3.654 \(\int \cos ^3(x) (a+b \cos ^2(x))^3 \sin (x) \, dx\)

Optimal. Leaf size=36 \[ \frac{a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac{\left (a+b \cos ^2(x)\right )^5}{10 b^2} \]

[Out]

(a*(a + b*Cos[x]^2)^4)/(8*b^2) - (a + b*Cos[x]^2)^5/(10*b^2)

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Rubi [A]  time = 0.0872477, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4335, 266, 43} \[ \frac{a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac{\left (a+b \cos ^2(x)\right )^5}{10 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a + b*Cos[x]^2)^4)/(8*b^2) - (a + b*Cos[x]^2)^5/(10*b^2)

Rule 4335

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[d/(
b*c), Subst[Int[SubstFor[1, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[c*(a
+ b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Sin] || EqQ[F, sin])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \cos ^3(x) \left (a+b \cos ^2(x)\right )^3 \sin (x) \, dx &=-\operatorname{Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=\frac{a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac{\left (a+b \cos ^2(x)\right )^5}{10 b^2}\\ \end{align*}

Mathematica [B]  time = 0.301158, size = 137, normalized size = 3.81 \[ \frac{1}{32} \left (-12 a^2 b \cos ^4(x)-4 a^2 b \cos (3 x) \cos ^3(x)-4 a^3 \cos (2 x)-a^3 \cos (4 x)-8 a b^2 \cos ^6(x)-\frac{1}{32} a b^2 (48 \cos (2 x)+36 \cos (4 x)+16 \cos (6 x)+3 \cos (8 x))-2 b^3 \cos ^8(x)-\frac{1}{320} b^3 (140 \cos (2 x)+100 \cos (4 x)+50 \cos (6 x)+15 \cos (8 x)+2 \cos (10 x))\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-12*a^2*b*Cos[x]^4 - 8*a*b^2*Cos[x]^6 - 2*b^3*Cos[x]^8 - 4*a^3*Cos[2*x] - 4*a^2*b*Cos[x]^3*Cos[3*x] - a^3*Cos
[4*x] - (a*b^2*(48*Cos[2*x] + 36*Cos[4*x] + 16*Cos[6*x] + 3*Cos[8*x]))/32 - (b^3*(140*Cos[2*x] + 100*Cos[4*x]
+ 50*Cos[6*x] + 15*Cos[8*x] + 2*Cos[10*x]))/320)/32

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Maple [A]  time = 0.009, size = 40, normalized size = 1.1 \begin{align*} -{\frac{{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{10}}{10}}-{\frac{3\,a{b}^{2} \left ( \cos \left ( x \right ) \right ) ^{8}}{8}}-{\frac{{a}^{2}b \left ( \cos \left ( x \right ) \right ) ^{6}}{2}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}{a}^{3}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*b^3*cos(x)^10-3/8*a*b^2*cos(x)^8-1/2*a^2*b*cos(x)^6-1/4*cos(x)^4*a^3

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Maxima [B]  time = 0.980933, size = 139, normalized size = 3.86 \begin{align*} \frac{1}{10} \, b^{3} \sin \left (x\right )^{10} - \frac{1}{8} \,{\left (3 \, a b^{2} + 4 \, b^{3}\right )} \sin \left (x\right )^{8} + \frac{1}{2} \,{\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \sin \left (x\right )^{6} - \frac{1}{4} \,{\left (a^{3} + 6 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \sin \left (x\right )^{4} + \frac{1}{2} \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*b^3*sin(x)^10 - 1/8*(3*a*b^2 + 4*b^3)*sin(x)^8 + 1/2*(a^2*b + 3*a*b^2 + 2*b^3)*sin(x)^6 - 1/4*(a^3 + 6*a^
2*b + 9*a*b^2 + 4*b^3)*sin(x)^4 + 1/2*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*sin(x)^2

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Fricas [A]  time = 2.21531, size = 111, normalized size = 3.08 \begin{align*} -\frac{1}{10} \, b^{3} \cos \left (x\right )^{10} - \frac{3}{8} \, a b^{2} \cos \left (x\right )^{8} - \frac{1}{2} \, a^{2} b \cos \left (x\right )^{6} - \frac{1}{4} \, a^{3} \cos \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*b^3*cos(x)^10 - 3/8*a*b^2*cos(x)^8 - 1/2*a^2*b*cos(x)^6 - 1/4*a^3*cos(x)^4

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Sympy [B]  time = 16.4708, size = 97, normalized size = 2.69 \begin{align*} \frac{a^{3} \sin ^{4}{\left (x \right )}}{4} + \frac{a^{3} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} + \frac{a^{2} b \sin ^{6}{\left (x \right )}}{2} + \frac{3 a^{2} b \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} + \frac{3 a^{2} b \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{2} - \frac{3 a b^{2} \cos ^{8}{\left (x \right )}}{8} - \frac{b^{3} \cos ^{10}{\left (x \right )}}{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**3*sin(x)**4/4 + a**3*sin(x)**2*cos(x)**2/2 + a**2*b*sin(x)**6/2 + 3*a**2*b*sin(x)**4*cos(x)**2/2 + 3*a**2*b
*sin(x)**2*cos(x)**4/2 - 3*a*b**2*cos(x)**8/8 - b**3*cos(x)**10/10

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Giac [A]  time = 1.08834, size = 53, normalized size = 1.47 \begin{align*} -\frac{1}{10} \, b^{3} \cos \left (x\right )^{10} - \frac{3}{8} \, a b^{2} \cos \left (x\right )^{8} - \frac{1}{2} \, a^{2} b \cos \left (x\right )^{6} - \frac{1}{4} \, a^{3} \cos \left (x\right )^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*b^3*cos(x)^10 - 3/8*a*b^2*cos(x)^8 - 1/2*a^2*b*cos(x)^6 - 1/4*a^3*cos(x)^4