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

Optimal. Leaf size=175 \[ \frac {a x}{8 \left (a^2+b^2\right )}-\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}-\frac {a^2 b \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}-\frac {a \sin (x) \cos ^3(x)}{4 \left (a^2+b^2\right )}+\frac {a \sin (x) \cos (x)}{8 \left (a^2+b^2\right )}-\frac {a b^2 \sin (x) \cos (x)}{2 \left (a^2+b^2\right )^2}+\frac {a^2 b^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}+\frac {a^3 b^2 x}{\left (a^2+b^2\right )^3} \]

[Out]

a^3*b^2*x/(a^2+b^2)^3-1/2*a*b^2*x/(a^2+b^2)^2+1/8*a*x/(a^2+b^2)-1/4*b*cos(x)^4/(a^2+b^2)+a^2*b^3*ln(a*cos(x)+b
*sin(x))/(a^2+b^2)^3-1/2*a*b^2*cos(x)*sin(x)/(a^2+b^2)^2+1/8*a*cos(x)*sin(x)/(a^2+b^2)-1/4*a*cos(x)^3*sin(x)/(
a^2+b^2)-1/2*a^2*b*sin(x)^2/(a^2+b^2)^2

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Rubi [A]  time = 0.28, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {3109, 2565, 30, 2568, 2635, 8, 2564, 3098, 3133} \[ \frac {a x}{8 \left (a^2+b^2\right )}-\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {a^3 b^2 x}{\left (a^2+b^2\right )^3}-\frac {a^2 b \sin ^2(x)}{2 \left (a^2+b^2\right )^2}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}-\frac {a \sin (x) \cos ^3(x)}{4 \left (a^2+b^2\right )}+\frac {a \sin (x) \cos (x)}{8 \left (a^2+b^2\right )}-\frac {a b^2 \sin (x) \cos (x)}{2 \left (a^2+b^2\right )^2}+\frac {a^2 b^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*b^2*x)/(a^2 + b^2)^3 - (a*b^2*x)/(2*(a^2 + b^2)^2) + (a*x)/(8*(a^2 + b^2)) - (b*Cos[x]^4)/(4*(a^2 + b^2))
+ (a^2*b^3*Log[a*Cos[x] + b*Sin[x]])/(a^2 + b^2)^3 - (a*b^2*Cos[x]*Sin[x])/(2*(a^2 + b^2)^2) + (a*Cos[x]*Sin[
x])/(8*(a^2 + b^2)) - (a*Cos[x]^3*Sin[x])/(4*(a^2 + b^2)) - (a^2*b*Sin[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 2565

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

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*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 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 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 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]

Rubi steps

\begin {align*} \int \frac {\cos ^3(x) \sin ^2(x)}{a \cos (x)+b \sin (x)} \, dx &=\frac {a \int \cos ^2(x) \sin ^2(x) \, dx}{a^2+b^2}+\frac {b \int \cos ^3(x) \sin (x) \, dx}{a^2+b^2}-\frac {(a b) \int \frac {\cos ^2(x) \sin (x)}{a \cos (x)+b \sin (x)} \, dx}{a^2+b^2}\\ &=-\frac {a \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )}-\frac {\left (a^2 b\right ) \int \cos (x) \sin (x) \, dx}{\left (a^2+b^2\right )^2}-\frac {\left (a b^2\right ) \int \cos ^2(x) \, dx}{\left (a^2+b^2\right )^2}+\frac {\left (a^2 b^2\right ) \int \frac {\cos (x)}{a \cos (x)+b \sin (x)} \, dx}{\left (a^2+b^2\right )^2}+\frac {a \int \cos ^2(x) \, dx}{4 \left (a^2+b^2\right )}-\frac {b \operatorname {Subst}\left (\int x^3 \, dx,x,\cos (x)\right )}{a^2+b^2}\\ &=\frac {a^3 b^2 x}{\left (a^2+b^2\right )^3}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}-\frac {a b^2 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac {a \cos (x) \sin (x)}{8 \left (a^2+b^2\right )}-\frac {a \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )}+\frac {\left (a^2 b^3\right ) \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\right ) \operatorname {Subst}(\int x \, dx,x,\sin (x))}{\left (a^2+b^2\right )^2}-\frac {\left (a b^2\right ) \int 1 \, dx}{2 \left (a^2+b^2\right )^2}+\frac {a \int 1 \, dx}{8 \left (a^2+b^2\right )}\\ &=\frac {a^3 b^2 x}{\left (a^2+b^2\right )^3}-\frac {a b^2 x}{2 \left (a^2+b^2\right )^2}+\frac {a x}{8 \left (a^2+b^2\right )}-\frac {b \cos ^4(x)}{4 \left (a^2+b^2\right )}+\frac {a^2 b^3 \log (a \cos (x)+b \sin (x))}{\left (a^2+b^2\right )^3}-\frac {a b^2 \cos (x) \sin (x)}{2 \left (a^2+b^2\right )^2}+\frac {a \cos (x) \sin (x)}{8 \left (a^2+b^2\right )}-\frac {a \cos ^3(x) \sin (x)}{4 \left (a^2+b^2\right )}-\frac {a^2 b \sin ^2(x)}{2 \left (a^2+b^2\right )^2}\\ \end {align*}

