3.186 \(\int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx\)

Optimal. Leaf size=138 \[ -\frac {3 \cosh (a+x (b-2 d)-2 c)}{16 (b-2 d)}+\frac {\cosh (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}-\frac {3 \cosh (a+x (b+2 d)+2 c)}{16 (b+2 d)}+\frac {\cosh (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac {3 \cosh (a+b x)}{8 b}+\frac {\cosh (3 a+3 b x)}{24 b} \]

[Out]

-3/8*cosh(b*x+a)/b+1/24*cosh(3*b*x+3*a)/b-3/16*cosh(a-2*c+(b-2*d)*x)/(b-2*d)+1/16*cosh(3*a-2*c+(3*b-2*d)*x)/(3
*b-2*d)-3/16*cosh(a+2*c+(b+2*d)*x)/(b+2*d)+1/16*cosh(3*a+2*c+(3*b+2*d)*x)/(3*b+2*d)

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Rubi [A]  time = 0.11, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5618, 2638} \[ -\frac {3 \cosh (a+x (b-2 d)-2 c)}{16 (b-2 d)}+\frac {\cosh (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}-\frac {3 \cosh (a+x (b+2 d)+2 c)}{16 (b+2 d)}+\frac {\cosh (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac {3 \cosh (a+b x)}{8 b}+\frac {\cosh (3 a+3 b x)}{24 b} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[c + d*x]^2*Sinh[a + b*x]^3,x]

[Out]

(-3*Cosh[a + b*x])/(8*b) + Cosh[3*a + 3*b*x]/(24*b) - (3*Cosh[a - 2*c + (b - 2*d)*x])/(16*(b - 2*d)) + Cosh[3*
a - 2*c + (3*b - 2*d)*x]/(16*(3*b - 2*d)) - (3*Cosh[a + 2*c + (b + 2*d)*x])/(16*(b + 2*d)) + Cosh[3*a + 2*c +
(3*b + 2*d)*x]/(16*(3*b + 2*d))

Rule 2638

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

Rule 5618

Int[Cosh[w_]^(q_.)*Sinh[v_]^(p_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Cosh[w]^q, x], x] /; IGtQ[p, 0]
&& IGtQ[q, 0] && ((PolynomialQ[v, x] && PolynomialQ[w, x]) || (BinomialQ[{v, w}, x] && IndependentQ[Cancel[v/
w], x]))

Rubi steps

\begin {align*} \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx &=\int \left (-\frac {3}{8} \sinh (a+b x)+\frac {1}{8} \sinh (3 a+3 b x)-\frac {3}{16} \sinh (a-2 c+(b-2 d) x)+\frac {1}{16} \sinh (3 a-2 c+(3 b-2 d) x)-\frac {3}{16} \sinh (a+2 c+(b+2 d) x)+\frac {1}{16} \sinh (3 a+2 c+(3 b+2 d) x)\right ) \, dx\\ &=\frac {1}{16} \int \sinh (3 a-2 c+(3 b-2 d) x) \, dx+\frac {1}{16} \int \sinh (3 a+2 c+(3 b+2 d) x) \, dx+\frac {1}{8} \int \sinh (3 a+3 b x) \, dx-\frac {3}{16} \int \sinh (a-2 c+(b-2 d) x) \, dx-\frac {3}{16} \int \sinh (a+2 c+(b+2 d) x) \, dx-\frac {3}{8} \int \sinh (a+b x) \, dx\\ &=-\frac {3 \cosh (a+b x)}{8 b}+\frac {\cosh (3 a+3 b x)}{24 b}-\frac {3 \cosh (a-2 c+(b-2 d) x)}{16 (b-2 d)}+\frac {\cosh (3 a-2 c+(3 b-2 d) x)}{16 (3 b-2 d)}-\frac {3 \cosh (a+2 c+(b+2 d) x)}{16 (b+2 d)}+\frac {\cosh (3 a+2 c+(3 b+2 d) x)}{16 (3 b+2 d)}\\ \end {align*}

