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

Optimal. Leaf size=129 \[ -\frac {\left (a^2-b^2\right ) \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {a^2 b \sin (x)}{\left (a^2+b^2\right )^2 (a \cos (x)+b \sin (x))}-\frac {a b \sin (x) \cos (x)}{\left (a^2+b^2\right )^2}-\frac {a^2 \left (a^2-3 b^2\right ) \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac {b x \left (3 a^3-a b^2\right )}{\left (a^2+b^2\right )^3} \]

[Out]

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

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Rubi [A]  time = 0.51, antiderivative size = 198, 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, 3109, 2564, 30, 2635, 8, 3097, 3133, 3099, 3085, 3483, 3531, 3530} \[ \frac {a^3 b x}{\left (a^2+b^2\right )^3}+\frac {a b x}{\left (a^2+b^2\right )^2}-\frac {a b^3 x}{\left (a^2+b^2\right )^3}+\frac {a b x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^3}-\frac {a^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}+\frac {b^2 \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {a^2 b}{\left (a^2+b^2\right )^2 (a \cot (x)+b)}-\frac {a b \sin (x) \cos (x)}{\left (a^2+b^2\right )^2}-\frac {a^4 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac {3 a^2 b^2 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[x]*Sin[x]^3)/(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
- (a^2*b)/((a^2 + b^2)^2*(b + a*Cot[x])) - (a^4*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 + (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*Sin[x]^2)/(2*(a^2 + b^2)^2) + (b^2*S
in[x]^2)/(2*(a^2 + b^2)^2)