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Mathematica [C]  time = 0.79, size = 287, normalized size = 1.64 \[ -\frac {-4 a^5 x+a^5 \sin (4 x)+4 b \left (b^4-a^4\right ) \cos (2 x)+4 i a^4 b x+a^4 b \cos (4 x)-4 a^4 b \log (a \cos (x)+b \sin (x))+2 a^4 b \log \left ((a \cos (x)+b \sin (x))^2\right )-24 a^3 b^2 x+8 a^3 b^2 \sin (2 x)+2 a^3 b^2 \sin (4 x)-24 i a^2 b^3 x+2 a^2 b^3 \cos (4 x)-8 a^2 b^3 \log (a \cos (x)+b \sin (x))-12 a^2 b^3 \log \left ((a \cos (x)+b \sin (x))^2\right )-4 i b \left (a^4-6 a^2 b^2+b^4\right ) \tan ^{-1}(\tan (x))-4 b^5 \log (a \cos (x)+b \sin (x))+2 b^5 \log \left ((a \cos (x)+b \sin (x))^2\right )+12 a b^4 x+8 a b^4 \sin (2 x)+a b^4 \sin (4 x)+4 i b^5 x+b^5 \cos (4 x)}{32 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(-4*a^5*x + (4*I)*a^4*b*x - 24*a^3*b^2*x - (24*I)*a^2*b^3*x + 12*a*b^4*x + (4*I)*b^5*x - (4*I)*b*(a^4 -
6*a^2*b^2 + b^4)*ArcTan[Tan[x]] + 4*b*(-a^4 + b^4)*Cos[2*x] + a^4*b*Cos[4*x] + 2*a^2*b^3*Cos[4*x] + b^5*Cos[4*
x] - 4*a^4*b*Log[a*Cos[x] + b*Sin[x]] - 8*a^2*b^3*Log[a*Cos[x] + b*Sin[x]] - 4*b^5*Log[a*Cos[x] + b*Sin[x]] +
2*a^4*b*Log[(a*Cos[x] + b*Sin[x])^2] - 12*a^2*b^3*Log[(a*Cos[x] + b*Sin[x])^2] + 2*b^5*Log[(a*Cos[x] + b*Sin[x
])^2] + 8*a^3*b^2*Sin[2*x] + 8*a*b^4*Sin[2*x] + a^5*Sin[4*x] + 2*a^3*b^2*Sin[4*x] + a*b^4*Sin[4*x])/(a^2 + b^2
)^3