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Mathematica [A]  time = 1.64, size = 153, normalized size = 1.11 \[ \frac {1}{48} \left (-\frac {9 \cosh (a+b x-2 c-2 d x)}{b-2 d}+\frac {3 \cosh (3 a+3 b x-2 c-2 d x)}{3 b-2 d}-\frac {9 \cosh (a+b x+2 c+2 d x)}{b+2 d}+\frac {3 \cosh (3 a+3 b x+2 c+2 d x)}{3 b+2 d}-\frac {18 \sinh (a) \sinh (b x)}{b}+\frac {2 \sinh (3 a) \sinh (3 b x)}{b}-\frac {18 \cosh (a) \cosh (b x)}{b}+\frac {2 \cosh (3 a) \cosh (3 b x)}{b}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[c + d*x]^2*Sinh[a + b*x]^3,x]

[Out]

((-18*Cosh[a]*Cosh[b*x])/b + (2*Cosh[3*a]*Cosh[3*b*x])/b - (9*Cosh[a - 2*c + b*x - 2*d*x])/(b - 2*d) + (3*Cosh
[3*a - 2*c + 3*b*x - 2*d*x])/(3*b - 2*d) - (9*Cosh[a + 2*c + b*x + 2*d*x])/(b + 2*d) + (3*Cosh[3*a + 2*c + 3*b
*x + 2*d*x])/(3*b + 2*d) - (18*Sinh[a]*Sinh[b*x])/b + (2*Sinh[3*a]*Sinh[3*b*x])/b)/48

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fricas [B]  time = 0.41, size = 443, normalized size = 3.21 \[ \frac {{\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left ({\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (9 \, {\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} + {\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 9 \, {\left ({\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (9 \, b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} - 9 \, {\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right ) - 12 \, {\left ({\left (b^{3} d - 4 \, b d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{3} d - 4 \, b d^{3} - {\left (b^{3} d - 4 \, b d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left ({\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/24*((9*b^4 - 40*b^2*d^2 + 16*d^4)*cosh(b*x + a)^3 + 9*((b^4 - 4*b^2*d^2)*cosh(b*x + a)^3 - (9*b^4 - 4*b^2*d^
2)*cosh(b*x + a))*cosh(d*x + c)^2 + 3*(9*(b^4 - 4*b^2*d^2)*cosh(b*x + a)*cosh(d*x + c)^2 + (9*b^4 - 40*b^2*d^2
+ 16*d^4)*cosh(b*x + a))*sinh(b*x + a)^2 + 9*((b^4 - 4*b^2*d^2)*cosh(b*x + a)^3 + 3*(b^4 - 4*b^2*d^2)*cosh(b*
x + a)*sinh(b*x + a)^2 - (9*b^4 - 4*b^2*d^2)*cosh(b*x + a))*sinh(d*x + c)^2 - 9*(9*b^4 - 40*b^2*d^2 + 16*d^4)*
cosh(b*x + a) - 12*((b^3*d - 4*b*d^3)*cosh(d*x + c)*sinh(b*x + a)^3 - 3*(9*b^3*d - 4*b*d^3 - (b^3*d - 4*b*d^3)
*cosh(b*x + a)^2)*cosh(d*x + c)*sinh(b*x + a))*sinh(d*x + c))/((9*b^5 - 40*b^3*d^2 + 16*b*d^4)*cosh(b*x + a)^4
- 2*(9*b^5 - 40*b^3*d^2 + 16*b*d^4)*cosh(b*x + a)^2*sinh(b*x + a)^2 + (9*b^5 - 40*b^3*d^2 + 16*b*d^4)*sinh(b*
x + a)^4)

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giac [B]  time = 0.15, size = 256, normalized size = 1.86 \[ \frac {e^{\left (3 \, b x + 2 \, d x + 3 \, a + 2 \, c\right )}}{32 \, {\left (3 \, b + 2 \, d\right )}} + \frac {e^{\left (3 \, b x - 2 \, d x + 3 \, a - 2 \, c\right )}}{32 \, {\left (3 \, b - 2 \, d\right )}} + \frac {e^{\left (3 \, b x + 3 \, a\right )}}{48 \, b} - \frac {3 \, e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{32 \, {\left (b + 2 \, d\right )}} - \frac {3 \, e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{32 \, {\left (b - 2 \, d\right )}} - \frac {3 \, e^{\left (b x + a\right )}}{16 \, b} - \frac {3 \, e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{32 \, {\left (b - 2 \, d\right )}} - \frac {3 \, e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{32 \, {\left (b + 2 \, d\right )}} - \frac {3 \, e^{\left (-b x - a\right )}}{16 \, b} + \frac {e^{\left (-3 \, b x + 2 \, d x - 3 \, a + 2 \, c\right )}}{32 \, {\left (3 \, b - 2 \, d\right )}} + \frac {e^{\left (-3 \, b x - 2 \, d x - 3 \, a - 2 \, c\right )}}{32 \, {\left (3 \, b + 2 \, d\right )}} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

