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

Optimal. Leaf size=98 \[ \frac{6 \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{5 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[Out]

(-2*Cos[a + b*Log[c*x^n]])/(5*b*n*Csc[a + b*Log[c*x^n]]^(3/2)) + (6*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(a -
Pi/2 + b*Log[c*x^n])/2, 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(5*b*n)

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Rubi [A]  time = 0.0583067, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3769, 3771, 2639} \[ \frac{6 \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{5 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[a + b*Log[c*x^n]])/(5*b*n*Csc[a + b*Log[c*x^n]]^(3/2)) + (6*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(a -
Pi/2 + b*Log[c*x^n])/2, 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(5*b*n)

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2639

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

Rubi steps

\begin{align*} \int \frac{1}{x \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\csc ^{\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 )}{5 b n \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{\csc (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\left (3 \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int \sqrt{\sin (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{6 \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )}}{5 b n}\\ \end{align*}

Mathematica [A]  time = 0.220296, size = 88, normalized size = 0.9 \[ -\frac{2 \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \left (\sin ^2\left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )+3 \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right )\right |2\right )\right )}{5 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*(3*EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]] +
Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2))/(5*b*n)

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Maple [A]  time = 1.183, size = 205, normalized size = 2.1 \begin{align*}{\frac{1}{n\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) b} \left ({\frac{2\, \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}}{5}}-{\frac{2\, \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{5}}-{\frac{6}{5}\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 EllipticE} \left ( \sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +1},{\frac{\sqrt{2}}{2}} \right ) }+{\frac{3}{5}\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 ) } \right ){\frac{1}{\sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*(2/5*sin(a+b*ln(c*x^n))^4-2/5*sin(a+b*ln(c*x^n))^2-6/5*(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)*EllipticE((sin(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))+3/5*(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)))/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., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \csc \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/csc(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*csc(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}{x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(x*csc(b*log(c*x^n) + a)^(5/2)), 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/csc(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

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Giac [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/csc(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

Timed out