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)*ln(a*cos(x)+b*sin(x))/(a^2+b^2)^3+a*b*cos(x)*sin(x)/(a^2+b^2)^2
+1/2*(a^2-b^2)*sin(x)^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 = 0.41, 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 8

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

Rule 30

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

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 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 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 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 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 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 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 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 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 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]

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]

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*}

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Mathematica [C]  time = 1.33, size = 221, normalized size = 1.73 \[ \frac {-4 i b^2 \left (b^2-3 a^2\right ) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))-a \cos (x) \left (\left (a^4-b^4\right ) \cos (2 x)+2 b \left (-a \left (a^2+b^2\right ) \sin (2 x)-b \left (b^2-3 a^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+2 x (a+i b)^3\right )\right )+b \sin (x) \left (\left (b^4-a^4\right ) \cos (2 x)+2 b \left (\left (b^3-3 a^2 b\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )-2 (a+i b) \left (a^2 x+a (b+2 i b x)-b^2 (x+i)\right )+a \left (a^2+b^2\right ) \sin (2 x)\right )\right )}{4 \left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]

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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fricas [A]  time = 0.65, size = 252, normalized size = 1.97 \[ -\frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \relax (x)^{3} - {\left (a^{5} + 4 \, a^{3} b^{2} + 7 \, a b^{4} - 4 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} x\right )} \cos \relax (x) + 2 \, {\left ({\left (3 \, a^{3} b^{2} - a b^{4}\right )} \cos \relax (x) + {\left (3 \, a^{2} b^{3} - b^{5}\right )} \sin \relax (x)\right )} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) - {\left (a^{4} b - 4 \, a^{2} b^{3} - b^{5} + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \relax (x)^{2} - 4 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} x\right )} \sin \relax (x)}{4 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \relax (x) + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \relax (x)\right )}} \]

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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giac [A]  time = 0.19, size = 214, normalized size = 1.67 \[ -\frac {{\left (a^{3} b - 3 \, a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (3 \, a^{2} b^{3} - b^{5}\right )} \log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} + \frac {4 \, a b^{2} \tan \relax (x)^{2} + a^{2} b \tan \relax (x) + b^{3} \tan \relax (x) - a^{3} + 3 \, a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \relax (x)^{3} + a \tan \relax (x)^{2} + b \tan \relax (x) + a\right )}} \]

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))

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maple [A]  time = 0.12, size = 241, normalized size = 1.88 \[ \frac {\tan \relax (x ) a^{3} b}{\left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}+\frac {\tan \relax (x ) a \,b^{3}}{\left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}-\frac {a^{4}}{2 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}+\frac {b^{4}}{2 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}+\frac {3 \ln \left (\tan ^{2}\relax (x )+1\right ) a^{2} b^{2}}{2 \left (a^{2}+b^{2}\right )^{3}}-\frac {\ln \left (\tan ^{2}\relax (x )+1\right ) b^{4}}{2 \left (a^{2}+b^{2}\right )^{3}}-\frac {\arctan \left (\tan \relax (x )\right ) a^{3} b}{\left (a^{2}+b^{2}\right )^{3}}+\frac {3 a \arctan \left (\tan \relax (x )\right ) b^{3}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a \,b^{2}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \relax (x )\right )}-\frac {3 a^{2} \ln \left (a +b \tan \relax (x )\right ) b^{2}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {b^{4} \ln \left (a +b \tan \relax (x )\right )}{\left (a^{2}+b^{2}\right )^{3}} \]

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]

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+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))

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maxima [B]  time = 0.46, size = 256, normalized size = 2.00 \[ -\frac {{\left (a^{3} b - 3 \, a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (b \tan \relax (x) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {4 \, a b^{2} \tan \relax (x)^{2} - a^{3} + 3 \, a b^{2} + {\left (a^{2} b + b^{3}\right )} \tan \relax (x)}{2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \relax (x)^{3} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \relax (x)^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \relax (x)\right )}} \]

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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mupad [B]  time = 8.16, size = 5428, normalized size = 42.41 \[ \text {result too large to display} \]

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]

