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]

-2/5*cos(b*x+a)^(5/2)/b+4/9*cos(b*x+a)^(9/2)/b-2/13*cos(b*x+a)^(13/2)/b

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Rubi [A]  time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, 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 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]

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])

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*}

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Mathematica [B]  time = 0.26, 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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fricas [A]  time = 0.45, size = 44, normalized size = 0.85 \[ -\frac {2 \, {\left (45 \, \cos \left (b x + a\right )^{6} - 130 \, \cos \left (b x + a\right )^{4} + 117 \, \cos \left (b x + a\right )^{2}\right )} \sqrt {\cos \left (b x + a\right )}}{585 \, b} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (b x + a\right )^{\frac {3}{2}} \sin \left (b x + a\right )^{5}\,{d x} \]

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)

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maple [B]  time = 0.13, size = 103, normalized size = 1.98 \[ -\frac {32 \sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+1}\, \left (180 \left (\sin ^{12}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-540 \left (\sin ^{10}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+545 \left (\sin ^{8}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-190 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+3 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+2\right )}{585 b} \]

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 = 0.31, size = 36, normalized size = 0.69 \[ -\frac {2 \, {\left (45 \, \cos \left (b x + a\right )^{\frac {13}{2}} - 130 \, \cos \left (b x + a\right )^{\frac {9}{2}} + 117 \, \cos \left (b x + a\right )^{\frac {5}{2}}\right )}}{585 \, b} \]

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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mupad [B]  time = 0.64, size = 35, normalized size = 0.67 \[ -\frac {2\,{\cos \left (a+b\,x\right )}^{5/2}\,\left (\frac {5\,{\cos \left (a+b\,x\right )}^4}{13}-\frac {10\,{\cos \left (a+b\,x\right )}^2}{9}+1\right )}{5\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*cos(a + b*x)^(5/2)*((5*cos(a + b*x)^4)/13 - (10*cos(a + b*x)^2)/9 + 1))/(5*b)

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

[Out]

Timed out

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