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fricas [A]  time = 0.64, size = 175, normalized size = 1.00 \[ \frac {4 \, a^{2} b^{3} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) + {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + b^{2}\right ) - 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \relax (x)^{4} + 4 \, {\left (a^{4} b + a^{2} b^{3}\right )} \cos \relax (x)^{2} + {\left (a^{5} + 6 \, a^{3} b^{2} - 3 \, a b^{4}\right )} x - {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \relax (x)^{3} - {\left (a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \relax (x)\right )} \sin \relax (x)}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.21, size = 273, normalized size = 1.56 \[ \frac {a^{2} b^{4} \log \left ({\left | b \tan \relax (x) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {a^{2} b^{3} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{5} + 6 \, a^{3} b^{2} - 3 \, a b^{4}\right )} x}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {6 \, a^{2} b^{3} \tan \relax (x)^{4} + a^{5} \tan \relax (x)^{3} - 2 \, a^{3} b^{2} \tan \relax (x)^{3} - 3 \, a b^{4} \tan \relax (x)^{3} + 4 \, a^{4} b \tan \relax (x)^{2} + 16 \, a^{2} b^{3} \tan \relax (x)^{2} - a^{5} \tan \relax (x) - 6 \, a^{3} b^{2} \tan \relax (x) - 5 \, a b^{4} \tan \relax (x) + 2 \, a^{4} b + 6 \, a^{2} b^{3} - 2 \, b^{5}}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \relax (x)^{2} + 1\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*b^4*log(abs(b*tan(x) + a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - 1/2*a^2*b^3*log(tan(x)^2 + 1)/(a^6 + 3*
a^4*b^2 + 3*a^2*b^4 + b^6) + 1/8*(a^5 + 6*a^3*b^2 - 3*a*b^4)*x/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/8*(6*a^
2*b^3*tan(x)^4 + a^5*tan(x)^3 - 2*a^3*b^2*tan(x)^3 - 3*a*b^4*tan(x)^3 + 4*a^4*b*tan(x)^2 + 16*a^2*b^3*tan(x)^2
- a^5*tan(x) - 6*a^3*b^2*tan(x) - 5*a*b^4*tan(x) + 2*a^4*b + 6*a^2*b^3 - 2*b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*(tan(x)^2 + 1)^2)

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maple [B]  time = 0.09, size = 363, normalized size = 2.07 \[ \frac {\left (\tan ^{3}\relax (x )\right ) a^{5}}{8 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {\left (\tan ^{3}\relax (x )\right ) a^{3} b^{2}}{4 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {3 \left (\tan ^{3}\relax (x )\right ) a \,b^{4}}{8 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}+\frac {\left (\tan ^{2}\relax (x )\right ) a^{4} b}{2 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}+\frac {\left (\tan ^{2}\relax (x )\right ) a^{2} b^{3}}{2 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {3 \tan \relax (x ) a^{3} b^{2}}{4 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {5 \tan \relax (x ) a \,b^{4}}{8 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {\tan \relax (x ) a^{5}}{8 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}+\frac {a^{4} b}{4 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {b^{5}}{4 \left (a^{2}+b^{2}\right )^{3} \left (\tan ^{2}\relax (x )+1\right )^{2}}-\frac {\ln \left (\tan ^{2}\relax (x )+1\right ) a^{2} b^{3}}{2 \left (a^{2}+b^{2}\right )^{3}}+\frac {\arctan \left (\tan \relax (x )\right ) a^{5}}{8 \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \arctan \left (\tan \relax (x )\right ) a^{3} b^{2}}{4 \left (a^{2}+b^{2}\right )^{3}}-\frac {3 \arctan \left (\tan \relax (x )\right ) a \,b^{4}}{8 \left (a^{2}+b^{2}\right )^{3}}+\frac {a^{2} b^{3} \ln \left (a +b \tan \relax (x )\right )}{\left (a^{2}+b^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8/(a^2+b^2)^3/(tan(x)^2+1)^2*tan(x)^3*a^5-1/4/(a^2+b^2)^3/(tan(x)^2+1)^2*tan(x)^3*a^3*b^2-3/8/(a^2+b^2)^3/(t
an(x)^2+1)^2*tan(x)^3*a*b^4+1/2/(a^2+b^2)^3/(tan(x)^2+1)^2*tan(x)^2*a^4*b+1/2/(a^2+b^2)^3/(tan(x)^2+1)^2*tan(x
)^2*a^2*b^3-3/4/(a^2+b^2)^3/(tan(x)^2+1)^2*tan(x)*a^3*b^2-5/8/(a^2+b^2)^3/(tan(x)^2+1)^2*tan(x)*a*b^4-1/8/(a^2
+b^2)^3/(tan(x)^2+1)^2*tan(x)*a^5+1/4/(a^2+b^2)^3/(tan(x)^2+1)^2*a^4*b-1/4/(a^2+b^2)^3/(tan(x)^2+1)^2*b^5-1/2/
(a^2+b^2)^3*ln(tan(x)^2+1)*a^2*b^3+1/8/(a^2+b^2)^3*arctan(tan(x))*a^5+3/4/(a^2+b^2)^3*arctan(tan(x))*a^3*b^2-3
/8/(a^2+b^2)^3*arctan(tan(x))*a*b^4+a^2*b^3/(a^2+b^2)^3*ln(a+b*tan(x))

