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

Optimal. Leaf size=144 \[ -\frac {\cosh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \cosh (2 a+x (2 b-d)-c)}{16 (2 b-d)}-\frac {3 \cosh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\cosh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac {3 \cosh (c+d x)}{8 d}-\frac {\cosh (3 c+3 d x)}{24 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 144, 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 = {5613, 2638} \[ -\frac {\cosh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \cosh (2 a+x (2 b-d)-c)}{16 (2 b-d)}-\frac {3 \cosh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\cosh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac {3 \cosh (c+d x)}{8 d}-\frac {\cosh (3 c+3 d x)}{24 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2638

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

Rule 5613

Int[Sinh[v_]^(p_.)*Sinh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Sinh[v]^p*Sinh[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 \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx &=\int \left (-\frac {1}{16} \sinh (2 a-3 c+(2 b-3 d) x)+\frac {3}{16} \sinh (2 a-c+(2 b-d) x)+\frac {3}{8} \sinh (c+d x)-\frac {1}{8} \sinh (3 c+3 d x)-\frac {3}{16} \sinh (2 a+c+(2 b+d) x)+\frac {1}{16} \sinh (2 a+3 c+(2 b+3 d) x)\right ) \, dx\\ &=-\left (\frac {1}{16} \int \sinh (2 a-3 c+(2 b-3 d) x) \, dx\right )+\frac {1}{16} \int \sinh (2 a+3 c+(2 b+3 d) x) \, dx-\frac {1}{8} \int \sinh (3 c+3 d x) \, dx+\frac {3}{16} \int \sinh (2 a-c+(2 b-d) x) \, dx-\frac {3}{16} \int \sinh (2 a+c+(2 b+d) x) \, dx+\frac {3}{8} \int \sinh (c+d x) \, dx\\ &=-\frac {\cosh (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \cosh (2 a-c+(2 b-d) x)}{16 (2 b-d)}+\frac {3 \cosh (c+d x)}{8 d}-\frac {\cosh (3 c+3 d x)}{24 d}-\frac {3 \cosh (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\cosh (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.60, size = 158, normalized size = 1.10 \[ \frac {1}{48} \left (-\frac {3 \cosh (2 a+2 b x-3 c-3 d x)}{2 b-3 d}+\frac {9 \cosh (2 a+2 b x-c-d x)}{2 b-d}-\frac {9 \cosh (2 a+2 b x+c+d x)}{2 b+d}+\frac {3 \cosh (2 a+2 b x+3 c+3 d x)}{2 b+3 d}+\frac {18 \sinh (c) \sinh (d x)}{d}-\frac {2 \sinh (3 c) \sinh (3 d x)}{d}+\frac {18 \cosh (c) \cosh (d x)}{d}-\frac {2 \cosh (3 c) \cosh (3 d x)}{d}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.44, size = 414, normalized size = 2.88 \[ \frac {12 \, {\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{3} - {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{3} - 9 \, {\left ({\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right )^{3} - {\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right )^{2} + 36 \, {\left ({\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - {\left (4 \, b^{3} d - 9 \, b d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) \sinh \left (d x + c\right ) - 3 \, {\left (9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 9 \, {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + {\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )}{24 \, {\left ({\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} - {\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(12*(4*b^3*d - b*d^3)*cosh(b*x + a)*sinh(b*x + a)*sinh(d*x + c)^3 - (16*b^4 - 40*b^2*d^2 + 9*d^4 + 9*(4*b
^2*d^2 - d^4)*cosh(b*x + a)^2)*cosh(d*x + c)^3 - 9*((4*b^2*d^2 - d^4)*cosh(d*x + c)^3 - (4*b^2*d^2 - 9*d^4)*co
sh(d*x + c))*sinh(b*x + a)^2 + 36*((4*b^3*d - b*d^3)*cosh(b*x + a)*cosh(d*x + c)^2 - (4*b^3*d - 9*b*d^3)*cosh(
b*x + a))*sinh(b*x + a)*sinh(d*x + c) - 3*(9*(4*b^2*d^2 - d^4)*cosh(d*x + c)*sinh(b*x + a)^2 + (16*b^4 - 40*b^
2*d^2 + 9*d^4 + 9*(4*b^2*d^2 - d^4)*cosh(b*x + a)^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 9*(16*b^4 - 40*b^2*d^2 +
9*d^4 + (4*b^2*d^2 - 9*d^4)*cosh(b*x + a)^2)*cosh(d*x + c))/((16*b^4*d - 40*b^2*d^3 + 9*d^5)*cosh(b*x + a)^2
- (16*b^4*d - 40*b^2*d^3 + 9*d^5)*sinh(b*x + a)^2)

