\(\int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx\) [15]

Optimal result

Integrand size = 16, antiderivative size = 107 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {6 a^2 b \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {3 a \left (a^2-b^2\right )+a \left (a^2+b^2\right ) \cos (2 x)-b \left (a^2+b^2\right ) \sin (2 x)}{2 \left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]

[Out]

6*a^2*b*arctanh((-b+a*tan(1/2*x))/(a^2+b^2)^(1/2))/(a^2+b^2)^(5/2)+1/2*(3*a*(a^2-b^2)+a*(a^2+b^2)*cos(2*x)-b*(
a^2+b^2)*sin(2*x))/(a^2+b^2)^2/(a*cos(x)+b*sin(x))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(283\) vs. \(2(107)=214\).

Time = 1.53 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.64, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {4486, 2717, 2718, 6874, 653, 209, 652, 632, 212, 3179, 3153} \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {2 a^2 \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{5/2}}-\frac {2 a^2 b \text {arctanh}\left (\frac {b-a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {3 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}+\frac {3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}+\frac {2 a^2 \left (a+b \tan \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (-a \tan ^2\left (\frac {x}{2}\right )+a+2 b \tan \left (\frac {x}{2}\right )\right )}+\frac {3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac {2 a^3 \cos ^2\left (\frac {x}{2}\right ) \left (\left (a^2-b^2\right ) \tan \left (\frac {x}{2}\right )+2 a b\right )}{b^3 \left (a^2+b^2\right )^2}-\frac {2 a \sin (x)}{b^3}-\frac {\cos (x)}{b^2} \]

[In]

Int[Sin[x]^3/(a*Cos[x] + b*Sin[x])^2,x]

[Out]

(-3*a^2*ArcTanh[(b*Cos[x] - a*Sin[x])/Sqrt[a^2 + b^2]])/(b*(a^2 + b^2)^(3/2)) - (2*a^2*b*ArcTanh[(b - a*Tan[x/
2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (2*a^2*(3*a^2 + b^2)*ArcTanh[(b - a*Tan[x/2])/Sqrt[a^2 + b^2]])/(b*(
a^2 + b^2)^(5/2)) - Cos[x]/b^2 + (3*a^2*Cos[x])/(b^2*(a^2 + b^2)) - (2*a*Sin[x])/b^3 + (3*a^3*Sin[x])/(b^3*(a^
2 + b^2)) - (2*a^3*Cos[x/2]^2*(2*a*b + (a^2 - b^2)*Tan[x/2]))/(b^3*(a^2 + b^2)^2) + (2*a^2*(a + b*Tan[x/2]))/(
(a^2 + b^2)^2*(a + 2*b*Tan[x/2] - a*Tan[x/2]^2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 652

