3.326 \(\int x^3 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx\)

Optimal. Leaf size=143 \[ \frac {9 \sinh (2 a+2 b x)}{256 b^4}-\frac {\sinh (6 a+6 b x)}{6912 b^4}-\frac {9 x \cosh (2 a+2 b x)}{128 b^3}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x^3 \cosh (6 a+6 b x)}{192 b} \]

[Out]

-9/128*x*cosh(2*b*x+2*a)/b^3-3/64*x^3*cosh(2*b*x+2*a)/b+1/1152*x*cosh(6*b*x+6*a)/b^3+1/192*x^3*cosh(6*b*x+6*a)
/b+9/256*sinh(2*b*x+2*a)/b^4+9/128*x^2*sinh(2*b*x+2*a)/b^2-1/6912*sinh(6*b*x+6*a)/b^4-1/384*x^2*sinh(6*b*x+6*a
)/b^2

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5448, 3296, 2637} \[ \frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}+\frac {9 \sinh (2 a+2 b x)}{256 b^4}-\frac {\sinh (6 a+6 b x)}{6912 b^4}-\frac {9 x \cosh (2 a+2 b x)}{128 b^3}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x^3 \cosh (6 a+6 b x)}{192 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-9*x*Cosh[2*a + 2*b*x])/(128*b^3) - (3*x^3*Cosh[2*a + 2*b*x])/(64*b) + (x*Cosh[6*a + 6*b*x])/(1152*b^3) + (x^
3*Cosh[6*a + 6*b*x])/(192*b) + (9*Sinh[2*a + 2*b*x])/(256*b^4) + (9*x^2*Sinh[2*a + 2*b*x])/(128*b^2) - Sinh[6*
a + 6*b*x]/(6912*b^4) - (x^2*Sinh[6*a + 6*b*x])/(384*b^2)

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^3 \cosh ^3(a+b x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{32} x^3 \sinh (2 a+2 b x)+\frac {1}{32} x^3 \sinh (6 a+6 b x)\right ) \, dx\\ &=\frac {1}{32} \int x^3 \sinh (6 a+6 b x) \, dx-\frac {3}{32} \int x^3 \sinh (2 a+2 b x) \, dx\\ &=-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}-\frac {\int x^2 \cosh (6 a+6 b x) \, dx}{64 b}+\frac {9 \int x^2 \cosh (2 a+2 b x) \, dx}{64 b}\\ &=-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}+\frac {\int x \sinh (6 a+6 b x) \, dx}{192 b^2}-\frac {9 \int x \sinh (2 a+2 b x) \, dx}{64 b^2}\\ &=-\frac {9 x \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}-\frac {\int \cosh (6 a+6 b x) \, dx}{1152 b^3}+\frac {9 \int \cosh (2 a+2 b x) \, dx}{128 b^3}\\ &=-\frac {9 x \cosh (2 a+2 b x)}{128 b^3}-\frac {3 x^3 \cosh (2 a+2 b x)}{64 b}+\frac {x \cosh (6 a+6 b x)}{1152 b^3}+\frac {x^3 \cosh (6 a+6 b x)}{192 b}+\frac {9 \sinh (2 a+2 b x)}{256 b^4}+\frac {9 x^2 \sinh (2 a+2 b x)}{128 b^2}-\frac {\sinh (6 a+6 b x)}{6912 b^4}-\frac {x^2 \sinh (6 a+6 b x)}{384 b^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.90, size = 90, normalized size = 0.63 \[ -\frac {-3 \left (6 b^3 x^3+b x\right ) \cosh (6 (a+b x))+81 b x \left (2 b^2 x^2+3\right ) \cosh (2 (a+b x))+\sinh (2 (a+b x)) \left (\left (18 b^2 x^2+1\right ) \cosh (4 (a+b x))-234 b^2 x^2-121\right )}{3456 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3456*(81*b*x*(3 + 2*b^2*x^2)*Cosh[2*(a + b*x)] - 3*(b*x + 6*b^3*x^3)*Cosh[6*(a + b*x)] + (-121 - 234*b^2*x^
2 + (1 + 18*b^2*x^2)*Cosh[4*(a + b*x)])*Sinh[2*(a + b*x)])/b^4

________________________________________________________________________________________

fricas [A]  time = 0.81, size = 248, normalized size = 1.73 \[ \frac {3 \, {\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{6} - 10 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{3} + 45 \, {\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{4} - 3 \, {\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + 3 \, {\left (6 \, b^{3} x^{3} + b x\right )} \sinh \left (b x + a\right )^{6} - 81 \, {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} - 9 \, {\left (18 \, b^{3} x^{3} - 5 \, {\left (6 \, b^{3} x^{3} + b x\right )} \cosh \left (b x + a\right )^{4} + 27 \, b x\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left ({\left (18 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{5} - 81 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{3456 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3456*(3*(6*b^3*x^3 + b*x)*cosh(b*x + a)^6 - 10*(18*b^2*x^2 + 1)*cosh(b*x + a)^3*sinh(b*x + a)^3 + 45*(6*b^3*
x^3 + b*x)*cosh(b*x + a)^2*sinh(b*x + a)^4 - 3*(18*b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^5 + 3*(6*b^3*x^3 +
b*x)*sinh(b*x + a)^6 - 81*(2*b^3*x^3 + 3*b*x)*cosh(b*x + a)^2 - 9*(18*b^3*x^3 - 5*(6*b^3*x^3 + b*x)*cosh(b*x
+ a)^4 + 27*b*x)*sinh(b*x + a)^2 - 3*((18*b^2*x^2 + 1)*cosh(b*x + a)^5 - 81*(2*b^2*x^2 + 1)*cosh(b*x + a))*sin
h(b*x + a))/b^4