________________________________________________________________________________________

giac [A]  time = 0.14, size = 260, normalized size = 1.81 \[ \frac {e^{\left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} - \frac {3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{32 \, {\left (2 \, b + d\right )}} + \frac {3 \, e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{32 \, {\left (2 \, b - d\right )}} - \frac {e^{\left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {e^{\left (-2 \, b x + 3 \, d x - 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} + \frac {3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{32 \, {\left (2 \, b - d\right )}} - \frac {3 \, e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{32 \, {\left (2 \, b + d\right )}} + \frac {e^{\left (-2 \, b x - 3 \, d x - 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} - \frac {e^{\left (3 \, d x + 3 \, c\right )}}{48 \, d} + \frac {3 \, e^{\left (d x + c\right )}}{16 \, d} + \frac {3 \, e^{\left (-d x - c\right )}}{16 \, d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 133, normalized size = 0.92 \[ -\frac {\cosh \left (2 a -3 c +\left (2 b -3 d \right ) x \right )}{16 \left (2 b -3 d \right )}+\frac {3 \cosh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 \left (2 b -d \right )}+\frac {3 \cosh \left (d x +c \right )}{8 d}-\frac {\cosh \left (3 d x +3 c \right )}{24 d}-\frac {3 \cosh \left (2 a +c +\left (2 b +d \right ) x \right )}{16 \left (2 b +d \right )}+\frac {\cosh \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{32 b +48 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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(sinh(b*x+a)^2*sinh(d*x+c)^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(1-(3*d)/b>0)', see `assume?` f
or more details)Is 1-(3*d)/b equal to -1?

________________________________________________________________________________________

mupad [B]  time = 1.94, size = 337, normalized size = 2.34 \[ \frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (8\,b^4-26\,b^2\,d^2+9\,d^4\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}+\frac {1}{3\,d}\right )-\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (7\,b\,d^2-4\,b^3\right )}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (8\,b^4-14\,b^2\,d^2\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {1}{3\,d}\right )+\frac {12\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{16\,b^4-40\,b^2\,d^2+9\,d^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cosh(c + d*x)*sinh(a + b*x)^2*sinh(c + d*x)^2*(8*b^4 + 9*d^4 - 26*b^2*d^2))/(d*(16*b^4 + 9*d^4 - 40*b^2*d^2))
- cosh(c + d*x)^3*sinh(a + b*x)^2*((3*d^3)/(16*b^4 + 9*d^4 - 40*b^2*d^2) + 1/(3*d)) - (2*cosh(a + b*x)*sinh(a
+ b*x)*sinh(c + d*x)^3*(7*b*d^2 - 4*b^3))/(16*b^4 + 9*d^4 - 40*b^2*d^2) - (cosh(a + b*x)^2*cosh(c + d*x)*sinh
(c + d*x)^2*(8*b^4 - 14*b^2*d^2))/(d*(16*b^4 + 9*d^4 - 40*b^2*d^2)) - cosh(a + b*x)^2*cosh(c + d*x)^3*((3*d^3)
/(16*b^4 + 9*d^4 - 40*b^2*d^2) - 1/(3*d)) + (12*b*d^2*cosh(a + b*x)*cosh(c + d*x)^2*sinh(a + b*x)*sinh(c + d*x
))/(16*b^4 + 9*d^4 - 40*b^2*d^2)