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

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3153

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

Rule 3179

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

Rule 4486

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 a \cos (x)}{b^3}+\frac {\sin (x)}{b^2}-\frac {a^3 \cos ^3(x)}{b^3 (a \cos (x)+b \sin (x))^2}+\frac {3 a^2 \cos ^2(x)}{b^3 (a \cos (x)+b \sin (x))}\right ) \, dx \\ & = -\frac {(2 a) \int \cos (x) \, dx}{b^3}+\frac {\left (3 a^2\right ) \int \frac {\cos ^2(x)}{a \cos (x)+b \sin (x)} \, dx}{b^3}-\frac {a^3 \int \frac {\cos ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx}{b^3}+\frac {\int \sin (x) \, dx}{b^2} \\ & = -\frac {\cos (x)}{b^2}+\frac {3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac {2 a \sin (x)}{b^3}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \frac {\left (1-x^2\right )^3}{\left (1+x^2\right )^2 \left (a+2 b x-a x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^3}+\frac {\left (3 a^3\right ) \int \cos (x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac {\left (3 a^2\right ) \int \frac {1}{a \cos (x)+b \sin (x)} \, dx}{b \left (a^2+b^2\right )} \\ & = -\frac {\cos (x)}{b^2}+\frac {3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac {2 a \sin (x)}{b^3}+\frac {3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac {\left (2 a^3\right ) \text {Subst}\left (\int \left (\frac {2 \left (a^2-b^2-2 a b x\right )}{\left (a^2+b^2\right )^2 \left (1+x^2\right )^2}+\frac {-a^2+b^2}{\left (a^2+b^2\right )^2 \left (1+x^2\right )}-\frac {2 b^3 x}{a \left (a^2+b^2\right ) \left (-a-2 b x+a x^2\right )^2}-\frac {b^2 \left (3 a^2+b^2\right )}{a \left (a^2+b^2\right )^2 \left (-a-2 b x+a x^2\right )}\right ) \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^3}-\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (x)-a \sin (x)\right )}{b \left (a^2+b^2\right )} \\ & = -\frac {3 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}-\frac {\cos (x)}{b^2}+\frac {3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac {2 a \sin (x)}{b^3}+\frac {3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac {\left (4 a^3\right ) \text {Subst}\left (\int \frac {a^2-b^2-2 a b x}{\left (1+x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac {\left (2 a^3 \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac {\left (4 a^2\right ) \text {Subst}\left (\int \frac {x}{\left (-a-2 b x+a x^2\right )^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{a^2+b^2}+\frac {\left (2 a^2 \left (3 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-a-2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b \left (a^2+b^2\right )^2} \\ & = \frac {a^3 \left (a^2-b^2\right ) x}{b^3 \left (a^2+b^2\right )^2}-\frac {3 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}-\frac {\cos (x)}{b^2}+\frac {3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac {2 a \sin (x)}{b^3}+\frac {3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac {2 a^3 \cos ^2\left (\frac {x}{2}\right ) \left (2 a b+\left (a^2-b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac {2 a^2 \left (a+b \tan \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )}-\frac {\left (2 a^2 b\right ) \text {Subst}\left (\int \frac {1}{-a-2 b x+a x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2}-\frac {\left (2 a^3 \left (a^2-b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}-\frac {\left (4 a^2 \left (3 a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{b \left (a^2+b^2\right )^2} \\ & = -\frac {3 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}+\frac {2 a^2 \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{5/2}}-\frac {\cos (x)}{b^2}+\frac {3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac {2 a \sin (x)}{b^3}+\frac {3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac {2 a^3 \cos ^2\left (\frac {x}{2}\right ) \left (2 a b+\left (a^2-b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac {2 a^2 \left (a+b \tan \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )}+\frac {\left (4 a^2 b\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 b+2 a \tan \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2} \\ & = -\frac {3 a^2 \text {arctanh}\left (\frac {b \cos (x)-a \sin (x)}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{3/2}}-\frac {2 a^2 b \text {arctanh}\left (\frac {b-a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {2 a^2 \left (3 a^2+b^2\right ) \text {arctanh}\left (\frac {b-a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right )^{5/2}}-\frac {\cos (x)}{b^2}+\frac {3 a^2 \cos (x)}{b^2 \left (a^2+b^2\right )}-\frac {2 a \sin (x)}{b^3}+\frac {3 a^3 \sin (x)}{b^3 \left (a^2+b^2\right )}-\frac {2 a^3 \cos ^2\left (\frac {x}{2}\right ) \left (2 a b+\left (a^2-b^2\right ) \tan \left (\frac {x}{2}\right )\right )}{b^3 \left (a^2+b^2\right )^2}+\frac {2 a^2 \left (a+b \tan \left (\frac {x}{2}\right )\right )}{\left (a^2+b^2\right )^2 \left (a+2 b \tan \left (\frac {x}{2}\right )-a \tan ^2\left (\frac {x}{2}\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {6 a^2 b \text {arctanh}\left (\frac {-b+a \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}+\frac {3 a \left (a^2-b^2\right )+a \left (a^2+b^2\right ) \cos (2 x)-b \left (a^2+b^2\right ) \sin (2 x)}{2 \left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))} \]

[In]

Integrate[Sin[x]^3/(a*Cos[x] + b*Sin[x])^2,x]

[Out]

(6*a^2*b*ArcTanh[(-b + a*Tan[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) + (3*a*(a^2 - b^2) + a*(a^2 + b^2)*Cos[
2*x] - b*(a^2 + b^2)*Sin[2*x])/(2*(a^2 + b^2)^2*(a*Cos[x] + b*Sin[x]))

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.31

method	result	size
default	\(-\frac {4 \left (\tan \left (\frac {x}{2}\right ) a b -\frac {a^{2}}{2}+\frac {b^{2}}{2}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+\tan \left (\frac {x}{2}\right )^{2}\right )}+\frac {4 a^{2} \left (\frac {-\frac {b \tan \left (\frac {x}{2}\right )}{2}-\frac {a}{2}}{\tan \left (\frac {x}{2}\right )^{2} a -2 b \tan \left (\frac {x}{2}\right )-a}+\frac {3 b \,\operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\)	\(140\)
risch	\(\frac {{\mathrm e}^{i x}}{-4 i b a +2 a^{2}-2 b^{2}}+\frac {{\mathrm e}^{-i x}}{4 i b a +2 a^{2}-2 b^{2}}+\frac {2 a^{3} {\mathrm e}^{i x}}{\left (-i b \,{\mathrm e}^{2 i x}+a \,{\mathrm e}^{2 i x}+i b +a \right ) \left (i b +a \right )^{2} \left (-i b +a \right )^{2}}-\frac {3 i b \,a^{2} \ln \left ({\mathrm e}^{i x}+\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2}}+\frac {3 i b \,a^{2} \ln \left ({\mathrm e}^{i x}-\frac {i b +a}{\sqrt {-a^{2}-b^{2}}}\right )}{\sqrt {-a^{2}-b^{2}}\, \left (a^{2}+b^{2}\right )^{2}}\)	\(211\)

[In]

int(sin(x)^3/(a*cos(x)+b*sin(x))^2,x,method=_RETURNVERBOSE)

[Out]

-4/(a^4+2*a^2*b^2+b^4)*(tan(1/2*x)*a*b-1/2*a^2+1/2*b^2)/(1+tan(1/2*x)^2)+4*a^2/(a^4+2*a^2*b^2+b^4)*((-1/2*b*ta
n(1/2*x)-1/2*a)/(tan(1/2*x)^2*a-2*b*tan(1/2*x)-a)+3/2*b/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*x)-2*b)/(a^2+
b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (99) = 198\).