1/32*e^(3*b*x + 2*d*x + 3*a + 2*c)/(3*b + 2*d) + 1/32*e^(3*b*x - 2*d*x + 3*a - 2*c)/(3*b - 2*d) + 1/48*e^(3*b*
x + 3*a)/b - 3/32*e^(b*x + 2*d*x + a + 2*c)/(b + 2*d) - 3/32*e^(b*x - 2*d*x + a - 2*c)/(b - 2*d) - 3/16*e^(b*x
+ a)/b - 3/32*e^(-b*x + 2*d*x - a + 2*c)/(b - 2*d) - 3/32*e^(-b*x - 2*d*x - a - 2*c)/(b + 2*d) - 3/16*e^(-b*x
- a)/b + 1/32*e^(-3*b*x + 2*d*x - 3*a + 2*c)/(3*b - 2*d) + 1/32*e^(-3*b*x - 2*d*x - 3*a - 2*c)/(3*b + 2*d) +
1/48*e^(-3*b*x - 3*a)/b

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maple [A]  time = 0.10, size = 127, normalized size = 0.92 \[ -\frac {3 \cosh \left (b x +a \right )}{8 b}+\frac {\cosh \left (3 b x +3 a \right )}{24 b}-\frac {3 \cosh \left (a -2 c +\left (b -2 d \right ) x \right )}{16 \left (b -2 d \right )}+\frac {\cosh \left (3 a -2 c +\left (3 b -2 d \right ) x \right )}{48 b -32 d}-\frac {3 \cosh \left (a +2 c +\left (b +2 d \right ) x \right )}{16 \left (b +2 d \right )}+\frac {\cosh \left (3 a +2 c +\left (3 b +2 d \right ) x \right )}{48 b +32 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(d*x+c)^2*sinh(b*x+a)^3,x)

[Out]

-3/8*cosh(b*x+a)/b+1/24*cosh(3*b*x+3*a)/b-3/16*cosh(a-2*c+(b-2*d)*x)/(b-2*d)+1/16*cosh(3*a-2*c+(3*b-2*d)*x)/(3
*b-2*d)-3/16*cosh(a+2*c+(b+2*d)*x)/(b+2*d)+1/16*cosh(3*a+2*c+(3*b+2*d)*x)/(3*b+2*d)

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)^2*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(-(2*d)/b>0)', see `assume?` fo
r more details)Is -(2*d)/b equal to -1?

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mupad [B]  time = 2.00, size = 337, normalized size = 2.44 \[ \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^4-26\,b^2\,d^2+8\,d^4\right )}{b\,\left (9\,b^4-40\,b^2\,d^2+16\,d^4\right )}-{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (\frac {3\,b^3}{9\,b^4-40\,b^2\,d^2+16\,d^4}-\frac {1}{3\,b}\right )-{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (\frac {3\,b^3}{9\,b^4-40\,b^2\,d^2+16\,d^4}+\frac {1}{3\,b}\right )-\frac {2\,d\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (7\,b^2-4\,d^2\right )}{9\,b^4-40\,b^2\,d^2+16\,d^4}+\frac {12\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{9\,b^4-40\,b^2\,d^2+16\,d^4}+\frac {2\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (7\,b^2-4\,d^2\right )}{b\,\left (9\,b^4-40\,b^2\,d^2+16\,d^4\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(c + d*x)^2*sinh(a + b*x)^3,x)

[Out]

