\(\int x^5 \cosh ^7(a+b x^3) \sinh (a+b x^3) \, dx\) [709]

Optimal result

Integrand size = 22, antiderivative size = 129 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx=-\frac {35 x^3}{3072 b}+\frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {35 \cosh \left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{3072 b^2}-\frac {35 \cosh ^3\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{4608 b^2}-\frac {7 \cosh ^5\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{1152 b^2}-\frac {\cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right )}{192 b^2} \] Output:

-35/3072*x^3/b+1/24*x^3*cosh(b*x^3+a)^8/b-35/3072*cosh(b*x^3+a)*sinh(b*x^3 
+a)/b^2-35/4608*cosh(b*x^3+a)^3*sinh(b*x^3+a)/b^2-7/1152*cosh(b*x^3+a)^5*s 
inh(b*x^3+a)/b^2-1/192*cosh(b*x^3+a)^7*sinh(b*x^3+a)/b^2
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.93 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx=\frac {1344 b x^3 \cosh \left (2 \left (a+b x^3\right )\right )+672 b x^3 \cosh \left (4 \left (a+b x^3\right )\right )+192 b x^3 \cosh \left (6 \left (a+b x^3\right )\right )+24 b x^3 \cosh \left (8 \left (a+b x^3\right )\right )-672 \sinh \left (2 \left (a+b x^3\right )\right )-168 \sinh \left (4 \left (a+b x^3\right )\right )-32 \sinh \left (6 \left (a+b x^3\right )\right )-3 \sinh \left (8 \left (a+b x^3\right )\right )}{73728 b^2} \] Input:

Integrate[x^5*Cosh[a + b*x^3]^7*Sinh[a + b*x^3],x]
 

Output:

(1344*b*x^3*Cosh[2*(a + b*x^3)] + 672*b*x^3*Cosh[4*(a + b*x^3)] + 192*b*x^ 
3*Cosh[6*(a + b*x^3)] + 24*b*x^3*Cosh[8*(a + b*x^3)] - 672*Sinh[2*(a + b*x 
^3)] - 168*Sinh[4*(a + b*x^3)] - 32*Sinh[6*(a + b*x^3)] - 3*Sinh[8*(a + b* 
x^3)])/(73728*b^2)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.16, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5896, 5844, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 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.

\(\displaystyle \int x^5 \sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right ) \, dx\)

\(\Big \downarrow \) 5896

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\int x^2 \cosh ^8\left (b x^3+a\right )dx}{8 b}\)

\(\Big \downarrow \) 5844

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\int \cosh ^8\left (b x^3+a\right )dx^3}{24 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\int \sin \left (i b x^3+i a+\frac {\pi }{2}\right )^8dx^3}{24 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {7}{8} \int \cosh ^6\left (b x^3+a\right )dx^3+\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}}{24 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}+\frac {7}{8} \int \sin \left (i b x^3+i a+\frac {\pi }{2}\right )^6dx^3}{24 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {7}{8} \left (\frac {5}{6} \int \cosh ^4\left (b x^3+a\right )dx^3+\frac {\sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{6 b}\right )+\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}}{24 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}+\frac {7}{8} \left (\frac {\sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{6 b}+\frac {5}{6} \int \sin \left (i b x^3+i a+\frac {\pi }{2}\right )^4dx^3\right )}{24 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cosh ^2\left (b x^3+a\right )dx^3+\frac {\sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4 b}\right )+\frac {\sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{6 b}\right )+\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}}{24 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}+\frac {7}{8} \left (\frac {\sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{6 b}+\frac {5}{6} \left (\frac {\sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4 b}+\frac {3}{4} \int \sin \left (i b x^3+i a+\frac {\pi }{2}\right )^2dx^3\right )\right )}{24 b}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx^3}{2}+\frac {\sinh \left (a+b x^3\right ) \cosh \left (a+b x^3\right )}{2 b}\right )+\frac {\sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4 b}\right )+\frac {\sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{6 b}\right )+\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}}{24 b}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {x^3 \cosh ^8\left (a+b x^3\right )}{24 b}-\frac {\frac {\sinh \left (a+b x^3\right ) \cosh ^7\left (a+b x^3\right )}{8 b}+\frac {7}{8} \left (\frac {\sinh \left (a+b x^3\right ) \cosh ^5\left (a+b x^3\right )}{6 b}+\frac {5}{6} \left (\frac {\sinh \left (a+b x^3\right ) \cosh ^3\left (a+b x^3\right )}{4 b}+\frac {3}{4} \left (\frac {\sinh \left (a+b x^3\right ) \cosh \left (a+b x^3\right )}{2 b}+\frac {x^3}{2}\right )\right )\right )}{24 b}\)

