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

Optimal. Leaf size=128 \[ -\frac{a b x \left (a^2-3 b^2\right )}{\left (a^2+b^2\right )^3}+\frac{\left (a^2-b^2\right ) \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{a b^2 \cos (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}+\frac{a b \sin (x) \cos (x)}{\left (a^2+b^2\right )^2}-\frac{b^2 \left (3 a^2-b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.407634, antiderivative size = 196, normalized size of antiderivative = 1.53, number of steps used = 17, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {3111, 3100, 2635, 8, 3098, 3133, 3109, 2564, 30, 3086, 3483, 3531, 3530} \[ \frac{a b^3 x}{\left (a^2+b^2\right )^3}+\frac{a b x}{\left (a^2+b^2\right )^2}-\frac{a^3 b x}{\left (a^2+b^2\right )^3}-\frac{a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3}+\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{b^2 \cos ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{a b^2}{\left (a^2+b^2\right )^2 (a+b \tan (x))}+\frac{a b \sin (x) \cos (x)}{\left (a^2+b^2\right )^2}+\frac{b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}-\frac{3 a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3111

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

Rule 3100

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 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 8

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

Rule 3098

Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(a*x)/(a^2 + b^2), x] + Dist[b/(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 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]

Rule 3109

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

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3086

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

Rule 3483

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 3531

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]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), 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]

Rubi steps

\begin{align*} \int \frac{\cos ^3(x) \sin (x)}{(a \cos (x)+b \sin (x))^2} \, dx &=\frac{a \int \frac{\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}+\frac{b \int \frac{\cos ^3(x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}-\frac{(a b) \int \frac{\cos ^2(x)}{(a \cos (x)+b \sin (x))^2} \, dx}{a^2+b^2}\\ &=\frac{b^2 \cos ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{a^2 \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}+2 \frac{(a b) \int \cos ^2(x) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (a^2 b\right ) \int \frac{\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac{b^3 \int \frac{\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}-\frac{(a b) \int \frac{1}{(a+b \tan (x))^2} \, dx}{a^2+b^2}\\ &=-\frac{a^3 b x}{\left (a^2+b^2\right )^3}+\frac{a b^3 x}{\left (a^2+b^2\right )^3}+\frac{b^2 \cos ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac{a b^2}{\left (a^2+b^2\right )^2 (a+b \tan (x))}-\frac{\left (a^2 b^2\right ) \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}+\frac{b^4 \int \frac{b \cos (x)-a \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^3}+\frac{a^2 \operatorname{Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^2}+2 \left (\frac{a b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac{(a b) \int 1 \, dx}{2 \left (a^2+b^2\right )^2}\right )-\frac{(a b) \int \frac{a-b \tan (x)}{a+b \tan (x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=-\frac{a^3 b x}{\left (a^2+b^2\right )^3}+\frac{a b^3 x}{\left (a^2+b^2\right )^3}-\frac{a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac{b^2 \cos ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac{a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+2 \left (\frac{a b x}{2 \left (a^2+b^2\right )^2}+\frac{a b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}\right )+\frac{a b^2}{\left (a^2+b^2\right )^2 (a+b \tan (x))}-\frac{\left (2 a^2 b^2\right ) \int \frac{b-a \tan (x)}{a+b \tan (x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=-\frac{a^3 b x}{\left (a^2+b^2\right )^3}+\frac{a b^3 x}{\left (a^2+b^2\right )^3}-\frac{a b \left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^3}+\frac{b^2 \cos ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac{3 a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{b^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac{a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+2 \left (\frac{a b x}{2 \left (a^2+b^2\right )^2}+\frac{a b \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}\right )+\frac{a b^2}{\left (a^2+b^2\right )^2 (a+b \tan (x))}\\ \end{align*}

