3.317 \(\int \frac{1}{x \csc ^{\frac{3}{2}}(a+b \log (c x^n))} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0589235, 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, 2641} \[ \frac{2 \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{3 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cos[a + b*Log[c*x^n]])/(3*b*n*Sqrt[Csc[a + b*Log[c*x^n]]]) + (2*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticF[(a -
Pi/2 + b*Log[c*x^n])/2, 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(3*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 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 \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\csc ^{\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 )}{3 b n \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac{\operatorname{Subst}\left (\int \sqrt{\csc (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac{\left (\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 \frac{1}{\sqrt{\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac{2 \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} F\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 )}}{3 b n}\\ \end{align*}

Mathematica [A]  time = 0.165752, size = 76, normalized size = 0.78 \[ -\frac{\sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \left (2 \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right ),2\right )+\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Csc[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

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

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Maple [A]  time = 1.423, size = 131, normalized size = 1.3 \begin{align*}{\frac{1}{n\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) b} \left ({\frac{1}{3}\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 ) }-{\frac{2\, \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{3}} \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))^(3/2),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*cos(a+b*ln(c*x^n))^2*sin(a+b*ln(c*x^n)))/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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/(x*csc(b*log(c*x^n) + a)^(3/2)), 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{3}{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))^(3/2),x, algorithm="fricas")

[Out]

integral(1/(x*csc(b*log(c*x^n) + a)^(3/2)), 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))**(3/2),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))^(3/2),x, algorithm="giac")

[Out]

Timed out