Input:

Int[x^5*Cosh[a + b*x^3]^7*Sinh[a + b*x^3],x]
 

Output:

(x^3*Cosh[a + b*x^3]^8)/(24*b) - ((Cosh[a + b*x^3]^7*Sinh[a + b*x^3])/(8*b 
) + (7*((Cosh[a + b*x^3]^5*Sinh[a + b*x^3])/(6*b) + (5*((Cosh[a + b*x^3]^3 
*Sinh[a + b*x^3])/(4*b) + (3*(x^3/2 + (Cosh[a + b*x^3]*Sinh[a + b*x^3])/(2 
*b)))/4))/6))/8)/(24*b)
 

Defintions of rubi rules used

rule 24

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 5844

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x] 
)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplif 
y[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplif 
y[(m + 1)/n], 0]))
 

rule 5896

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_) 
^(n_.)], x_Symbol] :> Simp[x^(m - n + 1)*(Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 
 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Cosh[a + b*x^n]^ 
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.50

\[\frac {\left (8 b \,x^{3}-1\right ) {\mathrm e}^{8 b \,x^{3}+8 a}}{49152 b^{2}}+\frac {\left (6 b \,x^{3}-1\right ) {\mathrm e}^{6 b \,x^{3}+6 a}}{4608 b^{2}}+\frac {7 \left (4 b \,x^{3}-1\right ) {\mathrm e}^{4 b \,x^{3}+4 a}}{6144 b^{2}}+\frac {7 \left (2 b \,x^{3}-1\right ) {\mathrm e}^{2 b \,x^{3}+2 a}}{1536 b^{2}}+\frac {7 \left (2 b \,x^{3}+1\right ) {\mathrm e}^{-2 b \,x^{3}-2 a}}{1536 b^{2}}+\frac {7 \left (4 b \,x^{3}+1\right ) {\mathrm e}^{-4 b \,x^{3}-4 a}}{6144 b^{2}}+\frac {\left (6 b \,x^{3}+1\right ) {\mathrm e}^{-6 b \,x^{3}-6 a}}{4608 b^{2}}+\frac {\left (8 b \,x^{3}+1\right ) {\mathrm e}^{-8 b \,x^{3}-8 a}}{49152 b^{2}}\]

Input:

int(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x)
 

Output:

1/49152*(8*b*x^3-1)/b^2*exp(8*b*x^3+8*a)+1/4608*(6*b*x^3-1)/b^2*exp(6*b*x^ 
3+6*a)+7/6144*(4*b*x^3-1)/b^2*exp(4*b*x^3+4*a)+7/1536*(2*b*x^3-1)/b^2*exp( 
2*b*x^3+2*a)+7/1536*(2*b*x^3+1)/b^2*exp(-2*b*x^3-2*a)+7/6144*(4*b*x^3+1)/b 
^2*exp(-4*b*x^3-4*a)+1/4608*(6*b*x^3+1)/b^2*exp(-6*b*x^3-6*a)+1/49152*(8*b 
*x^3+1)/b^2*exp(-8*b*x^3-8*a)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (117) = 234\).

