3.269 \(\int \sinh ^4(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=191 \[ \frac{12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4-20 b^2 n^2+1}+\frac{x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac{4 b n x \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac{24 b^3 n^3 x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4-20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4-20 b^2 n^2+1} \]

[Out]

(24*b^4*n^4*x)/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(1
- 20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Sinh[a + b*Log[c*x^n]]^2)/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (4*b*n*x
*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^3)/(1 - 16*b^2*n^2) + (x*Sinh[a + b*Log[c*x^n]]^4)/(1 - 16*b^2*
n^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0512167, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5519, 8} \[ \frac{12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4-20 b^2 n^2+1}+\frac{x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac{4 b n x \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac{24 b^3 n^3 x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4-20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4-20 b^2 n^2+1} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*Log[c*x^n]]^4,x]

[Out]

(24*b^4*n^4*x)/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (24*b^3*n^3*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/(1
- 20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Sinh[a + b*Log[c*x^n]]^2)/(1 - 20*b^2*n^2 + 64*b^4*n^4) - (4*b*n*x
*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]]^3)/(1 - 16*b^2*n^2) + (x*Sinh[a + b*Log[c*x^n]]^4)/(1 - 16*b^2*
n^2)

Rule 5519

Int[Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_), x_Symbol] :> -Simp[(x*Sinh[d*(a + b*Log[c*x^n])]^p
)/(b^2*d^2*n^2*p^2 - 1), x] + (-Dist[(b^2*d^2*n^2*p*(p - 1))/(b^2*d^2*n^2*p^2 - 1), Int[Sinh[d*(a + b*Log[c*x^
n])]^(p - 2), x], x] + Simp[(b*d*n*p*x*Cosh[d*(a + b*Log[c*x^n])]*Sinh[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2
*n^2*p^2 - 1), x]) /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac{4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac{x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac{\left (12 b^2 n^2\right ) \int \sinh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{1-16 b^2 n^2}\\ &=-\frac{24 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}+\frac{12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}-\frac{4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac{x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac{\left (24 b^4 n^4\right ) \int 1 \, dx}{1-20 b^2 n^2+64 b^4 n^4}\\ &=\frac{24 b^4 n^4 x}{1-20 b^2 n^2+64 b^4 n^4}-\frac{24 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}+\frac{12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}-\frac{4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac{x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}\\ \end{align*}

Mathematica [A]  time = 0.41787, size = 167, normalized size = 0.87 \[ \frac{x \left (-128 b^3 n^3 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+\left (64 b^2 n^2-4\right ) \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (1-4 b^2 n^2\right ) \cosh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+192 b^4 n^4-60 b^2 n^2+3\right )}{8 \left (64 b^4 n^4-20 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*Log[c*x^n]]^4,x]

[Out]

(x*(3 - 60*b^2*n^2 + 192*b^4*n^4 + (-4 + 64*b^2*n^2)*Cosh[2*(a + b*Log[c*x^n])] + (1 - 4*b^2*n^2)*Cosh[4*(a +
b*Log[c*x^n])] + 8*b*n*Sinh[2*(a + b*Log[c*x^n])] - 128*b^3*n^3*Sinh[2*(a + b*Log[c*x^n])] - 4*b*n*Sinh[4*(a +
b*Log[c*x^n])] + 16*b^3*n^3*Sinh[4*(a + b*Log[c*x^n])]))/(8*(1 - 20*b^2*n^2 + 64*b^4*n^4))

________________________________________________________________________________________

Maple [F]  time = 0.131, size = 0, normalized size = 0. \begin{align*} \int \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.25793, size = 174, normalized size = 0.91 \begin{align*} \frac{c^{4 \, b} x e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )}}{16 \,{\left (4 \, b n + 1\right )}} - \frac{c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}}{4 \,{\left (2 \, b n + 1\right )}} + \frac{3}{8} \, x + \frac{x e^{\left (-2 \, b \log \left (x^{n}\right ) - 2 \, a\right )}}{4 \,{\left (2 \, b c^{2 \, b} n - c^{2 \, b}\right )}} - \frac{x e^{\left (-4 \, a\right )}}{16 \,{\left (4 \, b c^{4 \, b} n - c^{4 \, b}\right )}{\left (x^{n}\right )}^{4 \, b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*c^(4*b)*x*e^(4*b*log(x^n) + 4*a)/(4*b*n + 1) - 1/4*c^(2*b)*x*e^(2*b*log(x^n) + 2*a)/(2*b*n + 1) + 3/8*x +
1/4*x*e^(-2*b*log(x^n) - 2*a)/(2*b*c^(2*b)*n - c^(2*b)) - 1/16*x*e^(-4*a)/((4*b*c^(4*b)*n - c^(4*b))*(x^n)^(4
*b))