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 3085

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

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Mathematica [C]  time = 1.48, size = 226, normalized size = 1.75 \[ \frac {4 i a^2 \left (a^2-3 b^2\right ) \tan ^{-1}(\tan (x)) (a \cos (x)+b \sin (x))+a \cos (x) \left (\left (a^4-b^4\right ) \cos (2 x)+2 a \left (-b \left (a^2+b^2\right ) \sin (2 x)-a \left (a^2-3 b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+2 x (-b+i a)^3\right )\right )-b \sin (x) \left (\left (b^4-a^4\right ) \cos (2 x)+2 a \left (b \left (a^2+b^2\right ) \sin (2 x)+a \left (a^2-3 b^2\right ) \log \left ((a \cos (x)+b \sin (x))^2\right )+2 \left (a^3 (1+i x)-3 a^2 b x+a b^2 (1-3 i x)+b^3 x\right )\right )\right )}{4 \left (a^2+b^2\right )^3 (a \cos (x)+b \sin (x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*I)*a^2*(a^2 - 3*b^2)*ArcTan[Tan[x]]*(a*Cos[x] + b*Sin[x]) + a*Cos[x]*((a^4 - b^4)*Cos[2*x] + 2*a*(2*(I*a -
b)^3*x - a*(a^2 - 3*b^2)*Log[(a*Cos[x] + b*Sin[x])^2] - b*(a^2 + b^2)*Sin[2*x])) - b*Sin[x]*((-a^4 + b^4)*Cos
[2*x] + 2*a*(2*(a^3*(1 + I*x) + a*b^2*(1 - (3*I)*x) - 3*a^2*b*x + b^3*x) + a*(a^2 - 3*b^2)*Log[(a*Cos[x] + b*S
in[x])^2] + b*(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 = 0.74, size = 236, normalized size = 1.83 \[ \frac {2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \relax (x)^{3} - {\left (a^{5} + 3 \, a b^{4} - 4 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} x\right )} \cos \relax (x) - 2 \, {\left ({\left (a^{5} - 3 \, a^{3} b^{2}\right )} \cos \relax (x) + {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \sin \relax (x)\right )} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) - {\left (5 \, a^{4} b - b^{5} + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \relax (x)^{2} - 4 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} x\right )} \sin \relax (x)}{4 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \relax (x) + {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \sin \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*sin(x)^3/(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 + 3*a*b^4 - 4*(3*a^4*b - a^2*b^3)*x)*cos(x) - 2*((a^5 - 3*a^3
*b^2)*cos(x) + (a^4*b - 3*a^2*b^3)*sin(x))*log(2*a*b*cos(x)*sin(x) + (a^2 - b^2)*cos(x)^2 + b^2) - (5*a^4*b -
b^5 + 2*(a^4*b + 2*a^2*b^3 + b^5)*cos(x)^2 - 4*(3*a^3*b^2 - 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 = 0.45, size = 223, normalized size = 1.73 \[ \frac {{\left (3 \, a^{3} b - a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} + \frac {2 \, a^{3} \tan \relax (x)^{2} - 2 \, a b^{2} \tan \relax (x)^{2} - a^{2} b \tan \relax (x) - b^{3} \tan \relax (x) + 3 \, a^{3} - a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \tan \relax (x)^{3} + a \tan \relax (x)^{2} + b \tan \relax (x) + a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a^3*b - a*b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(a^4 - 3*a^2*b^2)*log(tan(x)^2 + 1)/(a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6) - (a^4*b - 3*a^2*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*(2*a^3*tan(x)^2 - 2*a*b^2*tan(x)^2 - a^2*b*tan(x) - b^3*tan(x) + 3*a^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 = 0.11, size = 243, normalized size = 1.88 \[ -\frac {\tan \relax (x ) a^{3} b}{\left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}-\frac {\tan \relax (x ) a \,b^{3}}{\left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}+\frac {a^{4}}{2 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}-\frac {b^{4}}{2 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )}+\frac {\ln \left (\tan ^{2}\relax (x )+1\right ) a^{4}}{2 \left (a^{2}+b^{2}\right )^{3}}-\frac {3 \ln \left (\tan ^{2}\relax (x )+1\right ) a^{2} b^{2}}{2 \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \arctan \left (\tan \relax (x )\right ) a^{3} b}{\left (a^{2}+b^{2}\right )^{3}}-\frac {a \arctan \left (\tan \relax (x )\right ) b^{3}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {a^{3}}{\left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \relax (x )\right )}-\frac {a^{4} \ln \left (a +b \tan \relax (x )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {3 a^{2} \ln \left (a +b \tan \relax (x )\right ) b^{2}}{\left (a^{2}+b^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [B]  time = 0.46, size = 259, normalized size = 2.01 \[ \frac {{\left (3 \, a^{3} b - a b^{3}\right )} x}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (b \tan \relax (x) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {3 \, a^{3} - a b^{2} + 2 \, {\left (a^{3} - a b^{2}\right )} \tan \relax (x)^{2} - {\left (a^{2} b + b^{3}\right )} \tan \relax (x)}{2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \relax (x)^{3} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \tan \relax (x)^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \tan \relax (x)\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a^3*b - a*b^3)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^4 - 3*a^2*b^2)*log(b*tan(x) + a)/(a^6 + 3*a^4*b^2