Mathematica [C]  time = 1.28507, size = 221, normalized size = 1.73 \[ \frac{b \sin (x) \left (\left (b^4-a^4\right ) \cos (2 x)+2 b \left (-2 (a+i b) \left (a^2 x+a (b+2 i b x)-b^2 (x+i)\right )+a \left (a^2+b^2\right ) \sin (2 x)+\left (b^3-3 a^2 b\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )\right )\right )-a \cos (x) \left (\left (a^4-b^4\right ) \cos (2 x)+2 b \left (-a \left (a^2+b^2\right ) \sin (2 x)-b \left (b^2-3 a^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+2 x (a+i b)^3\right )\right )-4 i b^2 \left (b^2-3 a^2\right ) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))}{4 \left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.098, size = 241, normalized size = 1.9 \begin{align*}{\frac{a{b}^{2}}{ \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( x \right ) \right ) }}-3\,{\frac{{a}^{2}\ln \left ( a+b\tan \left ( x \right ) \right ){b}^{2}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{{b}^{4}\ln \left ( a+b\tan \left ( x \right ) \right ) }{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\tan \left ( x \right ){a}^{3}b}{ \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{a\tan \left ( x \right ){b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}-{\frac{{a}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{{b}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3} \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ) }}+{\frac{3\,\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ){a}^{2}{b}^{2}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\ln \left ( \left ( \tan \left ( x \right ) \right ) ^{2}+1 \right ){b}^{4}}{2\, \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\arctan \left ( \tan \left ( x \right ) \right ){a}^{3}b}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{a\arctan \left ( \tan \left ( x \right ) \right ){b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*b^2/(a^2+b^2)^2/(a+b*tan(x))-3*a^2/(a^2+b^2)^3*ln(a+b*tan(x))*b^2+b^4/(a^2+b^2)^3*ln(a+b*tan(x))+1/(a^2+b^2)
^3/(tan(x)^2+1)*tan(x)*a^3*b+1/(a^2+b^2)^3/(tan(x)^2+1)*tan(x)*a*b^3-1/2/(a^2+b^2)^3/(tan(x)^2+1)*a^4+1/2/(a^2
+b^2)^3/(tan(x)^2+1)*b^4+3/2/(a^2+b^2)^3*ln(tan(x)^2+1)*a^2*b^2-1/2/(a^2+b^2)^3*ln(tan(x)^2+1)*b^4-1/(a^2+b^2)
^3*arctan(tan(x))*a^3*b+3/(a^2+b^2)^3*a*arctan(tan(x))*b^3

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Maxima [B]  time = 1.70046, size = 346, normalized size = 2.7 \begin{align*} -\frac{{\left (a^{3} b - 3 \, a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (b \tan \left (x\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left (3 \, a^{2} b^{2} - b^{4}\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{4 \, a b^{2} \tan \left (x\right )^{2} - a^{3} + 3 \, a b^{2} +{\left (a^{2} b + b^{3}\right )} \tan \left (x\right )}{2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )^{3} +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \left (x\right )^{2} +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^3*b - 3*a*b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (3*a^2*b^2 - b^4)*log(b*tan(x) + a)/(a^6 + 3*a^4*b^
2 + 3*a^2*b^4 + b^6) + 1/2*(3*a^2*b^2 - b^4)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(4*a*
b^2*tan(x)^2 - a^3 + 3*a*b^2 + (a^2*b + b^3)*tan(x))/(a^5 + 2*a^3*b^2 + a*b^4 + (a^4*b + 2*a^2*b^3 + b^5)*tan(
x)^3 + (a^5 + 2*a^3*b^2 + a*b^4)*tan(x)^2 + (a^4*b + 2*a^2*b^3 + b^5)*tan(x))

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Fricas [A]  time = 0.559767, size = 562, normalized size = 4.39 \begin{align*} -\frac{2 \,{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (x\right )^{3} -{\left (a^{5} + 4 \, a^{3} b^{2} + 7 \, a b^{4} - 4 \,{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} x\right )} \cos \left (x\right ) + 2 \,{\left ({\left (3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (x\right ) +{\left (3 \, a^{2} b^{3} - b^{5}\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 ) -{\left (a^{4} b - 4 \, a^{2} b^{3} - b^{5} + 2 \,{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (x\right )^{2} - 4 \,{\left (a^{3} b^{2} - 3 \, a b^{4}\right )} x\right )} \sin \left (x\right )}{4 \,{\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.13159, size = 289, normalized size = 2.26 \begin{align*} -\frac{{\left (a^{3} b - 3 \, a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left (3 \, a^{2} b^{2} - b^{4}\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 (3 \, a^{2} b^{3} - b^{5}\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{4 \, a b^{2} \tan \left (x\right )^{2} + a^{2} b \tan \left (x\right ) + b^{3} \tan \left (x\right ) - a^{3} + 3 \, a b^{2}}{2 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}{\left (b \tan \left (x\right )^{3} + a \tan \left (x\right )^{2} + b \tan \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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