Integrand size = 13, antiderivative size = 191 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {24 b^4 n^4 x}{1-20 b^2 n^2+64 b^4 n^4}-\frac {24 b^3 n^3 x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}+\frac {12 b^2 n^2 x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-20 b^2 n^2+64 b^4 n^4}-\frac {4 b n x \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2} \] Output:
24*b^4*n^4*x/(64*b^4*n^4-20*b^2*n^2+1)-24*b^3*n^3*x*cosh(a+b*ln(c*x^n))*si nh(a+b*ln(c*x^n))/(64*b^4*n^4-20*b^2*n^2+1)+12*b^2*n^2*x*sinh(a+b*ln(c*x^n ))^2/(64*b^4*n^4-20*b^2*n^2+1)-4*b*n*x*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c*x ^n))^3/(-16*b^2*n^2+1)+x*sinh(a+b*ln(c*x^n))^4/(-16*b^2*n^2+1)
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.87 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (3-60 b^2 n^2+192 b^4 n^4+\left (-4+64 b^2 n^2\right ) \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (1-4 b^2 n^2\right ) \cosh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-128 b^3 n^3 \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-4 b n \sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sinh \left (4 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 \left (1-20 b^2 n^2+64 b^4 n^4\right )} \] Input:
Integrate[Sinh[a + b*Log[c*x^n]]^4,x]
Output:
(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*L og[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))
Time = 0.37 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {6045, 6045, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx\) |
| \(\Big \downarrow \) 6045 |
| \(\displaystyle \frac {12 b^2 n^2 \int \sinh ^2\left (a+b \log \left (c x^n\right )\right )dx}{1-16 b^2 n^2}+\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac {4 b n x \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}\) |
| \(\Big \downarrow \) 6045 |
| \(\displaystyle \frac {12 b^2 n^2 \left (\frac {2 b^2 n^2 \int 1dx}{1-4 b^2 n^2}+\frac {x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}-\frac {2 b n x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}\right )}{1-16 b^2 n^2}+\frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac {4 b n x \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {x \sinh ^4\left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}-\frac {4 b n x \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-16 b^2 n^2}+\frac {12 b^2 n^2 \left (\frac {x \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}-\frac {2 b n x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{1-4 b^2 n^2}+\frac {2 b^2 n^2 x}{1-4 b^2 n^2}\right )}{1-16 b^2 n^2}\) |
Input:
Int[Sinh[a + b*Log[c*x^n]]^4,x]
Output:
(-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) + (12*b^2*n^2*((2*b^2*n^ 2*x)/(1 - 4*b^2*n^2) - (2*b*n*x*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^ n]])/(1 - 4*b^2*n^2) + (x*Sinh[a + b*Log[c*x^n]]^2)/(1 - 4*b^2*n^2)))/(1 - 16*b^2*n^2)
Int[Sinh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_), x_Symbol] :> Si mp[(-x)*(Sinh[d*(a + b*Log[c*x^n])]^p/(b^2*d^2*n^2*p^2 - 1)), 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] - Simp[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]) /; FreeQ[{a, b, c, d, n} , x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - 1, 0]
Time = 20.38 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.96
| method | result | size |
| parallelrisch | \(\frac {x \left (4 \left (16 b^{2} n^{2}-1\right ) \cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+192 b^{4} n^{4}-128 b^{3} n^{3} \sinh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+16 b^{3} n^{3} \sinh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )-4 b^{2} n^{2} \cosh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )-60 b^{2} n^{2}+8 b n \sinh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )-4 b n \sinh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+\cosh \left (4 b \ln \left (c \,x^{n}\right )+4 a \right )+3\right )}{512 b^{4} n^{4}-160 b^{2} n^{2}+8}\) | \(184\) |
Input:
int(sinh(a+b*ln(c*x^n))^4,x,method=_RETURNVERBOSE)
Output:
1/8*x*(4*(16*b^2*n^2-1)*cosh(2*b*ln(c*x^n)+2*a)+192*b^4*n^4-128*b^3*n^3*si nh(2*b*ln(c*x^n)+2*a)+16*b^3*n^3*sinh(4*b*ln(c*x^n)+4*a)-4*b^2*n^2*cosh(4* b*ln(c*x^n)+4*a)-60*b^2*n^2+8*b*n*sinh(2*b*ln(c*x^n)+2*a)-4*b*n*sinh(4*b*l n(c*x^n)+4*a)+cosh(4*b*ln(c*x^n)+4*a)+3)/(64*b^4*n^4-20*b^2*n^2+1)
