3.281 \(\int \frac{\sqrt{\sinh (a+b \log (c x^n))}}{x} \, dx\)

Optimal. Leaf size=72 \[ -\frac{2 i \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

((-2*I)*EllipticE[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2]*Sqrt[Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[I*Sinh[a + b*Log
[c*x^n]]])

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Rubi [A]  time = 0.043666, antiderivative size = 72, 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 = {2640, 2639} \[ -\frac{2 i \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sinh[a + b*Log[c*x^n]]]/x,x]

[Out]

((-2*I)*EllipticE[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2]*Sqrt[Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[I*Sinh[a + b*Log
[c*x^n]]])

Rule 2640

Int[Sqrt[(b_)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[b*Sin[c + d*x]]/Sqrt[Sin[c + d*x]], Int[Sqrt[Si
n[c + d*x]], x], x] /; FreeQ[{b, c, d}, x]

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

Mathematica [A]  time = 0.0420745, size = 68, normalized size = 0.94 \[ \frac{2 \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (\frac{\pi }{2}-i \left (a+b \log \left (c x^n\right )\right )\right )\right |2\right )}{b n \sqrt{\sinh \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sinh[a + b*Log[c*x^n]]]/x,x]

[Out]

(2*EllipticE[(Pi/2 - I*(a + b*Log[c*x^n]))/2, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(b*n*Sqrt[Sinh[a + b*Log[c*x^
n]]])

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Maple [A]  time = 0.046, size = 146, normalized size = 2. \begin{align*}{\frac{\sqrt{2}}{bn\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{-i \left ( i+\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }\sqrt{-i \left ( -\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +i \right ) }\sqrt{i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) \right ){\frac{1}{\sqrt{\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*(-I*(I+sinh(a+b*ln(c*x^n))))^(1/2)*2^(1/2)*(-I*(-sinh(a+b*ln(c*x^n))+I))^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/
2)*(2*EllipticE((1-I*sinh(a+b*ln(c*x^n)))^(1/2),1/2*2^(1/2))-EllipticF((1-I*sinh(a+b*ln(c*x^n)))^(1/2),1/2*2^(
1/2)))/cosh(a+b*ln(c*x^n))/sinh(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{\sqrt{\sinh \left (b \log \left (c x^{n}\right ) + a\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sinh(b*log(c*x^n) + a))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(sinh(b*log(c*x^n) + a))/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sinh{\left (a + b \log{\left (c x^{n} \right )} \right )}}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*ln(c*x**n))**(1/2)/x,x)

[Out]

Integral(sqrt(sinh(a + b*log(c*x**n)))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sinh(b*log(c*x^n) + a))/x, x)