________________________________________________________________________________________

sympy [A]  time = 109.06, size = 2006, normalized size = 13.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((x*sinh(a)**2*sinh(c)**3, Eq(b, 0) & Eq(d, 0)), (x*sinh(a - 3*d*x/2)**2*sinh(c + d*x)**3/16 + 3*x*si
nh(a - 3*d*x/2)**2*sinh(c + d*x)*cosh(c + d*x)**2/16 + 3*x*sinh(a - 3*d*x/2)*sinh(c + d*x)**2*cosh(a - 3*d*x/2
)*cosh(c + d*x)/8 + x*sinh(a - 3*d*x/2)*cosh(a - 3*d*x/2)*cosh(c + d*x)**3/8 + x*sinh(c + d*x)**3*cosh(a - 3*d
*x/2)**2/16 + 3*x*sinh(c + d*x)*cosh(a - 3*d*x/2)**2*cosh(c + d*x)**2/16 + sinh(a - 3*d*x/2)**2*sinh(c + d*x)*
*2*cosh(c + d*x)/d - 5*sinh(a - 3*d*x/2)**2*cosh(c + d*x)**3/(48*d) + sinh(a - 3*d*x/2)*sinh(c + d*x)**3*cosh(
a - 3*d*x/2)/(24*d) + 5*sinh(a - 3*d*x/2)*sinh(c + d*x)*cosh(a - 3*d*x/2)*cosh(c + d*x)**2/(4*d) + 9*cosh(a -
3*d*x/2)**2*cosh(c + d*x)**3/(16*d), Eq(b, -3*d/2)), (3*x*sinh(a - d*x/2)**2*sinh(c + d*x)**3/16 - 3*x*sinh(a
- d*x/2)**2*sinh(c + d*x)*cosh(c + d*x)**2/16 + 3*x*sinh(a - d*x/2)*sinh(c + d*x)**2*cosh(a - d*x/2)*cosh(c +
d*x)/8 - 3*x*sinh(a - d*x/2)*cosh(a - d*x/2)*cosh(c + d*x)**3/8 + 3*x*sinh(c + d*x)**3*cosh(a - d*x/2)**2/16 -
3*x*sinh(c + d*x)*cosh(a - d*x/2)**2*cosh(c + d*x)**2/16 + 17*sinh(a - d*x/2)**2*cosh(c + d*x)**3/(48*d) - 13
*sinh(a - d*x/2)*sinh(c + d*x)**3*cosh(a - d*x/2)/(8*d) + 7*sinh(a - d*x/2)*sinh(c + d*x)*cosh(a - d*x/2)*cosh
(c + d*x)**2/(4*d) - sinh(c + d*x)**2*cosh(a - d*x/2)**2*cosh(c + d*x)/d + 49*cosh(a - d*x/2)**2*cosh(c + d*x)
**3/(48*d), Eq(b, -d/2)), (3*x*sinh(a + d*x/2)**2*sinh(c + d*x)**3/16 - 3*x*sinh(a + d*x/2)**2*sinh(c + d*x)*c
osh(c + d*x)**2/16 - 3*x*sinh(a + d*x/2)*sinh(c + d*x)**2*cosh(a + d*x/2)*cosh(c + d*x)/8 + 3*x*sinh(a + d*x/2
)*cosh(a + d*x/2)*cosh(c + d*x)**3/8 + 3*x*sinh(c + d*x)**3*cosh(a + d*x/2)**2/16 - 3*x*sinh(c + d*x)*cosh(a +
d*x/2)**2*cosh(c + d*x)**2/16 + 17*sinh(a + d*x/2)**2*cosh(c + d*x)**3/(48*d) + 13*sinh(a + d*x/2)*sinh(c + d
*x)**3*cosh(a + d*x/2)/(8*d) - 7*sinh(a + d*x/2)*sinh(c + d*x)*cosh(a + d*x/2)*cosh(c + d*x)**2/(4*d) - sinh(c
+ d*x)**2*cosh(a + d*x/2)**2*cosh(c + d*x)/d + 49*cosh(a + d*x/2)**2*cosh(c + d*x)**3/(48*d), Eq(b, d/2)), (x
