3.60 \(\int \frac {\sin ^{\frac {3}{2}}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=68 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n}-\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

[Out]

-2/3*(sin(1/2*a+1/4*Pi+1/2*b*ln(c*x^n))^2)^(1/2)/sin(1/2*a+1/4*Pi+1/2*b*ln(c*x^n))*EllipticF(cos(1/2*a+1/4*Pi+
1/2*b*ln(c*x^n)),2^(1/2))/b/n-2/3*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))^(1/2)/b/n

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Rubi [A]  time = 0.04, antiderivative size = 68, 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 = {2635, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n}-\frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sin[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

(2*EllipticF[(a - Pi/2 + b*Log[c*x^n])/2, 2])/(3*b*n) - (2*Cos[a + b*Log[c*x^n]]*Sqrt[Sin[a + b*Log[c*x^n]]])/
(3*b*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 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \sin ^{\frac {3}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{3 b n}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{3 b n}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 58, normalized size = 0.85 \[ -\frac {2 \left (F\left (\left .\frac {1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right )\right |2\right )+\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \cos \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2] + Cos[a + b*Log[c*x^n]]*Sqrt[Sin[a + b*Log[c*x^n]]]))/(3*b*n
)

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fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sin(b*log(c*x^n) + a)^(3/2)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*log(c*x^n) + a)^(3/2)/x, x)

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maple [A]  time = 0.06, size = 131, normalized size = 1.93 \[ \frac {\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right ) \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a+b*ln(c*x^n))^(3/2)/x,x)

[Out]

1/n*(1/3*(sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))+2)^(1/2)*(-sin(a+b*ln(c*x^n)))^(1/2)*EllipticF((s
in(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))-2/3*sin(a+b*ln(c*x^n))*cos(a+b*ln(c*x^n))^2)/cos(a+b*ln(c*x^n))/sin(a+
b*ln(c*x^n))^(1/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sin(b*log(c*x^n) + a)^(3/2)/x, x)

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mupad [B]  time = 2.53, size = 65, normalized size = 0.96 \[ -\frac {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,{\left ({\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}^{5/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(a + b*log(c*x^n))^(3/2)/x,x)

[Out]

-(cos(a + b*log(c*x^n))*sin(a + b*log(c*x^n))^(5/2)*hypergeom([-1/4, 1/2], 3/2, cos(a + b*log(c*x^n))^2))/(b*n
*(sin(a + b*log(c*x^n))^2)^(5/4))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(a+b*ln(c*x**n))**(3/2)/x,x)

[Out]

Integral(sin(a + b*log(c*x**n))**(3/2)/x, x)

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