^2)^2 + (2*a*b
*(10*a^6 - 7*b^6 + 68*a^2*b^4 - 59*a^4*b^2)*((32*(2*a^8 - 6*a^2*b^6 + 2*a^4*b^4 - 6*a^6*b^2))/(a^12 + b^12 + 6
*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - ((6*a*b^2 - 2*a^3)*((32*(2*a*b^10 - 24*a^3*b^
8 - 36*a^5*b^6 + 8*a^7*b^4 + 18*a^9*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6
*a^10*b^2) - ((6*a*b^2 - 2*a^3)*((32*(a^2*b^12 - 2*a^14 + 15*a^4*b^10 + 48*a^6*b^8 + 62*a^8*b^6 + 33*a^10*b^4
+ 3*a^12*b^2))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - (16*(6*a*b^2 -
2*a^3)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^1
5*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4
+ 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (b*((b*((
32*(a^2*b^12 - 2*a^14 + 15*a^4*b^10 + 48*a^6*b^8 + 62*a^8*b^6 + 33*a^10*b^4 + 3*a^12*b^2))/(a^12 + b^12 + 6*a^
2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - (16*(6*a*b^2 - 2*a^3)*(3*a*b^16 + 21*a^3*b^14 +
63*a^5*b^12 + 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 +
3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^2 - b^2))/(a^
6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*b*(6*a*b^2 - 2*a^3)*(3*a^2 - b^2)*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12
+ 105*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^
2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^2 - b^2))/(a^6 + b^6 +
3*a^2*b^4 + 3*a^4*b^2) + (16*b^2*(6*a*b^2 - 2*a^3)*(3*a^2 - b^2)^2*(3*a*b^16 + 21*a^3*b^14 + 63*a^5*b^12 + 10
5*a^7*b^10 + 105*a^9*b^8 + 63*a^11*b^6 + 21*a^13*b^4 + 3*a^15*b^2))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^
12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(4*a^8 + b^8 + 31*a^2*b^6 + 15*
a^4*b^4 - 11*a^6*b^2)^2)*(a^16 + b^16 + 8*a^2*b^14 + 28*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 56*a^10*b^6 + 28
*a^12*b^4 + 8*a^14*b^2))/(32*a*b^3 - 96*a^3*b) + (((b*((32*(5*a^2*b^9 - 3*a^10*b + 12*a^4*b^7 + 6*a^6*b^5 - 4*
a^8*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) - ((6*a*b^2 - 2*a^3)*
((32*(3*a^3*b^11 - 4*a^13*b - a*b^13 + 18*a^5*b^9 + 22*a^7*b^7 + 3*a^9*b^5 - 9*a^11*b^3))/(a^12 + b^12 + 6*a^2
*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a*b^2 - 2*a^3)*(3*a^16*b + 3*a^2*b^15 + 21
*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3
*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*
a^2*b^4 + 3*a^4*b^2)))*(3*a^2 - b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - ((6*a*b^2 - 2*a^3)*((b*((32*(3*a^3
*b^11 - 4*a^13*b - a*b^13 + 18*a^5*b^9 + 22*a^7*b^7 + 3*a^9*b^5 - 9*a^11*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*
a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a*b^2 - 2*a^3)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 +
63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(
a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^2 - b^2))/(a^6 + b^6 + 3*
a^2*b^4 + 3*a^4*b^2) + (16*b*(6*a*b^2 - 2*a^3)*(3*a^2 - b^2)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^1
1 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^2*(a^12 + b^
12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)
) + (32*b^3*(3*a^2 - b^2)^3*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 +
63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20
*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(4*a^8 + b^8 - 67*a^2*b^6 + 155*a^4*b^4 - 61*a^6*b^2)*(a^16 + b^16 + 8*a
^2*b^14 + 28*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 56*a^10*b^6 + 28*a^12*b^4 + 8*a^14*b^2))/((32*a*b^3 - 96*a^
3*b)*(4*a^8 + b^8 + 31*a^2*b^6 + 15*a^4*b^4 - 11*a^6*b^2)^2) + (2*a*b*((32*(2*a^7*b + 6*a^3*b^5 - 8*a^5*b^3))/
(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + ((6*a*b^2 - 2*a^3)*((32*(5*a^
2*b^9 - 3*a^10*b + 12*a^4*b^7 + 6*a^6*b^5 - 4*a^8*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 +
15*a^8*b^4 + 6*a^10*b^2) - ((6*a*b^2 - 2*a^3)*((32*(3*a^3*b^11 - 4*a^13*b - a*b^13 + 18*a^5*b^9 + 22*a^7*b^7 +
3*a^9*b^5 - 9*a^11*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16
*(6*a*b^2 - 2*a^3)*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b
^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 +
15*a^8*b^4 + 6*a^10*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2))
+ (b*((b*((32*(3*a^3*b^11 - 4*a^13*b - a*b^13 + 18*a^5*b^9 + 22*a^7*b^7 + 3*a^9*b^5 - 9*a^11*b^3))/(a^12 + b^1
2 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (16*(6*a*b^2 - 2*a^3)*(3*a^16*b + 3*a^2*
b^15 + 21*a^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^
2*b^4 + 3*a^4*b^2)*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^2 - b
^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*b*(6*a*b^2 - 2*a^3)*(3*a^2 - b^2)*(3*a^16*b + 3*a^2*b^15 + 21*a
^4*b^13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a
^4*b^2)^2*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(3*a^2 - b^2))/(a^6
+ b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*b^2*(6*a*b^2 - 2*a^3)*(3*a^2 - b^2)^2*(3*a^16*b + 3*a^2*b^15 + 21*a^4*b^
13 + 63*a^6*b^11 + 105*a^8*b^9 + 105*a^10*b^7 + 63*a^12*b^5 + 21*a^14*b^3))/((a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^
2)^3*(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2)))*(10*a^6 - 7*b^6 + 68*a^2
*b^4 - 59*a^4*b^2)*(a^16 + b^16 + 8*a^2*b^14 + 28*a^4*b^12 + 56*a^6*b^10 + 70*a^8*b^8 + 56*a^10*b^6 + 28*a^12*
b^4 + 8*a^14*b^2))/((32*a*b^3 - 96*a^3*b)*(4*a^8 + b^8 + 31*a^2*b^6 + 15*a^4*b^4 - 11*a^6*b^2)^2))*(3*a^2 - b^
2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)