*sinh(a + 3*d*x/2)**2*sinh(c + d*x)**3/16 + 3*x*sinh(a + 3*d*x/2)**2*sinh(c + d*x)*cosh(c + d*x)**2/16 - 3*x*s
inh(a + 3*d*x/2)*sinh(c + d*x)**2*cosh(a + 3*d*x/2)*cosh(c + d*x)/8 - x*sinh(a + 3*d*x/2)*cosh(a + 3*d*x/2)*co
sh(c + d*x)**3/8 + x*sinh(c + d*x)**3*cosh(a + 3*d*x/2)**2/16 + 3*x*sinh(c + d*x)*cosh(a + 3*d*x/2)**2*cosh(c
+ d*x)**2/16 + sinh(a + 3*d*x/2)**2*sinh(c + d*x)**2*cosh(c + d*x)/d - 5*sinh(a + 3*d*x/2)**2*cosh(c + d*x)**3
/(48*d) - sinh(a + 3*d*x/2)*sinh(c + d*x)**3*cosh(a + 3*d*x/2)/(24*d) - 5*sinh(a + 3*d*x/2)*sinh(c + d*x)*cosh
(a + 3*d*x/2)*cosh(c + d*x)**2/(4*d) + 9*cosh(a + 3*d*x/2)**2*cosh(c + d*x)**3/(16*d), Eq(b, 3*d/2)), ((x*sinh
(a + b*x)**2/2 - x*cosh(a + b*x)**2/2 + sinh(a + b*x)*cosh(a + b*x)/(2*b))*sinh(c)**3, Eq(d, 0)), (24*b**4*sin
h(a + b*x)**2*sinh(c + d*x)**2*cosh(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 16*b**4*sinh(a + b*x)**2*
cosh(c + d*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 24*b**4*sinh(c + d*x)**2*cosh(a + b*x)**2*cosh(c + d*
x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 16*b**4*cosh(a + b*x)**2*cosh(c + d*x)**3/(48*b**4*d - 120*b**2*d**
3 + 27*d**5) + 24*b**3*d*sinh(a + b*x)*sinh(c + d*x)**3*cosh(a + b*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) -
78*b**2*d**2*sinh(a + b*x)**2*sinh(c + d*x)**2*cosh(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 40*b**2*d
**2*sinh(a + b*x)**2*cosh(c + d*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) + 42*b**2*d**2*sinh(c + d*x)**2*co
sh(a + b*x)**2*cosh(c + d*x)/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 40*b**2*d**2*cosh(a + b*x)**2*cosh(c + d*
x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5) - 42*b*d**3*sinh(a + b*x)*sinh(c + d*x)**3*cosh(a + b*x)/(48*b**4*
d - 120*b**2*d**3 + 27*d**5) + 36*b*d**3*sinh(a + b*x)*sinh(c + d*x)*cosh(a + b*x)*cosh(c + d*x)**2/(48*b**4*d
- 120*b**2*d**3 + 27*d**5) + 27*d**4*sinh(a + b*x)**2*sinh(c + d*x)**2*cosh(c + d*x)/(48*b**4*d - 120*b**2*d*
*3 + 27*d**5) - 18*d**4*sinh(a + b*x)**2*cosh(c + d*x)**3/(48*b**4*d - 120*b**2*d**3 + 27*d**5), True))

________________________________________________________________________________________