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maxima [B]  time = 0.45, size = 424, normalized size = 2.42 \[ \frac {a^{2} b^{3} \log \left (-a - \frac {2 \, b \sin \relax (x)}{\cos \relax (x) + 1} + \frac {a \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {a^{2} b^{3} \log \left (\frac {\sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{5} + 6 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \arctan \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right )}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {\frac {8 \, b^{3} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} - \frac {16 \, a^{2} b \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {8 \, b^{3} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} - \frac {{\left (a^{3} + 5 \, a b^{2}\right )} \sin \relax (x)}{\cos \relax (x) + 1} + \frac {{\left (7 \, a^{3} + 3 \, a b^{2}\right )} \sin \relax (x)^{3}}{{\left (\cos \relax (x) + 1\right )}^{3}} - \frac {{\left (7 \, a^{3} + 3 \, a b^{2}\right )} \sin \relax (x)^{5}}{{\left (\cos \relax (x) + 1\right )}^{5}} + \frac {{\left (a^{3} + 5 \, a b^{2}\right )} \sin \relax (x)^{7}}{{\left (\cos \relax (x) + 1\right )}^{7}}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \relax (x)^{2}}{{\left (\cos \relax (x) + 1\right )}^{2}} + \frac {6 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \relax (x)^{4}}{{\left (\cos \relax (x) + 1\right )}^{4}} + \frac {4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \relax (x)^{6}}{{\left (\cos \relax (x) + 1\right )}^{6}} + \frac {{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sin \relax (x)^{8}}{{\left (\cos \relax (x) + 1\right )}^{8}}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*b^3*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^
2*b^3*log(sin(x)^2/(cos(x) + 1)^2 + 1)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*(a^5 + 6*a^3*b^2 - 3*a*b^4)*a
rctan(sin(x)/(cos(x) + 1))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*(8*b^3*sin(x)^2/(cos(x) + 1)^2 - 16*a^2*b
*sin(x)^4/(cos(x) + 1)^4 + 8*b^3*sin(x)^6/(cos(x) + 1)^6 - (a^3 + 5*a*b^2)*sin(x)/(cos(x) + 1) + (7*a^3 + 3*a*
b^2)*sin(x)^3/(cos(x) + 1)^3 - (7*a^3 + 3*a*b^2)*sin(x)^5/(cos(x) + 1)^5 + (a^3 + 5*a*b^2)*sin(x)^7/(cos(x) +
1)^7)/(a^4 + 2*a^2*b^2 + b^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*sin(x)^2/(cos(x) + 1)^2 + 6*(a^4 + 2*a^2*b^2 + b^4)*s
in(x)^4/(cos(x) + 1)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*sin(x)^6/(cos(x) + 1)^6 + (a^4 + 2*a^2*b^2 + b^4)*sin(x)^8/
(cos(x) + 1)^8)