Time = 0.27 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.24 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\frac {2 \, a^{5} - 2 \, a^{3} b^{2} - 4 \, a b^{4} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{2} - 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right ) \sin \left (x\right ) + 3 \, {\left (a^{3} b \cos \left (x\right ) + a^{2} b^{2} \sin \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (-\frac {2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (x\right ) - a \sin \left (x\right )\right )}}{2 \, a b \cos \left (x\right ) \sin \left (x\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + b^{2}}\right )}{2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right ) + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \left (x\right )\right )}} \]

[In]

integrate(sin(x)^3/(a*cos(x)+b*sin(x))^2,x, algorithm="fricas")

[Out]

1/2*(2*a^5 - 2*a^3*b^2 - 4*a*b^4 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*cos(x)^2 - 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(x)*s
in(x) + 3*(a^3*b*cos(x) + a^2*b^2*sin(x))*sqrt(a^2 + b^2)*log(-(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 - 2
*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(x) - a*sin(x)))/(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2)))/((a
^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cos(x) + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sin(x))

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=\text {Timed out} \]

[In]

integrate(sin(x)**3/(a*cos(x)+b*sin(x))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (99) = 198\).

Time = 0.31 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.36 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {3 \, a^{2} b \log \left (\frac {b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (x\right )}{\cos \left (x\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (2 \, a^{3} - a b^{2} - \frac {3 \, a b^{2} \sin \left (x\right )^{2}}{{\left (\cos \left (x\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} b \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1}\right )}}{a^{5} + 2 \, a^{3} b^{2} + a b^{4} + \frac {2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )}{\cos \left (x\right ) + 1} + \frac {2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (x\right )^{3}}{{\left (\cos \left (x\right ) + 1\right )}^{3}} - \frac {{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (x\right )^{4}}{{\left (\cos \left (x\right ) + 1\right )}^{4}}} \]

[In]

integrate(sin(x)^3/(a*cos(x)+b*sin(x))^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.74 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {3 \, a^{2} b \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, x\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (3 \, a^{2} b \tan \left (\frac {1}{2} \, x\right )^{3} - 3 \, a b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{2} b \tan \left (\frac {1}{2} \, x\right ) - 2 \, b^{3} \tan \left (\frac {1}{2} \, x\right ) + 2 \, a^{3} - a b^{2}\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, b \tan \left (\frac {1}{2} \, x\right )^{3} - 2 \, b \tan \left (\frac {1}{2} \, x\right ) - a\right )} {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} \]

[In]

integrate(sin(x)^3/(a*cos(x)+b*sin(x))^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 21.87 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.09 \[ \int \frac {\sin ^3(x)}{(a \cos (x)+b \sin (x))^2} \, dx=-\frac {\frac {2\,\left (a\,b^2-2\,a^3\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^2\,b-2\,b^3\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {6\,a\,b^2\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{a^4+2\,a^2\,b^2+b^4}-\frac {6\,a^2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3}{a^4+2\,a^2\,b^2+b^4}}{-a\,{\mathrm {tan}\left (\frac {x}{2}\right )}^4+2\,b\,{\mathrm {tan}\left (\frac {x}{2}\right )}^3+2\,b\,\mathrm {tan}\left (\frac {x}{2}\right )+a}-\frac {6\,a^2\,b\,\mathrm {atanh}\left (\frac {2\,a^4\,b+2\,b^5+4\,a^2\,b^3-2\,a\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}{2\,{\left (a^2+b^2\right )}^{5/2}}\right )}{{\left (a^2+b^2\right )}^{5/2}} \]

[In]

int(sin(x)^3/(a*cos(x) + b*sin(x))^2,x)

[Out]

- ((2*(a*b^2 - 2*a^3))/(a^4 + b^4 + 2*a^2*b^2) - (2*tan(x/2)*(a^2*b - 2*b^3))/(a^4 + b^4 + 2*a^2*b^2) + (6*a*b
^2*tan(x/2)^2)/(a^4 + b^4 + 2*a^2*b^2) - (6*a^2*b*tan(x/2)^3)/(a^4 + b^4 + 2*a^2*b^2))/(a + 2*b*tan(x/2) - a*t
an(x/2)^4 + 2*b*tan(x/2)^3) - (6*a^2*b*atanh((2*a^4*b + 2*b^5 + 4*a^2*b^3 - 2*a*tan(x/2)*(a^4 + b^4 + 2*a^2*b^
2))/(2*(a^2 + b^2)^(5/2))))/(a^2 + b^2)^(5/2)