+ 3*a^2*b^4 + b^6) + 1/2*(a^4 - 3*a^2*b^2)*log(tan(x)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(3*a^3
- a*b^2 + 2*(a^3 - a*b^2)*tan(x)^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 = 7.68, size = 5431, normalized size = 42.10 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(1/(cos(x) + 1))*(2*a^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (log(a + 2*b*tan(x/2) - a*
tan(x/2)^2)*(a^4 - 3*a^2*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - ((2*a*tan(x/2)^2)/(a^2 + b^2) - (2*a*tan(
x/2)^4)/(a^2 + b^2) + (4*a^2*b*tan(x/2))/(a^2 + b^2)^2 + (4*a^2*b*tan(x/2)^5)/(a^4 + b^4 + 2*a^2*b^2) + (4*b*t
an(x/2)^3*(a^2 - b^2))/(a^2 + b^2)^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) + (2*a*b*atan((((((a*b*((32*(3*a^4*b^11 - a^2*b^13 - 4*a^14*b + 18*a^6*b^9 + 22*a^8*
b^7 + 3*a^10*b^5 - 9*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*a^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 + 6
3*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*a*b*(2*a^4 - 6*a^2
*b^2)*(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)))*(2*a^4 - 6*a^2*b^2))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a*b*((32*(5*a^4*
b^9 - 3*a^12*b + 12*a^6*b^7 + 6*a^8*b^5 - 4*a^10*b^3))/(a^12 + b^12 + 6*a^2*b^10 + 15*a^4*b^8 + 20*a^6*b^6 + 1
5*a^8*b^4 + 6*a^10*b^2) + (((32*(3*a^4*b^11 - a^2*b^13 - 4*a^14*b + 18*a^6*b^9 + 22*a^8*b^7 + 3*a^10*b^5 - 9*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*a^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*a^4 - 6*a^2*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*a^3*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))/
((96*a^6*b - 32*a^4*b^3)*(4*a^8 + b^8 + 31*a^2*b^6 + 15*a^4*b^4 - 11*a^6*b^2)^2) - (tan(x/2)*(((((2*a^4 - 6*a^
2*b^2)*((a*b*(3*a^2 - b^2)*((32*(a^3*b^12 - 2*a^15 + 15*a^5*b^10 + 48*a^7*b^8 + 62*a^9*b^6 + 33*a^11*b^4 + 3*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*a^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^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) + (16*a*b*(2*a^4 - 6*a^2*b^2)*(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)) + (a*b*((32*(2*a^3*b^10 - 24*a^5*b^8 - 36*a^7*b^6 + 8*a^9*b^4 + 18*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) + ((2*a^4 - 6*a^2*b^2)*((32*
(a^3*b^12 - 2*a^15 + 15*a^5*b^10 + 48*a^7*b^8 + 62*a^9*b^6 + 33*a^11*b^4 + 3*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*a^4 - 6*a^2*b^2)*(3*a*b^16 + 21*a^3*b^14 + 6
3*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*a^3*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*((32*(2*a^11 - 6*a^5*b^6 + 2*a^7*b^4 - 6*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) + ((2*a^4 - 6*a^2*b^2)*((32*(2*a^3*b^10 - 24*a^5*b^8 - 36*a^7*b^6 + 8*a^9*b^4
+ 18*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) + ((2*a^4 - 6*
a^2*b^2)*((32*(a^3*b^12 - 2*a^15 + 15*a^5*b^10 + 48*a^7*b^8 + 62*a^9*b^6 + 33*a^11*b^4 + 3*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*a^4 - 6*a^2*b^2)*(3*a*b^16 + 2
1*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))))/(2*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (a*b*((a*b*(3*a^2 - b^2)*((32*
(a^3*b^12 - 2*a^15 + 15*a^5*b^10 + 48*a^7*b^8 + 62*a^9*b^6 + 33*a^11*b^4 + 3*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*a^4 - 6*a^2*b^2)*(3*a*b^16 + 21*a^3*b^14 + 6
3*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^6 + b^6 + 3*a^2
*b^4 + 3*a^4*b^2) + (16*a*b*(2*a^4 - 6*a^2*b^2)*(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*a^2*b^2*(2*a^4 - 6*a^2*b^2)*(3*a^2 - b^2)^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)))*(10*a^6 - 7*b^6 + 68*a^2*b^4 - 59*a^4
*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))/(96*a^6*b - 32*a^4*b^3) + (2*a*b*((32*(2*a^10*b
+ 6*a^6*b^5 - 8*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) - ((
2*a^4 - 6*a^2*b^2)*((32*(5*a^4*b^9 - 3*a^12*b + 12*a^6*b^7 + 6*a^8*b^5 - 4*a^10*b^3))/(a^12 + b^12 + 6*a^2*b^1
0 + 15*a^4*b^8 + 20*a^6*b^6 + 15*a^8*b^4 + 6*a^10*b^2) + (((32*(3*a^4*b^11 - a^2*b^13 - 4*a^14*b + 18*a^6*b^9
+ 22*a^8*b^7 + 3*a^10*b^5 - 9*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*a^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^1
0*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^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*b*((32*(3*a^4*b^11 - a^2*b^13 - 4*a^14*b + 18*a^6*b^9 + 22*a^8
*b^7 + 3*a^10*b^5 - 9*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*a^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)))*(3*a^2 - b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (16*a*b*(2*a^4 - 6*a^
2*b^2)*(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*a^2*b^2*(2*a^4 - 6*a^2
*b^2)*(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^1
2*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))/((96*a^6*b - 32*a^4*b^3)*(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)

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

[Out]

Timed out

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