Time = 0.09 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.07 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx=\frac {3 \, b x^{3} \cosh \left (b x^{3} + a\right )^{8} + 3 \, b x^{3} \sinh \left (b x^{3} + a\right )^{8} + 24 \, b x^{3} \cosh \left (b x^{3} + a\right )^{6} + 84 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} - 3 \, \cosh \left (b x^{3} + a\right ) \sinh \left (b x^{3} + a\right )^{7} + 12 \, {\left (7 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 2 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{6} + 168 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} - 3 \, {\left (7 \, \cosh \left (b x^{3} + a\right )^{3} + 8 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )^{5} + 6 \, {\left (35 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} + 60 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 14 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{4} - {\left (21 \, \cosh \left (b x^{3} + a\right )^{5} + 80 \, \cosh \left (b x^{3} + a\right )^{3} + 84 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )^{3} + 12 \, {\left (7 \, b x^{3} \cosh \left (b x^{3} + a\right )^{6} + 30 \, b x^{3} \cosh \left (b x^{3} + a\right )^{4} + 42 \, b x^{3} \cosh \left (b x^{3} + a\right )^{2} + 14 \, b x^{3}\right )} \sinh \left (b x^{3} + a\right )^{2} - 3 \, {\left (\cosh \left (b x^{3} + a\right )^{7} + 8 \, \cosh \left (b x^{3} + a\right )^{5} + 28 \, \cosh \left (b x^{3} + a\right )^{3} + 56 \, \cosh \left (b x^{3} + a\right )\right )} \sinh \left (b x^{3} + a\right )}{9216 \, b^{2}} \] Input:

integrate(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x, algorithm="fricas")
 

Output:

1/9216*(3*b*x^3*cosh(b*x^3 + a)^8 + 3*b*x^3*sinh(b*x^3 + a)^8 + 24*b*x^3*c 
osh(b*x^3 + a)^6 + 84*b*x^3*cosh(b*x^3 + a)^4 - 3*cosh(b*x^3 + a)*sinh(b*x 
^3 + a)^7 + 12*(7*b*x^3*cosh(b*x^3 + a)^2 + 2*b*x^3)*sinh(b*x^3 + a)^6 + 1 
68*b*x^3*cosh(b*x^3 + a)^2 - 3*(7*cosh(b*x^3 + a)^3 + 8*cosh(b*x^3 + a))*s 
inh(b*x^3 + a)^5 + 6*(35*b*x^3*cosh(b*x^3 + a)^4 + 60*b*x^3*cosh(b*x^3 + a 
)^2 + 14*b*x^3)*sinh(b*x^3 + a)^4 - (21*cosh(b*x^3 + a)^5 + 80*cosh(b*x^3 
+ a)^3 + 84*cosh(b*x^3 + a))*sinh(b*x^3 + a)^3 + 12*(7*b*x^3*cosh(b*x^3 + 
a)^6 + 30*b*x^3*cosh(b*x^3 + a)^4 + 42*b*x^3*cosh(b*x^3 + a)^2 + 14*b*x^3) 
*sinh(b*x^3 + a)^2 - 3*(cosh(b*x^3 + a)^7 + 8*cosh(b*x^3 + a)^5 + 28*cosh( 
b*x^3 + a)^3 + 56*cosh(b*x^3 + a))*sinh(b*x^3 + a))/b^2
 

Sympy [A] (verification not implemented)

Time = 3.27 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.87 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx=\begin {cases} - \frac {35 x^{3} \sinh ^{8}{\left (a + b x^{3} \right )}}{3072 b} + \frac {35 x^{3} \sinh ^{6}{\left (a + b x^{3} \right )} \cosh ^{2}{\left (a + b x^{3} \right )}}{768 b} - \frac {35 x^{3} \sinh ^{4}{\left (a + b x^{3} \right )} \cosh ^{4}{\left (a + b x^{3} \right )}}{512 b} + \frac {35 x^{3} \sinh ^{2}{\left (a + b x^{3} \right )} \cosh ^{6}{\left (a + b x^{3} \right )}}{768 b} + \frac {31 x^{3} \cosh ^{8}{\left (a + b x^{3} \right )}}{1024 b} + \frac {35 \sinh ^{7}{\left (a + b x^{3} \right )} \cosh {\left (a + b x^{3} \right )}}{3072 b^{2}} - \frac {385 \sinh ^{5}{\left (a + b x^{3} \right )} \cosh ^{3}{\left (a + b x^{3} \right )}}{9216 b^{2}} + \frac {511 \sinh ^{3}{\left (a + b x^{3} \right )} \cosh ^{5}{\left (a + b x^{3} \right )}}{9216 b^{2}} - \frac {31 \sinh {\left (a + b x^{3} \right )} \cosh ^{7}{\left (a + b x^{3} \right )}}{1024 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{6} \sinh {\left (a \right )} \cosh ^{7}{\left (a \right )}}{6} & \text {otherwise} \end {cases} \] Input:

integrate(x**5*cosh(b*x**3+a)**7*sinh(b*x**3+a),x)
 

