3.68 \(\int \frac{1}{x \sin ^{\frac{5}{2}}(a+b \log (c x^n))} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0421785, antiderivative size = 68, 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 = {2636, 2641} \[ \frac{2 F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{3 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Sin[a + b*Log[c*x^n]]^(5/2)),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]])/(3*b*n*Sin[a + b*Log[c*x^n]]
^(3/2))

Rule 2636

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Sin[c + d*x])^(n + 1))/(b*d*(n +
1)), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[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{1}{x \sin ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sin ^{\frac{5}{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 )}{3 b n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\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 )}{3 b n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.214424, size = 61, normalized size = 0.9 \[ \frac{2 \left (\text{EllipticF}\left (\frac{1}{4} \left (2 a+2 b \log \left (c x^n\right )-\pi \right ),2\right )-\frac{\cos \left (a+b \log \left (c x^n\right )\right )}{\sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sin[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

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

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Maple [A]  time = 1.235, size = 131, normalized size = 1.9 \begin{align*}{\frac{1}{3\,bn\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \left ( \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 ) }{\it EllipticF} \left ( \sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) -2\, \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \right ) \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/sin(a+b*ln(c*x^n))^(5/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*sin(b*log(c*x^n) + a)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{1}{{\left (x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{2} - x\right )} \sqrt{\sin \left (b \log \left (c x^{n}\right ) + a\right )}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

integral(-1/((x*cos(b*log(c*x^n) + a)^2 - x)*sqrt(sin(b*log(c*x^n) + a))), x)

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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(1/x/sin(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

integrate(1/(x*sin(b*log(c*x^n) + a)^(5/2)), x)