Time = 0.10 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.54 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {{\left (4 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 16 \, {\left (4 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (4 \, b^{2} n^{2} - 1\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 4 \, {\left (16 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, {\left (4 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, {\left (16 \, b^{2} n^{2} - 1\right )} x\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )} x - 16 \, {\left ({\left (4 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - {\left (16 \, b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{8 \, {\left (64 \, b^{4} n^{4} - 20 \, b^{2} n^{2} + 1\right )}} \] Input:
integrate(sinh(a+b*log(c*x^n))^4,x, algorithm="fricas")
Output:
-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*lo g(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*cosh( b*n*log(x) + b*log(c) + a)^3 - (16*b^3*n^3 - b*n)*x*cosh(b*n*log(x) + b*lo g(c) + a))*sinh(b*n*log(x) + b*log(c) + a))/(64*b^4*n^4 - 20*b^2*n^2 + 1)
\[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \int \sinh ^{4}{\left (a - \frac {\log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = - \frac {1}{2 n} \\\int \sinh ^{4}{\left (a - \frac {\log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = - \frac {1}{4 n} \\\int \sinh ^{4}{\left (a + \frac {\log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = \frac {1}{4 n} \\\int \sinh ^{4}{\left (a + \frac {\log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {1}{2 n} \\\frac {24 b^{4} n^{4} x \sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} - \frac {48 b^{4} n^{4} x \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\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 \cosh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} + \frac {40 b^{3} n^{3} x \sinh ^{3}{\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^{3} n^{3} x \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} - \frac {16 b^{2} n^{2} x \sinh ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} + \frac {12 b^{2} n^{2} x \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} - \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 )}}{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 )}}{64 b^{4} n^{4} - 20 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \] Input:
integrate(sinh(a+b*ln(c*x**n))**4,x)
Output:
Piecewise((Integral(sinh(a - log(c*x**n)/(2*n))**4, x), Eq(b, -1/(2*n))), (Integral(sinh(a - log(c*x**n)/(4*n))**4, x), Eq(b, -1/(4*n))), (Integral( sinh(a + log(c*x**n)/(4*n))**4, x), Eq(b, 1/(4*n))), (Integral(sinh(a + lo g(c*x**n)/(2*n))**4, x), Eq(b, 1/(2*n))), (24*b**4*n**4*x*sinh(a + b*log(c *x**n))**4/(64*b**4*n**4 - 20*b**2*n**2 + 1) - 48*b**4*n**4*x*sinh(a + b*l og(c*x**n))**2*cosh(a + b*log(c*x**n))**2/(64*b**4*n**4 - 20*b**2*n**2 + 1 ) + 24*b**4*n**4*x*cosh(a + b*log(c*x**n))**4/(64*b**4*n**4 - 20*b**2*n**2 + 1) + 40*b**3*n**3*x*sinh(a + b*log(c*x**n))**3*cosh(a + b*log(c*x**n))/ (64*b**4*n**4 - 20*b**2*n**2 + 1) - 24*b**3*n**3*x*sinh(a + b*log(c*x**n)) *cosh(a + b*log(c*x**n))**3/(64*b**4*n**4 - 20*b**2*n**2 + 1) - 16*b**2*n* *2*x*sinh(a + b*log(c*x**n))**4/(64*b**4*n**4 - 20*b**2*n**2 + 1) + 12*b** 2*n**2*x*sinh(a + b*log(c*x**n))**2*cosh(a + b*log(c*x**n))**2/(64*b**4*n* *4 - 20*b**2*n**2 + 1) - 4*b*n*x*sinh(a + b*log(c*x**n))**3*cosh(a + b*log (c*x**n))/(64*b**4*n**4 - 20*b**2*n**2 + 1) + x*sinh(a + b*log(c*x**n))**4 /(64*b**4*n**4 - 20*b**2*n**2 + 1), True))
Time = 0.09 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.68 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {c^{4 \, b} x e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )}}{16 \, {\left (4 \, b n + 1\right )}} - \frac {c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}}{4 \, {\left (2 \, b n + 1\right )}} + \frac {3}{8} \, x + \frac {x e^{\left (-2 \, b \log \left (x^{n}\right ) - 2 \, a\right )}}{4 \, {\left (2 \, b c^{2 \, b} n - c^{2 \, b}\right )}} - \frac {x e^{\left (-4 \, a\right )}}{16 \, {\left (4 \, b c^{4 \, b} n - c^{4 \, b}\right )} {\left (x^{n}\right )}^{4 \, b}} \] Input:
integrate(sinh(a+b*log(c*x^n))^4,x, algorithm="maxima")
Output:
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*l og(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 ))
Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (192) = 384\).