________________________________________________________________________________________

giac [A]  time = 0.15, size = 145, normalized size = 1.01 \[ \frac {{\left (36 \, b^{3} x^{3} - 18 \, b^{2} x^{2} + 6 \, b x - 1\right )} e^{\left (6 \, b x + 6 \, a\right )}}{13824 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{512 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{512 \, b^{4}} + \frac {{\left (36 \, b^{3} x^{3} + 18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{13824 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13824*(36*b^3*x^3 - 18*b^2*x^2 + 6*b*x - 1)*e^(6*b*x + 6*a)/b^4 - 3/512*(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*
e^(2*b*x + 2*a)/b^4 - 3/512*(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x - 2*a)/b^4 + 1/13824*(36*b^3*x^3 + 1
8*b^2*x^2 + 6*b*x + 1)*e^(-6*b*x - 6*a)/b^4

________________________________________________________________________________________

maple [B]  time = 0.43, size = 499, normalized size = 3.49 \[ \frac {\frac {\left (b x +a \right )^{3} \left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (b x +a \right )^{3} \left (\cosh ^{4}\left (b x +a \right )\right )}{12}-\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{12}+\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{12}+\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}+\frac {\left (b x +a \right )^{3}}{24}+\frac {\left (b x +a \right ) \left (\cosh ^{6}\left (b x +a \right )\right )}{36}-\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{216}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{216}+\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{72}+\frac {5 b x}{72}+\frac {5 a}{72}-\frac {\left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{24}-\frac {\left (b x +a \right ) \left (\cosh ^{2}\left (b x +a \right )\right )}{8}-3 a \left (\frac {\left (b x +a \right )^{2} \left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (b x +a \right )^{2} \left (\cosh ^{4}\left (b x +a \right )\right )}{12}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{18}+\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \left (\cosh ^{3}\left (b x +a \right )\right )}{18}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{12}+\frac {\left (b x +a \right )^{2}}{24}+\frac {\left (\cosh ^{6}\left (b x +a \right )\right )}{108}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{72}-\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{24}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \left (\sinh ^{2}\left (b x +a \right )\right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{6}-\frac {\left (b x +a \right ) \left (\cosh ^{4}\left (b x +a \right )\right )}{12}-\frac {\sinh \left (b x +a \right ) \left (\cosh ^{5}\left (b x +a \right )\right )}{36}+\frac {\left (\cosh ^{3}\left (b x +a \right )\right ) \sinh \left (b x +a \right )}{36}+\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{24}+\frac {b x}{24}+\frac {a}{24}\right )-a^{3} \left (\frac {\left (\cosh ^{4}\left (b x +a \right )\right ) \left (\sinh ^{2}\left (b x +a \right )\right )}{6}-\frac {\left (\cosh ^{4}\left (b x +a \right )\right )}{12}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^4*(1/6*(b*x+a)^3*sinh(b*x+a)^2*cosh(b*x+a)^4-1/12*(b*x+a)^3*cosh(b*x+a)^4-1/12*(b*x+a)^2*sinh(b*x+a)*cosh(
b*x+a)^5+1/12*(b*x+a)^2*sinh(b*x+a)*cosh(b*x+a)^3+1/8*(b*x+a)^2*cosh(b*x+a)*sinh(b*x+a)+1/24*(b*x+a)^3+1/36*(b
*x+a)*cosh(b*x+a)^6-1/216*sinh(b*x+a)*cosh(b*x+a)^5+1/216*cosh(b*x+a)^3*sinh(b*x+a)+5/72*cosh(b*x+a)*sinh(b*x+
a)+5/72*b*x+5/72*a-1/24*(b*x+a)*cosh(b*x+a)^4-1/8*(b*x+a)*cosh(b*x+a)^2-3*a*(1/6*(b*x+a)^2*sinh(b*x+a)^2*cosh(
b*x+a)^4-1/12*(b*x+a)^2*cosh(b*x+a)^4-1/18*(b*x+a)*sinh(b*x+a)*cosh(b*x+a)^5+1/18*(b*x+a)*sinh(b*x+a)*cosh(b*x
+a)^3+1/12*(b*x+a)*cosh(b*x+a)*sinh(b*x+a)+1/24*(b*x+a)^2+1/108*cosh(b*x+a)^6-1/72*cosh(b*x+a)^4-1/24*cosh(b*x
+a)^2)+3*a^2*(1/6*(b*x+a)*sinh(b*x+a)^2*cosh(b*x+a)^4-1/12*(b*x+a)*cosh(b*x+a)^4-1/36*sinh(b*x+a)*cosh(b*x+a)^
5+1/36*cosh(b*x+a)^3*sinh(b*x+a)+1/24*cosh(b*x+a)*sinh(b*x+a)+1/24*b*x+1/24*a)-a^3*(1/6*cosh(b*x+a)^4*sinh(b*x
+a)^2-1/12*cosh(b*x+a)^4))

