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

Optimal. Leaf size=29 \[ \frac{2 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n} \]

[Out]

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

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Rubi [A]  time = 0.0273205, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2639} \[ \frac{2 E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sin[a + b*Log[c*x^n]]]/x,x]

[Out]

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

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

Mathematica [A]  time = 0.0834006, size = 32, normalized size = 1.1 \[ -\frac{2 E\left (\left .\frac{1}{2} \left (-a-b \log \left (c x^n\right )+\frac{\pi }{2}\right )\right |2\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.938, size = 129, normalized size = 4.5 \begin{align*} -{\frac{1}{bn\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\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 ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +1},1/2\,\sqrt{2} \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(sin(a+b*ln(c*x^n))^(1/2)/x,x)

[Out]

-1/n*(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)*(2*EllipticE((si
n(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(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{\sqrt{\sin \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(sin(a+b*log(c*x^n))^(1/2)/x,x, algorithm="maxima")

[Out]

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

[Out]

integral(sqrt(sin(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{\sin{\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(sin(a+b*ln(c*x**n))**(1/2)/x,x)

[Out]

Integral(sqrt(sin(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{\sin \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(sin(a+b*log(c*x^n))^(1/2)/x,x, algorithm="giac")

[Out]

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