3.277 \(\int \frac{\sinh ^4(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=73 \[ \frac{\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{3 \log (x)}{8} \]

[Out]

(3*Log[x])/8 - (3*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(8*b*n) + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*
Log[c*x^n]]^3)/(4*b*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0465652, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2635, 8} \[ \frac{\sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{3 \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{3 \log (x)}{8} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*Log[c*x^n]]^4/x,x]

[Out]

(3*Log[x])/8 - (3*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(8*b*n) + (Cosh[a + b*Log[c*x^n]]*Sinh[a + b*
Log[c*x^n]]^3)/(4*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 ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sinh ^4(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 ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{3 \operatorname{Subst}\left (\int \sinh ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{4 n}\\ &=-\frac{3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{3 \operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{8 n}\\ &=\frac{3 \log (x)}{8}-\frac{3 \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{8 b n}+\frac{\cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{4 b n}\\ \end{align*}

Mathematica [A]  time = 0.0467006, size = 51, normalized size = 0.7 \[ \frac{12 \left (a+b \log \left (c x^n\right )\right )-8 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*Log[c*x^n]]^4/x,x]

[Out]

(12*(a + b*Log[c*x^n]) - 8*Sinh[2*(a + b*Log[c*x^n])] + Sinh[4*(a + b*Log[c*x^n])])/(32*b*n)

________________________________________________________________________________________

Maple [A]  time = 0.012, size = 84, normalized size = 1.2 \begin{align*}{\frac{\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}}{4\,bn}}-{\frac{3\,\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{8\,bn}}+{\frac{3\,\ln \left ( c{x}^{n} \right ) }{8\,n}}+{\frac{3\,a}{8\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a+b*ln(c*x^n))^4/x,x)

[Out]

1/4*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))^3/b/n-3/8*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x^n))/b/n+3/8*ln(c*x^n
)/n+3/8/b/n*a

________________________________________________________________________________________

Maxima [A]  time = 1.14188, size = 126, normalized size = 1.73 \begin{align*} \frac{e^{\left (4 \, b \log \left (c x^{n}\right ) + 4 \, a\right )}}{64 \, b n} - \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{e^{\left (-4 \, b \log \left (c x^{n}\right ) - 4 \, a\right )}}{64 \, b n} + \frac{3}{8} \, \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))^4/x,x, algorithm="maxima")

[Out]

1/64*e^(4*b*log(c*x^n) + 4*a)/(b*n) - 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)
- 1/64*e^(-4*b*log(c*x^n) - 4*a)/(b*n) + 3/8*log(x)

________________________________________________________________________________________

Fricas [A]  time = 2.08291, size = 270, normalized size = 3.7 \begin{align*} \frac{\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 )^{3} + 3 \, b n \log \left (x\right ) +{\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 4 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^4/x,x, algorithm="fricas")

[Out]

1/8*(cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + 3*b*n*log(x) + (cosh(b*n*log(x) + b*l
og(c) + a)^3 - 4*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))/(b*n)

________________________________________________________________________________________

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(sinh(a+b*ln(c*x**n))**4/x,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.18396, size = 154, normalized size = 2.11 \begin{align*} \frac{{\left (24 \, b c^{4 \, b} n e^{\left (4 \, a\right )} \log \left (x\right ) + c^{8 \, b} x^{4 \, b n} e^{\left (8 \, a\right )} - 8 \, c^{6 \, b} x^{2 \, b n} e^{\left (6 \, a\right )} - \frac{18 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 8 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 1}{x^{4 \, b n}}\right )} e^{\left (-4 \, a\right )}}{64 \, b c^{4 \, b} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(a+b*log(c*x^n))^4/x,x, algorithm="giac")

[Out]

1/64*(24*b*c^(4*b)*n*e^(4*a)*log(x) + c^(8*b)*x^(4*b*n)*e^(8*a) - 8*c^(6*b)*x^(2*b*n)*e^(6*a) - (18*c^(4*b)*x^
(4*b*n)*e^(4*a) - 8*c^(2*b)*x^(2*b*n)*e^(2*a) + 1)/x^(4*b*n))*e^(-4*a)/(b*c^(4*b)*n)