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 171, normalized size = 1.20 \[ \frac {{\left (36 \, b^{3} x^{3} e^{\left (6 \, a\right )} - 18 \, b^{2} x^{2} e^{\left (6 \, a\right )} + 6 \, b x e^{\left (6 \, a\right )} - e^{\left (6 \, a\right )}\right )} e^{\left (6 \, b x\right )}}{13824 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{512 \, b^{4}} - \frac {3 \, {\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{512 \, b^{4}} + \frac {{\left (36 \, b^{3} x^{3} + 18 \, b^{2} x^{2} + 6 \, b x + 1\right )} e^{\left (-6 \, b x - 6 \, a\right )}}{13824 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13824*(36*b^3*x^3*e^(6*a) - 18*b^2*x^2*e^(6*a) + 6*b*x*e^(6*a) - e^(6*a))*e^(6*b*x)/b^4 - 3/512*(4*b^3*x^3*e
^(2*a) - 6*b^2*x^2*e^(2*a) + 6*b*x*e^(2*a) - 3*e^(2*a))*e^(2*b*x)/b^4 - 3/512*(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x +
3)*e^(-2*b*x - 2*a)/b^4 + 1/13824*(36*b^3*x^3 + 18*b^2*x^2 + 6*b*x + 1)*e^(-6*b*x - 6*a)/b^4

________________________________________________________________________________________

mupad [B]  time = 1.72, size = 126, normalized size = 0.88 \[ \frac {\frac {9\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{128}-\frac {x^2\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{384}}{b^2}-\frac {\frac {3\,x^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{64}-\frac {x^3\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{192}}{b}-\frac {\frac {9\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{128}-\frac {x\,\mathrm {cosh}\left (6\,a+6\,b\,x\right )}{1152}}{b^3}+\frac {9\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{256\,b^4}-\frac {\mathrm {sinh}\left (6\,a+6\,b\,x\right )}{6912\,b^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((9*x^2*sinh(2*a + 2*b*x))/128 - (x^2*sinh(6*a + 6*b*x))/384)/b^2 - ((3*x^3*cosh(2*a + 2*b*x))/64 - (x^3*cosh(
6*a + 6*b*x))/192)/b - ((9*x*cosh(2*a + 2*b*x))/128 - (x*cosh(6*a + 6*b*x))/1152)/b^3 + (9*sinh(2*a + 2*b*x))/
(256*b^4) - sinh(6*a + 6*b*x)/(6912*b^4)

________________________________________________________________________________________

sympy [A]  time = 14.19, size = 314, normalized size = 2.20 \[ \begin {cases} - \frac {x^{3} \sinh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x^{3} \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{8 b} + \frac {x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{8 b} - \frac {x^{3} \cosh ^{6}{\left (a + b x \right )}}{24 b} + \frac {x^{2} \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b^{2}} - \frac {x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b^{2}} + \frac {x^{2} \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{8 b^{2}} - \frac {5 x \sinh ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac {x \sinh ^{4}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{12 b^{3}} + \frac {x \sinh ^{2}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{12 b^{3}} - \frac {5 x \cosh ^{6}{\left (a + b x \right )}}{72 b^{3}} + \frac {5 \sinh ^{5}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{72 b^{4}} - \frac {31 \sinh ^{3}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{216 b^{4}} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{72 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{3}{\relax (a )} \cosh ^{3}{\relax (a )}}{4} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-x**3*sinh(a + b*x)**6/(24*b) + x**3*sinh(a + b*x)**4*cosh(a + b*x)**2/(8*b) + x**3*sinh(a + b*x)**
2*cosh(a + b*x)**4/(8*b) - x**3*cosh(a + b*x)**6/(24*b) + x**2*sinh(a + b*x)**5*cosh(a + b*x)/(8*b**2) - x**2*
sinh(a + b*x)**3*cosh(a + b*x)**3/(3*b**2) + x**2*sinh(a + b*x)*cosh(a + b*x)**5/(8*b**2) - 5*x*sinh(a + b*x)*
*6/(72*b**3) + x*sinh(a + b*x)**4*cosh(a + b*x)**2/(12*b**3) + x*sinh(a + b*x)**2*cosh(a + b*x)**4/(12*b**3) -
5*x*cosh(a + b*x)**6/(72*b**3) + 5*sinh(a + b*x)**5*cosh(a + b*x)/(72*b**4) - 31*sinh(a + b*x)**3*cosh(a + b*
x)**3/(216*b**4) + 5*sinh(a + b*x)*cosh(a + b*x)**5/(72*b**4), Ne(b, 0)), (x**4*sinh(a)**3*cosh(a)**3/4, True)
)

________________________________________________________________________________________