3.221 \(\int \cos ^{\frac{3}{2}}(a+b x) \sin ^5(a+b x) \, dx\)

Optimal. Leaf size=52 \[ -\frac{2 \cos ^{\frac{13}{2}}(a+b x)}{13 b}+\frac{4 \cos ^{\frac{9}{2}}(a+b x)}{9 b}-\frac{2 \cos ^{\frac{5}{2}}(a+b x)}{5 b} \]

[Out]

(-2*Cos[a + b*x]^(5/2))/(5*b) + (4*Cos[a + b*x]^(9/2))/(9*b) - (2*Cos[a + b*x]^(13/2))/(13*b)

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Rubi [A]  time = 0.035061, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2565, 270} \[ -\frac{2 \cos ^{\frac{13}{2}}(a+b x)}{13 b}+\frac{4 \cos ^{\frac{9}{2}}(a+b x)}{9 b}-\frac{2 \cos ^{\frac{5}{2}}(a+b x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^(3/2)*Sin[a + b*x]^5,x]

[Out]

(-2*Cos[a + b*x]^(5/2))/(5*b) + (4*Cos[a + b*x]^(9/2))/(9*b) - (2*Cos[a + b*x]^(13/2))/(13*b)

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos ^{\frac{3}{2}}(a+b x) \sin ^5(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x^{3/2} \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{\operatorname{Subst}\left (\int \left (x^{3/2}-2 x^{7/2}+x^{11/2}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{2 \cos ^{\frac{5}{2}}(a+b x)}{5 b}+\frac{4 \cos ^{\frac{9}{2}}(a+b x)}{9 b}-\frac{2 \cos ^{\frac{13}{2}}(a+b x)}{13 b}\\ \end{align*}

Mathematica [B]  time = 0.279403, size = 111, normalized size = 2.13 \[ \frac{2 \sqrt{\cos (a+b x)} \left (-32 \sqrt [4]{\cos ^2(a+b x)}+45 \sin ^6(a+b x) \sqrt [4]{\cos ^2(a+b x)}-5 \sin ^4(a+b x) \sqrt [4]{\cos ^2(a+b x)}-8 \sin ^2(a+b x) \sqrt [4]{\cos ^2(a+b x)}+32\right )}{585 b \sqrt [4]{\cos ^2(a+b x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^(3/2)*Sin[a + b*x]^5,x]

[Out]

(2*Sqrt[Cos[a + b*x]]*(32 - 32*(Cos[a + b*x]^2)^(1/4) - 8*(Cos[a + b*x]^2)^(1/4)*Sin[a + b*x]^2 - 5*(Cos[a + b
*x]^2)^(1/4)*Sin[a + b*x]^4 + 45*(Cos[a + b*x]^2)^(1/4)*Sin[a + b*x]^6))/(585*b*(Cos[a + b*x]^2)^(1/4))

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Maple [B]  time = 0.088, size = 103, normalized size = 2. \begin{align*} -{\frac{32}{585\,b}\sqrt{-2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1} \left ( 180\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{12}-540\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{10}+545\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{8}-190\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{6}+3\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+2\, \left ( \sin \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+2 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^(3/2)*sin(b*x+a)^5,x)

[Out]

-32/585*(-2*sin(1/2*b*x+1/2*a)^2+1)^(1/2)*(180*sin(1/2*b*x+1/2*a)^12-540*sin(1/2*b*x+1/2*a)^10+545*sin(1/2*b*x
+1/2*a)^8-190*sin(1/2*b*x+1/2*a)^6+3*sin(1/2*b*x+1/2*a)^4+2*sin(1/2*b*x+1/2*a)^2+2)/b

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Maxima [A]  time = 0.972461, size = 49, normalized size = 0.94 \begin{align*} -\frac{2 \,{\left (45 \, \cos \left (b x + a\right )^{\frac{13}{2}} - 130 \, \cos \left (b x + a\right )^{\frac{9}{2}} + 117 \, \cos \left (b x + a\right )^{\frac{5}{2}}\right )}}{585 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(3/2)*sin(b*x+a)^5,x, algorithm="maxima")

[Out]

-2/585*(45*cos(b*x + a)^(13/2) - 130*cos(b*x + a)^(9/2) + 117*cos(b*x + a)^(5/2))/b

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Fricas [A]  time = 2.23869, size = 123, normalized size = 2.37 \begin{align*} -\frac{2 \,{\left (45 \, \cos \left (b x + a\right )^{6} - 130 \, \cos \left (b x + a\right )^{4} + 117 \, \cos \left (b x + a\right )^{2}\right )} \sqrt{\cos \left (b x + a\right )}}{585 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(3/2)*sin(b*x+a)^5,x, algorithm="fricas")

[Out]

-2/585*(45*cos(b*x + a)^6 - 130*cos(b*x + a)^4 + 117*cos(b*x + a)^2)*sqrt(cos(b*x + a))/b

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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(b*x+a)**(3/2)*sin(b*x+a)**5,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x + a\right )^{\frac{3}{2}} \sin \left (b x + a\right )^{5}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^(3/2)*sin(b*x+a)^5,x, algorithm="giac")

[Out]

integrate(cos(b*x + a)^(3/2)*sin(b*x + a)^5, x)