________________________________________________________________________________________

Fricas [A]  time = 2.13351, size = 801, normalized size = 4.19 \begin{align*} -\frac{{\left (4 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 16 \,{\left (4 \, b^{3} n^{3} - b n\right )} x \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} +{\left (4 \, b^{2} n^{2} - 1\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 4 \,{\left (16 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \,{\left (3 \,{\left (4 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \,{\left (16 \, b^{2} n^{2} - 1\right )} x\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \,{\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} x - 16 \,{\left ({\left (4 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} -{\left (16 \, b^{3} n^{3} - b n\right )} x \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 \,{\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*((4*b^2*n^2 - 1)*x*cosh(b*n*log(x) + b*log(c) + a)^4 - 16*(4*b^3*n^3 - b*n)*x*cosh(b*n*log(x) + b*log(c)
+ a)*sinh(b*n*log(x) + b*log(c) + a)^3 + (4*b^2*n^2 - 1)*x*sinh(b*n*log(x) + b*log(c) + a)^4 - 4*(16*b^2*n^2 -
1)*x*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*(3*(4*b^2*n^2 - 1)*x*cosh(b*n*log(x) + b*log(c) + a)^2 - 2*(16*b^2
*n^2 - 1)*x)*sinh(b*n*log(x) + b*log(c) + a)^2 - 3*(64*b^4*n^4 - 20*b^2*n^2 + 1)*x - 16*((4*b^3*n^3 - b*n)*x*c
osh(b*n*log(x) + b*log(c) + a)^3 - (16*b^3*n^3 - b*n)*x*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*l
og(c) + a))/(64*b^4*n^4 - 20*b^2*n^2 + 1)

________________________________________________________________________________________

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)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.2795, size = 1049, normalized size = 5.49 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^3*c^(4*b)*n^3*x*x^(4*b*n)*e^(4*a)/(64*b^4*n^4 - 20*b^2*n^2 + 1) - 8*b^3*c^(2*b)*n^3*x*x^(2*b*n)*e^(2*a)/(64*
b^4*n^4 - 20*b^2*n^2 + 1) + 24*b^4*n^4*x/(64*b^4*n^4 - 20*b^2*n^2 + 1) - 1/4*b^2*c^(4*b)*n^2*x*x^(4*b*n)*e^(4*
a)/(64*b^4*n^4 - 20*b^2*n^2 + 1) + 4*b^2*c^(2*b)*n^2*x*x^(2*b*n)*e^(2*a)/(64*b^4*n^4 - 20*b^2*n^2 + 1) - 1/4*b
*c^(4*b)*n*x*x^(4*b*n)*e^(4*a)/(64*b^4*n^4 - 20*b^2*n^2 + 1) + 1/2*b*c^(2*b)*n*x*x^(2*b*n)*e^(2*a)/(64*b^4*n^4
- 20*b^2*n^2 + 1) + 8*b^3*n^3*x*e^(-2*a)/((64*b^4*n^4 - 20*b^2*n^2 + 1)*c^(2*b)*x^(2*b*n)) - b^3*n^3*x*e^(-4*
a)/((64*b^4*n^4 - 20*b^2*n^2 + 1)*c^(4*b)*x^(4*b*n)) - 15/2*b^2*n^2*x/(64*b^4*n^4 - 20*b^2*n^2 + 1) + 1/16*c^(
4*b)*x*x^(4*b*n)*e^(4*a)/(64*b^4*n^4 - 20*b^2*n^2 + 1) - 1/4*c^(2*b)*x*x^(2*b*n)*e^(2*a)/(64*b^4*n^4 - 20*b^2*
n^2 + 1) + 4*b^2*n^2*x*e^(-2*a)/((64*b^4*n^4 - 20*b^2*n^2 + 1)*c^(2*b)*x^(2*b*n)) - 1/4*b^2*n^2*x*e^(-4*a)/((6
4*b^4*n^4 - 20*b^2*n^2 + 1)*c^(4*b)*x^(4*b*n)) - 1/2*b*n*x*e^(-2*a)/((64*b^4*n^4 - 20*b^2*n^2 + 1)*c^(2*b)*x^(
2*b*n)) + 1/4*b*n*x*e^(-4*a)/((64*b^4*n^4 - 20*b^2*n^2 + 1)*c^(4*b)*x^(4*b*n)) + 3/8*x/(64*b^4*n^4 - 20*b^2*n^
2 + 1) - 1/4*x*e^(-2*a)/((64*b^4*n^4 - 20*b^2*n^2 + 1)*c^(2*b)*x^(2*b*n)) + 1/16*x*e^(-4*a)/((64*b^4*n^4 - 20*
b^2*n^2 + 1)*c^(4*b)*x^(4*b*n))