Time = 0.15 (sec) , antiderivative size = 777, normalized size of antiderivative = 4.07 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx =\text {Too large to display} \] Input:
integrate(sinh(a+b*log(c*x^n))^4,x, algorithm="giac")
Output:
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 - 2 0*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)/((64*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))
Time = 1.60 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.53 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {3\,x}{8}+\frac {x\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}\,\left (8\,b\,n-4\right )}-\frac {x\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}{8\,b\,n+4}-\frac {x\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}\,\left (64\,b\,n-16\right )}+\frac {x\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}{64\,b\,n+16} \] Input:
int(sinh(a + b*log(c*x^n))^4,x)
Output:
(3*x)/8 + (x*exp(-2*a))/((c*x^n)^(2*b)*(8*b*n - 4)) - (x*exp(2*a)*(c*x^n)^ (2*b))/(8*b*n + 4) - (x*exp(-4*a))/((c*x^n)^(4*b)*(64*b*n - 16)) + (x*exp( 4*a)*(c*x^n)^(4*b))/(64*b*n + 16)
Time = 0.16 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.04 \[ \int \sinh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x \left (16 x^{8 b n} e^{8 a} c^{8 b} b^{3} n^{3}-4 x^{8 b n} e^{8 a} c^{8 b} b^{2} n^{2}-4 x^{8 b n} e^{8 a} c^{8 b} b n +x^{8 b n} e^{8 a} c^{8 b}-128 x^{6 b n} e^{6 a} c^{6 b} b^{3} n^{3}+64 x^{6 b n} e^{6 a} c^{6 b} b^{2} n^{2}+8 x^{6 b n} e^{6 a} c^{6 b} b n -4 x^{6 b n} e^{6 a} c^{6 b}+384 x^{4 b n} e^{4 a} c^{4 b} b^{4} n^{4}-120 x^{4 b n} e^{4 a} c^{4 b} b^{2} n^{2}+6 x^{4 b n} e^{4 a} c^{4 b}+128 x^{2 b n} e^{2 a} c^{2 b} b^{3} n^{3}+64 x^{2 b n} e^{2 a} c^{2 b} b^{2} n^{2}-8 x^{2 b n} e^{2 a} c^{2 b} b n -4 x^{2 b n} e^{2 a} c^{2 b}-16 b^{3} n^{3}-4 b^{2} n^{2}+4 b n +1\right )}{16 x^{4 b n} e^{4 a} c^{4 b} \left (64 b^{4} n^{4}-20 b^{2} n^{2}+1\right )} \] Input:
int(sinh(a+b*log(c*x^n))^4,x)
Output:
(x*(16*x**(8*b*n)*e**(8*a)*c**(8*b)*b**3*n**3 - 4*x**(8*b*n)*e**(8*a)*c**( 8*b)*b**2*n**2 - 4*x**(8*b*n)*e**(8*a)*c**(8*b)*b*n + x**(8*b*n)*e**(8*a)* c**(8*b) - 128*x**(6*b*n)*e**(6*a)*c**(6*b)*b**3*n**3 + 64*x**(6*b*n)*e**( 6*a)*c**(6*b)*b**2*n**2 + 8*x**(6*b*n)*e**(6*a)*c**(6*b)*b*n - 4*x**(6*b*n )*e**(6*a)*c**(6*b) + 384*x**(4*b*n)*e**(4*a)*c**(4*b)*b**4*n**4 - 120*x** (4*b*n)*e**(4*a)*c**(4*b)*b**2*n**2 + 6*x**(4*b*n)*e**(4*a)*c**(4*b) + 128 *x**(2*b*n)*e**(2*a)*c**(2*b)*b**3*n**3 + 64*x**(2*b*n)*e**(2*a)*c**(2*b)* b**2*n**2 - 8*x**(2*b*n)*e**(2*a)*c**(2*b)*b*n - 4*x**(2*b*n)*e**(2*a)*c** (2*b) - 16*b**3*n**3 - 4*b**2*n**2 + 4*b*n + 1))/(16*x**(4*b*n)*e**(4*a)*c **(4*b)*(64*b**4*n**4 - 20*b**2*n**2 + 1))