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mupad [B]  time = 11.14, size = 5870, normalized size = 33.54 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((tan(x/2)^3*(3*a*b^2 + 7*a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) - (tan(x/2)^5*(3*a*b^2 + 7*a^3))/(4*(a^4 + b^4 + 2
*a^2*b^2)) - (tan(x/2)*(5*a*b^2 + a^3))/(4*(a^4 + b^4 + 2*a^2*b^2)) + (tan(x/2)^7*(5*a*b^2 + a^3))/(4*(a^4 + b
^4 + 2*a^2*b^2)) + (2*b^3*tan(x/2)^2)/(a^4 + b^4 + 2*a^2*b^2) + (2*b^3*tan(x/2)^6)/(a^4 + b^4 + 2*a^2*b^2) - (
4*a^2*b*tan(x/2)^4)/(a^4 + b^4 + 2*a^2*b^2))/(4*tan(x/2)^2 + 6*tan(x/2)^4 + 4*tan(x/2)^6 + tan(x/2)^8 + 1) - (
a*atan((tan(x/2)*((((64*a^2*b^3*((a*((16*a^15*b + 16*a^3*b^13 + 288*a^5*b^11 + 1008*a^7*b^9 + 1472*a^9*b^7 + 1
008*a^11*b^5 + 288*a^13*b^3)/(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^2*b^3*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1
344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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^4 - 3*b^4 + 6*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b
^2)) - (4*a^3*b^3*(a^4 - 3*b^4 + 6*a^2*b^2)*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720
*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*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))
))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a*((a^15 + 18*a^3*b^12 - 141*a^5*b^10 - 327*a^7*b^8 - 146*
a^9*b^6 + 36*a^11*b^4 + 15*a^13*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)) - (64*a^2*b^3*((16*a^15*b + 16*a^3*b^13 + 288*a^5*b^11 + 1008*a^7*b^9 + 1472*a^9*b^7 + 1008*a^11*b^5
+ 288*a^13*b^3)/(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^2*
b^3*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4
+ 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2))*(a^4 - 3*b^4 + 6*a^2*b^2))
/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a^3*(a^4 - 3*b^4 + 6*a^2*b^2)^3*(192*a*b^16 + 1344*a^3*b^14 + 4032
*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/(1024*(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^10
- 9*b^10 + 493*a^2*b^8 - 706*a^4*b^6 - 46*a^6*b^4 + 11*a^8*b^2))/(a^10 + 9*b^10 + 229*a^2*b^8 + 250*a^4*b^6 +
42*a^6*b^4 + 13*a^8*b^2)^2 - (2*a*b*((18*a^5*b^9 + 13*a^7*b^7 + 12*a^9*b^5 + a^11*b^3)/(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*((a*((16*a^15*b + 16*a^3*b^13 + 288*a^5*b^11
+ 1008*a^7*b^9 + 1472*a^9*b^7 + 1008*a^11*b^5 + 288*a^13*b^3)/(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^2*b^3*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10
+ 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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^4 - 3*b^4 + 6*a^2*b^2))/(
8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*a^3*b^3*(a^4 - 3*b^4 + 6*a^2*b^2)*(192*a*b^16 + 1344*a^3*b^14 + 40
32*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6
+ 192*a^2*b^4 + 192*a^4*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^4 - 3*b^4 + 6*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (64*a
^2*b^3*((a^15 + 18*a^3*b^12 - 141*a^5*b^10 - 327*a^7*b^8 - 146*a^9*b^6 + 36*a^11*b^4 + 15*a^13*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)) - (64*a^2*b^3*((16*a^15*b + 16*a^3*b^
13 + 288*a^5*b^11 + 1008*a^7*b^9 + 1472*a^9*b^7 + 1008*a^11*b^5 + 288*a^13*b^3)/(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^2*b^3*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12
+ 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344*a^13*b^4 + 192*a^15*b^2))/((64*a^6 + 64*b^6 + 192*a^2*b
^4 + 192*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))))/(64*a^6 +
64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a^4*b^3*(a^4 - 3*b^4 +
6*a^2*b^2)^2*(192*a*b^16 + 1344*a^3*b^14 + 4032*a^5*b^12 + 6720*a^7*b^10 + 6720*a^9*b^8 + 4032*a^11*b^6 + 1344
*a^13*b^4 + 192*a^15*b^2))/(2*(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*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^8 + 57*b^8 - 436*a^2*b
^6 + 110*a^4*b^4 + 28*a^6*b^2))/(a^10 + 9*b^10 + 229*a^2*b^8 + 250*a^4*b^6 + 42*a^6*b^4 + 13*a^8*b^2)^2)*(16*a
^16 + 16*b^16 + 128*a^2*b^14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12*b^4 + 128*