Output:

Piecewise((-35*x**3*sinh(a + b*x**3)**8/(3072*b) + 35*x**3*sinh(a + b*x**3 
)**6*cosh(a + b*x**3)**2/(768*b) - 35*x**3*sinh(a + b*x**3)**4*cosh(a + b* 
x**3)**4/(512*b) + 35*x**3*sinh(a + b*x**3)**2*cosh(a + b*x**3)**6/(768*b) 
 + 31*x**3*cosh(a + b*x**3)**8/(1024*b) + 35*sinh(a + b*x**3)**7*cosh(a + 
b*x**3)/(3072*b**2) - 385*sinh(a + b*x**3)**5*cosh(a + b*x**3)**3/(9216*b* 
*2) + 511*sinh(a + b*x**3)**3*cosh(a + b*x**3)**5/(9216*b**2) - 31*sinh(a 
+ b*x**3)*cosh(a + b*x**3)**7/(1024*b**2), Ne(b, 0)), (x**6*sinh(a)*cosh(a 
)**7/6, True))
 

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.65 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx=\frac {{\left (8 \, b x^{3} e^{\left (8 \, a\right )} - e^{\left (8 \, a\right )}\right )} e^{\left (8 \, b x^{3}\right )}}{49152 \, b^{2}} + \frac {{\left (6 \, b x^{3} e^{\left (6 \, a\right )} - e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x^{3}\right )}}{4608 \, b^{2}} + \frac {7 \, {\left (4 \, b x^{3} e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x^{3}\right )}}{6144 \, b^{2}} + \frac {7 \, {\left (2 \, b x^{3} e^{\left (2 \, a\right )} - e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x^{3}\right )}}{1536 \, b^{2}} + \frac {7 \, {\left (2 \, b x^{3} + 1\right )} e^{\left (-2 \, b x^{3} - 2 \, a\right )}}{1536 \, b^{2}} + \frac {7 \, {\left (4 \, b x^{3} + 1\right )} e^{\left (-4 \, b x^{3} - 4 \, a\right )}}{6144 \, b^{2}} + \frac {{\left (6 \, b x^{3} + 1\right )} e^{\left (-6 \, b x^{3} - 6 \, a\right )}}{4608 \, b^{2}} + \frac {{\left (8 \, b x^{3} + 1\right )} e^{\left (-8 \, b x^{3} - 8 \, a\right )}}{49152 \, b^{2}} \] Input:

integrate(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x, algorithm="maxima")
 

Output:

1/49152*(8*b*x^3*e^(8*a) - e^(8*a))*e^(8*b*x^3)/b^2 + 1/4608*(6*b*x^3*e^(6 
*a) - e^(6*a))*e^(6*b*x^3)/b^2 + 7/6144*(4*b*x^3*e^(4*a) - e^(4*a))*e^(4*b 
*x^3)/b^2 + 7/1536*(2*b*x^3*e^(2*a) - e^(2*a))*e^(2*b*x^3)/b^2 + 7/1536*(2 
*b*x^3 + 1)*e^(-2*b*x^3 - 2*a)/b^2 + 7/6144*(4*b*x^3 + 1)*e^(-4*b*x^3 - 4* 
a)/b^2 + 1/4608*(6*b*x^3 + 1)*e^(-6*b*x^3 - 6*a)/b^2 + 1/49152*(8*b*x^3 + 
1)*e^(-8*b*x^3 - 8*a)/b^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 374 vs. \(2 (117) = 234\).