(cosh(a + b*x)*cosh(c + d*x)^2*sinh(a + b*x)^2*(9*b^4 + 8*d^4 - 26*b^2*d^2))/(b*(9*b^4 + 16*d^4 - 40*b^2*d^2))
- cosh(a + b*x)^3*sinh(c + d*x)^2*((3*b^3)/(9*b^4 + 16*d^4 - 40*b^2*d^2) - 1/(3*b)) - cosh(a + b*x)^3*cosh(c
+ d*x)^2*((3*b^3)/(9*b^4 + 16*d^4 - 40*b^2*d^2) + 1/(3*b)) - (2*d*cosh(c + d*x)*sinh(a + b*x)^3*sinh(c + d*x)*
(7*b^2 - 4*d^2))/(9*b^4 + 16*d^4 - 40*b^2*d^2) + (12*b^2*d*cosh(a + b*x)^2*cosh(c + d*x)*sinh(a + b*x)*sinh(c
+ d*x))/(9*b^4 + 16*d^4 - 40*b^2*d^2) + (2*d^2*cosh(a + b*x)*sinh(a + b*x)^2*sinh(c + d*x)^2*(7*b^2 - 4*d^2))/
(b*(9*b^4 + 16*d^4 - 40*b^2*d^2))

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sympy [A]  time = 109.99, size = 2030, normalized size = 14.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(d*x+c)**2*sinh(b*x+a)**3,x)

[Out]