a^14*b^2))/(a^8 - 3*a^4*b^4 + 6*a^6*b^2) + (((a*((39*a^4*b^11 - a^14*b + 123*a^6*b^9 + 134*a^8*b^7 + 54*a^10*b
^5 + 3*a^12*b^3)/(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)) + (64*a^2*
b^3*((8*a^16 + 24*a^2*b^14 - 40*a^4*b^12 - 360*a^6*b^10 - 552*a^8*b^8 - 248*a^10*b^6 + 72*a^12*b^4 + 72*a^14*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)) - (32*a^2*b^3*(192*a^16
*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b
^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2))*(a^4 - 3*b^4 + 6*a^2*b^2))/(8*(a^6 + b^
6 + 3*a^2*b^4 + 3*a^4*b^2)) + (64*a^2*b^3*((a*((8*a^16 + 24*a^2*b^14 - 40*a^4*b^12 - 360*a^6*b^10 - 552*a^8*b^
8 - 248*a^10*b^6 + 72*a^12*b^4 + 72*a^14*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)) - (32*a^2*b^3*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6
720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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^4 - 3*b^4 + 6*a^2*b^2))/(8*(a^6 + b^6 + 3*
a^2*b^4 + 3*a^4*b^2)) - (4*a^3*b^3*(a^4 - 3*b^4 + 6*a^2*b^2)*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032
*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 1
92*a^4*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))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2) + (a^3*(a^4 - 3*b^4 + 6*a^2*b^2)^3*(192*a^16
*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b
^3))/(1024*(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^10 - 9*b^10 + 493*a^2*b^8 - 706*a^4*b^6 - 46*a^6*b^4 + 11*a^8*b^2)*(16*a^16 + 16*b^16 +
128*a^2*b^14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12*b^4 + 128*a^14*b^2))/((a^
8 - 3*a^4*b^4 + 6*a^6*b^2)*(a^10 + 9*b^10 + 229*a^2*b^8 + 250*a^4*b^6 + 42*a^6*b^4 + 13*a^8*b^2)^2) - (2*a*b*(
(2*a^8*b^6 - 15*a^6*b^8 + a^10*b^4)/(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*((a*((8*a^16 + 24*a^2*b^14 - 40*a^4*b^12 - 360*a^6*b^10 - 552*a^8*b^8 - 248*a^10*b^6 + 72*a^12*b
^4 + 72*a^14*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)) - (32*a^2
*b^3*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5
+ 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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^4 - 3*b^4 + 6*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (4*a
^3*b^3*(a^4 - 3*b^4 + 6*a^2*b^2)*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6
720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*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^4 - 3*
b^4 + 6*a^2*b^2))/(8*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) - (64*a^2*b^3*((39*a^4*b^11 - a^14*b + 123*a^6*b^9 +
134*a^8*b^7 + 54*a^10*b^5 + 3*a^12*b^3)/(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)) + (64*a^2*b^3*((8*a^16 + 24*a^2*b^14 - 40*a^4*b^12 - 360*a^6*b^10 - 552*a^8*b^8 - 248*a^10*b^6 +
72*a^12*b^4 + 72*a^14*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))
- (32*a^2*b^3*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 403
2*a^12*b^5 + 1344*a^14*b^3))/((64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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))))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*a^4*b^2)))/(64*a^6 + 64*b
^6 + 192*a^2*b^4 + 192*a^4*b^2) - (a^4*b^3*(a^4 - 3*b^4 + 6*a^2*b^2)^2*(192*a^16*b + 192*a^2*b^15 + 1344*a^4*b
^13 + 4032*a^6*b^11 + 6720*a^8*b^9 + 6720*a^10*b^7 + 4032*a^12*b^5 + 1344*a^14*b^3))/(2*(64*a^6 + 64*b^6 + 192
*a^2*b^4 + 192*a^4*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^8 + 57*b^8 - 436*a^2*b^6 + 110*a^4*b^4 + 28*a^6*b^2)*(16*a^16 + 16*b^16 +
128*a^2*b^14 + 448*a^4*b^12 + 896*a^6*b^10 + 1120*a^8*b^8 + 896*a^10*b^6 + 448*a^12*b^4 + 128*a^14*b^2))/((a^8
- 3*a^4*b^4 + 6*a^6*b^2)*(a^10 + 9*b^10 + 229*a^2*b^8 + 250*a^4*b^6 + 42*a^6*b^4 + 13*a^8*b^2)^2))*(a^4 - 3*b
^4 + 6*a^2*b^2))/(4*(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2)) + (a^2*b^3*log(a + 2*b*tan(x/2) - a*tan(x/2)^2))/(a^6
+ b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (64*a^2*b^3*log(1/(cos(x) + 1)))/(64*a^6 + 64*b^6 + 192*a^2*b^4 + 192*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)**2/(a*cos(x)+b*sin(x)),x)

[Out]

Timed out

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