Time = 0.13 (sec) , antiderivative size = 374, normalized size of antiderivative = 2.90 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx=\frac {{\left (24 \, b x^{3} + 24 \, {\left (b x^{3} + a\right )} e^{\left (16 \, b x^{3} + 16 \, a\right )} + 192 \, {\left (b x^{3} + a\right )} e^{\left (14 \, b x^{3} + 14 \, a\right )} + 672 \, {\left (b x^{3} + a\right )} e^{\left (12 \, b x^{3} + 12 \, a\right )} + 1344 \, {\left (b x^{3} + a\right )} e^{\left (10 \, b x^{3} + 10 \, a\right )} + 1344 \, {\left (b x^{3} + a\right )} e^{\left (6 \, b x^{3} + 6 \, a\right )} + 672 \, {\left (b x^{3} + a\right )} e^{\left (4 \, b x^{3} + 4 \, a\right )} + 192 \, {\left (b x^{3} + a\right )} e^{\left (2 \, b x^{3} + 2 \, a\right )} + 24 \, a - 3 \, e^{\left (16 \, b x^{3} + 16 \, a\right )} - 32 \, e^{\left (14 \, b x^{3} + 14 \, a\right )} - 168 \, e^{\left (12 \, b x^{3} + 12 \, a\right )} - 672 \, e^{\left (10 \, b x^{3} + 10 \, a\right )} + 672 \, e^{\left (6 \, b x^{3} + 6 \, a\right )} + 168 \, e^{\left (4 \, b x^{3} + 4 \, a\right )} + 32 \, e^{\left (2 \, b x^{3} + 2 \, a\right )} + 3\right )} e^{\left (-8 \, b x^{3} - 8 \, a\right )}}{147456 \, b^{2}} - \frac {a {\left (e^{\left (2 \, b x^{3} + 2 \, a\right )} + e^{\left (-2 \, b x^{3} - 2 \, a\right )}\right )}^{4} + 8 \, a {\left (e^{\left (2 \, b x^{3} + 2 \, a\right )} + e^{\left (-2 \, b x^{3} - 2 \, a\right )}\right )}^{3} + 24 \, a {\left (e^{\left (2 \, b x^{3} + 2 \, a\right )} + e^{\left (-2 \, b x^{3} - 2 \, a\right )}\right )}^{2} + 32 \, a {\left (e^{\left (2 \, b x^{3} + 2 \, a\right )} + e^{\left (-2 \, b x^{3} - 2 \, a\right )}\right )}}{6144 \, b^{2}} \] Input:

integrate(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x, algorithm="giac")
 

Output:

1/147456*(24*b*x^3 + 24*(b*x^3 + a)*e^(16*b*x^3 + 16*a) + 192*(b*x^3 + a)* 
e^(14*b*x^3 + 14*a) + 672*(b*x^3 + a)*e^(12*b*x^3 + 12*a) + 1344*(b*x^3 + 
a)*e^(10*b*x^3 + 10*a) + 1344*(b*x^3 + a)*e^(6*b*x^3 + 6*a) + 672*(b*x^3 + 
 a)*e^(4*b*x^3 + 4*a) + 192*(b*x^3 + a)*e^(2*b*x^3 + 2*a) + 24*a - 3*e^(16 
*b*x^3 + 16*a) - 32*e^(14*b*x^3 + 14*a) - 168*e^(12*b*x^3 + 12*a) - 672*e^ 
(10*b*x^3 + 10*a) + 672*e^(6*b*x^3 + 6*a) + 168*e^(4*b*x^3 + 4*a) + 32*e^( 
2*b*x^3 + 2*a) + 3)*e^(-8*b*x^3 - 8*a)/b^2 - 1/6144*(a*(e^(2*b*x^3 + 2*a) 
+ e^(-2*b*x^3 - 2*a))^4 + 8*a*(e^(2*b*x^3 + 2*a) + e^(-2*b*x^3 - 2*a))^3 + 
 24*a*(e^(2*b*x^3 + 2*a) + e^(-2*b*x^3 - 2*a))^2 + 32*a*(e^(2*b*x^3 + 2*a) 
 + e^(-2*b*x^3 - 2*a)))/b^2
 