Piecewise((x*sinh(a)**3*cosh(c)**2, Eq(b, 0) & Eq(d, 0)), ((-x*sinh(c + d*x)**2/2 + x*cosh(c + d*x)**2/2 + sin
h(c + d*x)*cosh(c + d*x)/(2*d))*sinh(a)**3, Eq(b, 0)), (3*x*sinh(a - 2*d*x)**3*sinh(c + d*x)**2/16 + 3*x*sinh(
a - 2*d*x)**3*cosh(c + d*x)**2/16 + 3*x*sinh(a - 2*d*x)**2*sinh(c + d*x)*cosh(a - 2*d*x)*cosh(c + d*x)/8 - 3*x
*sinh(a - 2*d*x)*sinh(c + d*x)**2*cosh(a - 2*d*x)**2/16 - 3*x*sinh(a - 2*d*x)*cosh(a - 2*d*x)**2*cosh(c + d*x)
**2/16 - 3*x*sinh(c + d*x)*cosh(a - 2*d*x)**3*cosh(c + d*x)/8 + 13*sinh(a - 2*d*x)**3*sinh(c + d*x)*cosh(c + d
*x)/(16*d) + sinh(a - 2*d*x)**2*sinh(c + d*x)**2*cosh(a - 2*d*x)/(2*d) - 7*sinh(a - 2*d*x)*sinh(c + d*x)*cosh(
a - 2*d*x)**2*cosh(c + d*x)/(8*d) - 49*sinh(c + d*x)**2*cosh(a - 2*d*x)**3/(96*d) - 17*cosh(a - 2*d*x)**3*cosh
(c + d*x)**2/(96*d), Eq(b, -2*d)), (x*sinh(a - 2*d*x/3)**3*sinh(c + d*x)**2/16 + x*sinh(a - 2*d*x/3)**3*cosh(c
+ d*x)**2/16 + 3*x*sinh(a - 2*d*x/3)**2*sinh(c + d*x)*cosh(a - 2*d*x/3)*cosh(c + d*x)/8 + 3*x*sinh(a - 2*d*x/
3)*sinh(c + d*x)**2*cosh(a - 2*d*x/3)**2/16 + 3*x*sinh(a - 2*d*x/3)*cosh(a - 2*d*x/3)**2*cosh(c + d*x)**2/16 +
x*sinh(c + d*x)*cosh(a - 2*d*x/3)**3*cosh(c + d*x)/8 + 15*sinh(a - 2*d*x/3)**3*sinh(c + d*x)*cosh(c + d*x)/(1
6*d) + 3*sinh(a - 2*d*x/3)**2*sinh(c + d*x)**2*cosh(a - 2*d*x/3)/(2*d) + 9*sinh(a - 2*d*x/3)*sinh(c + d*x)*cos
h(a - 2*d*x/3)**2*cosh(c + d*x)/(8*d) - 11*sinh(c + d*x)**2*cosh(a - 2*d*x/3)**3/(32*d) + 21*cosh(a - 2*d*x/3)
**3*cosh(c + d*x)**2/(32*d), Eq(b, -2*d/3)), (x*sinh(a + 2*d*x/3)**3*sinh(c + d*x)**2/16 + x*sinh(a + 2*d*x/3)
**3*cosh(c + d*x)**2/16 - 3*x*sinh(a + 2*d*x/3)**2*sinh(c + d*x)*cosh(a + 2*d*x/3)*cosh(c + d*x)/8 + 3*x*sinh(
a + 2*d*x/3)*sinh(c + d*x)**2*cosh(a + 2*d*x/3)**2/16 + 3*x*sinh(a + 2*d*x/3)*cosh(a + 2*d*x/3)**2*cosh(c + d*
x)**2/16 - x*sinh(c + d*x)*cosh(a + 2*d*x/3)**3*cosh(c + d*x)/8 + 15*sinh(a + 2*d*x/3)**3*sinh(c + d*x)*cosh(c
+ d*x)/(16*d) - 3*sinh(a + 2*d*x/3)**2*sinh(c + d*x)**2*cosh(a + 2*d*x/3)/(2*d) + 9*sinh(a + 2*d*x/3)*sinh(c
+ d*x)*cosh(a + 2*d*x/3)**2*cosh(c + d*x)/(8*d) + 11*sinh(c + d*x)**2*cosh(a + 2*d*x/3)**3/(32*d) - 21*cosh(a
+ 2*d*x/3)**3*cosh(c + d*x)**2/(32*d), Eq(b, 2*d/3)), (3*x*sinh(a + 2*d*x)**3*sinh(c + d*x)**2/16 + 3*x*sinh(a
+ 2*d*x)**3*cosh(c + d*x)**2/16 - 3*x*sinh(a + 2*d*x)**2*sinh(c + d*x)*cosh(a + 2*d*x)*cosh(c + d*x)/8 - 3*x*
sinh(a + 2*d*x)*sinh(c + d*x)**2*cosh(a + 2*d*x)**2/16 - 3*x*sinh(a + 2*d*x)*cosh(a + 2*d*x)**2*cosh(c + d*x)*
*2/16 + 3*x*sinh(c + d*x)*cosh(a + 2*d*x)**3*cosh(c + d*x)/8 + 13*sinh(a + 2*d*x)**3*sinh(c + d*x)*cosh(c + d*
x)/(16*d) - sinh(a + 2*d*x)**2*sinh(c + d*x)**2*cosh(a + 2*d*x)/(2*d) - 7*sinh(a + 2*d*x)*sinh(c + d*x)*cosh(a
+ 2*d*x)**2*cosh(c + d*x)/(8*d) + 49*sinh(c + d*x)**2*cosh(a + 2*d*x)**3/(96*d) + 17*cosh(a + 2*d*x)**3*cosh(
c + d*x)**2/(96*d), Eq(b, 2*d)), (27*b**4*sinh(a + b*x)**2*cosh(a + b*x)*cosh(c + d*x)**2/(27*b**5 - 120*b**3*
d**2 + 48*b*d**4) - 18*b**4*cosh(a + b*x)**3*cosh(c + d*x)**2/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) - 42*b**3*
d*sinh(a + b*x)**3*sinh(c + d*x)*cosh(c + d*x)/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) + 36*b**3*d*sinh(a + b*x)
*sinh(c + d*x)*cosh(a + b*x)**2*cosh(c + d*x)/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) + 42*b**2*d**2*sinh(a + b*
x)**2*sinh(c + d*x)**2*cosh(a + b*x)/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) - 78*b**2*d**2*sinh(a + b*x)**2*cos
h(a + b*x)*cosh(c + d*x)**2/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) - 40*b**2*d**2*sinh(c + d*x)**2*cosh(a + b*x
)**3/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) + 40*b**2*d**2*cosh(a + b*x)**3*cosh(c + d*x)**2/(27*b**5 - 120*b**
3*d**2 + 48*b*d**4) + 24*b*d**3*sinh(a + b*x)**3*sinh(c + d*x)*cosh(c + d*x)/(27*b**5 - 120*b**3*d**2 + 48*b*d
**4) - 24*d**4*sinh(a + b*x)**2*sinh(c + d*x)**2*cosh(a + b*x)/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) + 24*d**4
*sinh(a + b*x)**2*cosh(a + b*x)*cosh(c + d*x)**2/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) + 16*d**4*sinh(c + d*x)
**2*cosh(a + b*x)**3/(27*b**5 - 120*b**3*d**2 + 48*b*d**4) - 16*d**4*cosh(a + b*x)**3*cosh(c + d*x)**2/(27*b**
5 - 120*b**3*d**2 + 48*b*d**4), True))

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