3.3.81 \(\int \frac {\sqrt {\sinh (a+b \log (c x^n))}}{x} \, dx\) [281]

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

[Out]

2*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))^2)^(1/2)/sin(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))*EllipticE(cos(1/2*I*
a+1/4*Pi+1/2*I*b*ln(c*x^n)),2^(1/2))*sinh(a+b*ln(c*x^n))^(1/2)/b/n/(I*sinh(a+b*ln(c*x^n)))^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 72, 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 = {2721, 2719} \begin {gather*} -\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 )}} \end {gather*}

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 2719

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

Rule 2721

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

Rubi steps

\begin {align*} \int \frac {\sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx &=\frac {\text {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 )} \text {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*}

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Mathematica [A]
time = 0.03, size = 68, normalized size = 0.94 \begin {gather*} \frac {2 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 ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \end {gather*}

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 = 6.96, size = 146, normalized size = 2.03

method	result	size
derivativedivides	\(\frac {\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(146\)
default	\(\frac {\sqrt {-i \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {2}\, \sqrt {-i \left (-\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+i\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(146\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n*(-I*(sinh(a+b*ln(c*x^n))+I))^(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 58, normalized size = 0.81 \begin {gather*} -\frac {2 \, {\left (\sqrt {2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + \sqrt {\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}\right )}}{b n} \end {gather*}

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]

-2*(sqrt(2)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x)
+ b*log(c) + a))) + sqrt(sinh(b*n*log(x) + b*log(c) + a)))/(b*n)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sinh {\left (a + b \log {\left (c x^{n} \right )} \right )}}}{x}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*log(c*x^n))^(1/2)/x,x)

[Out]

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

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