Mupad [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.65 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx={\mathrm {e}}^{-2\,b\,x^3-2\,a}\,\left (\frac {7}{1536\,b^2}+\frac {7\,x^3}{768\,b}\right )-{\mathrm {e}}^{2\,b\,x^3+2\,a}\,\left (\frac {7}{1536\,b^2}-\frac {7\,x^3}{768\,b}\right )+{\mathrm {e}}^{-6\,b\,x^3-6\,a}\,\left (\frac {1}{4608\,b^2}+\frac {x^3}{768\,b}\right )-{\mathrm {e}}^{6\,b\,x^3+6\,a}\,\left (\frac {1}{4608\,b^2}-\frac {x^3}{768\,b}\right )+{\mathrm {e}}^{-4\,b\,x^3-4\,a}\,\left (\frac {7}{6144\,b^2}+\frac {7\,x^3}{1536\,b}\right )-{\mathrm {e}}^{4\,b\,x^3+4\,a}\,\left (\frac {7}{6144\,b^2}-\frac {7\,x^3}{1536\,b}\right )+{\mathrm {e}}^{-8\,b\,x^3-8\,a}\,\left (\frac {1}{49152\,b^2}+\frac {x^3}{6144\,b}\right )-{\mathrm {e}}^{8\,b\,x^3+8\,a}\,\left (\frac {1}{49152\,b^2}-\frac {x^3}{6144\,b}\right ) \] Input:

int(x^5*cosh(a + b*x^3)^7*sinh(a + b*x^3),x)
 

Output:

exp(- 2*a - 2*b*x^3)*(7/(1536*b^2) + (7*x^3)/(768*b)) - exp(2*a + 2*b*x^3) 
*(7/(1536*b^2) - (7*x^3)/(768*b)) + exp(- 6*a - 6*b*x^3)*(1/(4608*b^2) + x 
^3/(768*b)) - exp(6*a + 6*b*x^3)*(1/(4608*b^2) - x^3/(768*b)) + exp(- 4*a 
- 4*b*x^3)*(7/(6144*b^2) + (7*x^3)/(1536*b)) - exp(4*a + 4*b*x^3)*(7/(6144 
*b^2) - (7*x^3)/(1536*b)) + exp(- 8*a - 8*b*x^3)*(1/(49152*b^2) + x^3/(614 
4*b)) - exp(8*a + 8*b*x^3)*(1/(49152*b^2) - x^3/(6144*b))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.95 \[ \int x^5 \cosh ^7\left (a+b x^3\right ) \sinh \left (a+b x^3\right ) \, dx=\frac {24 e^{16 b \,x^{3}+16 a} b \,x^{3}-3 e^{16 b \,x^{3}+16 a}+192 e^{14 b \,x^{3}+14 a} b \,x^{3}-32 e^{14 b \,x^{3}+14 a}+672 e^{12 b \,x^{3}+12 a} b \,x^{3}-168 e^{12 b \,x^{3}+12 a}+1344 e^{10 b \,x^{3}+10 a} b \,x^{3}-672 e^{10 b \,x^{3}+10 a}+1344 e^{6 b \,x^{3}+6 a} b \,x^{3}+672 e^{6 b \,x^{3}+6 a}+672 e^{4 b \,x^{3}+4 a} b \,x^{3}+168 e^{4 b \,x^{3}+4 a}+192 e^{2 b \,x^{3}+2 a} b \,x^{3}+32 e^{2 b \,x^{3}+2 a}+24 b \,x^{3}+3}{147456 e^{8 b \,x^{3}+8 a} b^{2}} \] Input:

int(x^5*cosh(b*x^3+a)^7*sinh(b*x^3+a),x)
 

Output:

(24*e**(16*a + 16*b*x**3)*b*x**3 - 3*e**(16*a + 16*b*x**3) + 192*e**(14*a 
+ 14*b*x**3)*b*x**3 - 32*e**(14*a + 14*b*x**3) + 672*e**(12*a + 12*b*x**3) 
*b*x**3 - 168*e**(12*a + 12*b*x**3) + 1344*e**(10*a + 10*b*x**3)*b*x**3 - 
672*e**(10*a + 10*b*x**3) + 1344*e**(6*a + 6*b*x**3)*b*x**3 + 672*e**(6*a 
+ 6*b*x**3) + 672*e**(4*a + 4*b*x**3)*b*x**3 + 168*e**(4*a + 4*b*x**3) + 1 
92*e**(2*a + 2*b*x**3)*b*x**3 + 32*e**(2*a + 2*b*x**3) + 24*b*x**3 + 3)/(1 
47456*e**(8*a + 8*b*x**3)*b**2)