3.275 \(\int \frac{\sinh ^2(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac{\log (x)}{2} \]

[Out]

-Log[x]/2 + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(2*b*n)

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Rubi [A]  time = 0.0292156, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac{\sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*Log[c*x^n]]^2/x,x]

[Out]

-Log[x]/2 + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(2*b*n)

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sinh ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sinh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=-\frac{\log (x)}{2}+\frac{\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end{align*}

Mathematica [A]  time = 0.0273888, size = 36, normalized size = 0.92 \[ \frac{\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 \left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*Log[c*x^n]]^2/x,x]

[Out]

(-2*(a + b*Log[c*x^n]) + Sinh[2*(a + b*Log[c*x^n])])/(4*b*n)

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Maple [A]  time = 0.01, size = 52, normalized size = 1.3 \begin{align*}{\frac{\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}-{\frac{\ln \left ( c{x}^{n} \right ) }{2\,n}}-{\frac{a}{2\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))/b/n-1/2*ln(c*x^n)/n-1/2/b/n*a

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Maxima [A]  time = 1.04754, size = 66, normalized size = 1.69 \begin{align*} \frac{e^{\left (2 \, b \log \left (c x^{n}\right ) + 2 \, a\right )}}{8 \, b n} - \frac{e^{\left (-2 \, b \log \left (c x^{n}\right ) - 2 \, a\right )}}{8 \, b n} - \frac{1}{2} \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*e^(2*b*log(c*x^n) + 2*a)/(b*n) - 1/8*e^(-2*b*log(c*x^n) - 2*a)/(b*n) - 1/2*log(x)

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Fricas [A]  time = 2.06571, size = 123, normalized size = 3.15 \begin{align*} -\frac{b n \log \left (x\right ) - \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 )}{2 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(b*n*log(x) - cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a))/(b*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\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))**2/x,x)

[Out]

Integral(sinh(a + b*log(c*x**n))**2/x, x)

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Giac [B]  time = 1.21402, size = 109, normalized size = 2.79 \begin{align*} -\frac{{\left (4 \, b c^{2 \, b} n e^{\left (2 \, a\right )} \log \left (x\right ) - c^{4 \, b} x^{2 \, b n} e^{\left (4 \, a\right )} - \frac{2 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1}{x^{2 \, b n}}\right )} e^{\left (-2 \, a\right )}}{8 \, b c^{2 \, b} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(4*b*c^(2*b)*n*e^(2*a)*log(x) - c^(4*b)*x^(2*b*n)*e^(4*a) - (2*c^(2*b)*x^(2*b*n)*e^(2*a) - 1)/x^(2*b*n))*
e^(-2*a)/(b*c^(2*b)*n)