((2*a*tan(x/2)^2)/(a^2 + b^2) - (8*b^3*tan(x/2)^3)/(a^2 + b^2)^2 - (2*a*tan(x/2)^4)/(a^2 + b^2) + (2*b*tan(x/2
)*(a^2 - b^2))/(a^2 + b^2)^2 + (2*b*tan(x/2)^5*(a^2 - b^2))/(a^4 + b^4 + 2*a^2*b^2))/(a + 2*b*tan(x/2) + a*tan
(x/2)^2 - a*tan(x/2)^4 - a*tan(x/2)^6 + 4*b*tan(x/2)^3 + 2*b*tan(x/2)^5) + (log(a + 2*b*tan(x/2) - a*tan(x/2)^
2)*(b^4 - 3*a^2*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (log(1/(cos(x) + 1))*(2*b^4 - 6*a^2*b^2))/(2*(a^6
+ b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (2*a*b*atan((tan(x/2)*((((((a*b*((32*(a*b^14 + 9*a^3*b^12 + 18*a^5*b^10 + 2*
a^7*b^8 - 27*a^9*b^6 - 27*a^11*b^4 - 8*a^13*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*(2*b^4 - 6*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)*(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)))*(a^2 - 3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*
a*b*(a^2 - 3*b^2)*(2*b^4 - 6*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*b^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) +
(a*b*(a^2 - 3*b^2)*((32*(3*a*b^12 - 21*a^3*b^10 - 34*a^5*b^8 + 6*a^7*b^6 + 15*a^9*b^4 - a^11*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) + (((32*(a*b^14 + 9*a^3*b^12 + 18*a^5*b^1
0 + 2*a^7*b^8 - 27*a^9*b^6 - 27*a^11*b^4 - 8*a^13*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*(2*b^4 - 6*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)*(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*b^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*
a^4*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a^3*b^3*(a^2 - 3*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)))*(a^8 + 4*b^8 - 61*a^2*
b^6 + 155*a^4*b^4 - 67*a^6*b^2))/(a^8 + 4*b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*a^6*b^2)^2 + (2*a*b*(7*a^6 - 10*b
^6 + 59*a^2*b^4 - 68*a^4*b^2)*((32*(a*b^10 + 3*a^3*b^8 - 17*a^5*b^6 - 3*a^7*b^4))/(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*b^4 - 6*a^2*b^2)*((32*(3*a*b^12 - 21*a^3*b^10 - 34*a^
5*b^8 + 6*a^7*b^6 + 15*a^9*b^4 - a^11*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) + (((32*(a*b^14 + 9*a^3*b^12 + 18*a^5*b^10 + 2*a^7*b^8 - 27*a^9*b^6 - 27*a^11*b^4 - 8*a^13*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*(2*b^4 - 6*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)*(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*b^4 - 6*a^2*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)) + (a
*b*(a^2 - 3*b^2)*((a*b*((32*(a*b^14 + 9*a^3*b^12 + 18*a^5*b^10 + 2*a^7*b^8 - 27*a^9*b^6 - 27*a^11*b^4 - 8*a^13
*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*(2*b^4 - 6*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)*(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)))*(a^2 - 3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*a*b*(a^2 - 3*b^2)*(2*b^4 - 6*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))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*a^2*b^2*(a^2 - 3*b^2)^2*(2*b^4 - 6*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)^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))))/(a^8 +
4*b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*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))/(96*a^2*b^5 - 32*a^4*b^3) + (((((a*b*((32*(3*a^6*b^9 - 4*a^2*
b^13 - 9*a^4*b^11 - a^14*b + 22*a^8*b^7 + 18*a^10*b^5 + 3*a^12*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*(2*b^4 - 6*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)*(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)))*(a^2 - 3*b^2))/(a^6 + b^6 + 3*a^2*b^4
+ 3*a^4*b^2) - (16*a*b*(a^2 - 3*b^2)*(2*b^4 - 6*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)))*(2*b^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*
b^4 + 3*a^4*b^2)) + (a*b*(a^2 - 3*b^2)*((32*(6*a^6*b^7 - 4*a^4*b^9 - 3*a^2*b^11 + 12*a^8*b^5 + 5*a^10*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) + (((32*(3*a^6*b^9 - 4*a^2*b^13 -
9*a^4*b^11 - a^14*b + 22*a^8*b^7 + 18*a^10*b^5 + 3*a^12*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*(2*b^4 - 6*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)*(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*b^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b
^4 + 3*a^4*b^2))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (32*a^3*b^3*(a^2 - 3*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)))*(a^8 + 4*b^8 -
61*a^2*b^6 + 155*a^4*b^4 - 67*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))/((96*a^2*b^5 - 32*a^4*b^3)*(a^8 + 4*b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*
a^6*b^2)^2) + (2*a*b*(7*a^6 - 10*b^6 + 59*a^2*b^4 - 68*a^4*b^2)*((32*(2*a^2*b^9 - 8*a^4*b^7 + 6*a^6*b^5))/(a^1
2 + 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*b^4 - 6*a^2*b^2)*((32*(6*a^6*
b^7 - 4*a^4*b^9 - 3*a^2*b^11 + 12*a^8*b^5 + 5*a^10*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) + (((32*(3*a^6*b^9 - 4*a^2*b^13 - 9*a^4*b^11 - a^14*b + 22*a^8*b^7 + 18*a^10*b^5 + 3
*a^12*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*(2*b^4 - 6*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)*(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*b^4 - 6*a^2*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)) + (a*b*(a^2 - 3*b^2)*((a*b*((32*(3*a^6*b^9 - 4*a^2*b^13 - 9*a^4*b^11 - a^14*b + 22*a^8*b^7 + 18*a^10*b
^5 + 3*a^12*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*(2*b^4
- 6*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 + 2
1*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)))*(a^2 - 3*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*a*b*(a^2 - 3*b^2)*(2*b^4 - 6*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))))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*a^2*b^2*(a^2 - 3*b^2)^2*(2*b^4 - 6*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)^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)
))*(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
))/((96*a^2*b^5 - 32*a^4*b^3)*(a^8 + 4*b^8 - 11*a^2*b^6 + 15*a^4*b^4 + 31*a^6*b^2)^2))*(a^2 - 3*b^2))/(a^6 + b
^6 + 3*a^2*b^4 + 3*a^4*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(cos(x)**3*sin(x)/(a*cos(x)+b*sin(x))**2,x)

[Out]

Timed out

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