Integrand size = 17, antiderivative size = 144 \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=-\frac {\cosh (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \cosh (2 a-c+(2 b-d) x)}{16 (2 b-d)}+\frac {3 \cosh (c+d x)}{8 d}-\frac {\cosh (3 c+3 d x)}{24 d}-\frac {3 \cosh (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\cosh (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)} \] Output:
-1/16*cosh(2*a-3*c+(2*b-3*d)*x)/(2*b-3*d)+3*cosh(2*a-c+(2*b-d)*x)/(32*b-16 *d)+3/8*cosh(d*x+c)/d-1/24*cosh(3*d*x+3*c)/d-3*cosh(2*a+c+(2*b+d)*x)/(32*b +16*d)+cosh(2*a+3*c+(2*b+3*d)*x)/(32*b+48*d)
Time = 1.18 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=\frac {1}{48} \left (\frac {18 \cosh (c) \cosh (d x)}{d}-\frac {2 \cosh (3 c) \cosh (3 d x)}{d}-\frac {3 \cosh (2 a-3 c+2 b x-3 d x)}{2 b-3 d}+\frac {9 \cosh (2 a-c+2 b x-d x)}{2 b-d}-\frac {9 \cosh (2 a+c+2 b x+d x)}{2 b+d}+\frac {3 \cosh (2 a+3 c+2 b x+3 d x)}{2 b+3 d}+\frac {18 \sinh (c) \sinh (d x)}{d}-\frac {2 \sinh (3 c) \sinh (3 d x)}{d}\right ) \] Input:
Integrate[Sinh[a + b*x]^2*Sinh[c + d*x]^3,x]
Output:
((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
Time = 0.37 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6147, 2009}
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.
| \(\displaystyle \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx\) |
| \(\Big \downarrow \) 6147 |
| \(\displaystyle \int \left (-\frac {1}{16} \sinh (2 a+x (2 b-3 d)-3 c)+\frac {3}{16} \sinh (2 a+x (2 b-d)-c)-\frac {3}{16} \sinh (2 a+x (2 b+d)+c)+\frac {1}{16} \sinh (2 a+x (2 b+3 d)+3 c)+\frac {3}{8} \sinh (c+d x)-\frac {1}{8} \sinh (3 c+3 d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\cosh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \cosh (2 a+x (2 b-d)-c)}{16 (2 b-d)}-\frac {3 \cosh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\cosh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac {3 \cosh (c+d x)}{8 d}-\frac {\cosh (3 c+3 d x)}{24 d}\) |
Input:
Int[Sinh[a + b*x]^2*Sinh[c + d*x]^3,x]
Output:
-1/16*Cosh[2*a - 3*c + (2*b - 3*d)*x]/(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))
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 ]))
Time = 9.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(\frac {3 \cosh \left (d x +c \right )}{8 d}-\frac {\cosh \left (3 d x +3 c \right )}{24 d}-\frac {\cosh \left (2 a -3 c +\left (2 b -3 d \right ) x \right )}{16 \left (2 b -3 d \right )}+\frac {3 \cosh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 \left (2 b -d \right )}-\frac {3 \cosh \left (2 a +c +\left (2 b +d \right ) x \right )}{16 \left (2 b +d \right )}+\frac {\cosh \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{32 b +48 d}\) | \(133\) |
| parallelrisch | \(\frac {\left (-24 b^{3} d -36 b^{2} d^{2}+6 b \,d^{3}+9 d^{4}\right ) \cosh \left (2 a -3 c +\left (2 b -3 d \right ) x \right )+\left (72 b^{3} d +36 b^{2} d^{2}-162 b \,d^{3}-81 d^{4}\right ) \cosh \left (2 a -c +\left (2 b -d \right ) x \right )+\left (24 b^{3} d -36 b^{2} d^{2}-6 b \,d^{3}+9 d^{4}\right ) \cosh \left (2 a +3 c +\left (2 b +3 d \right ) x \right )+\left (-72 b^{3} d +36 b^{2} d^{2}+162 b \,d^{3}-81 d^{4}\right ) \cosh \left (2 a +c +\left (2 b +d \right ) x \right )+\left (-32 b^{4}+80 b^{2} d^{2}-18 d^{4}\right ) \cosh \left (3 d x +3 c \right )+\left (288 b^{4}-720 b^{2} d^{2}+162 d^{4}\right ) \cosh \left (d x +c \right )+256 b^{4}-640 b^{2} d^{2}}{768 b^{4} d -1920 b^{2} d^{3}+432 d^{5}}\) | \(265\) |
| risch | \(-\frac {\left (-6 d \,{\mathrm e}^{4 b x +4 a} b +9 d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-18 d^{2} {\mathrm e}^{2 b x +2 a}+6 b d +9 d^{2}\right ) {\mathrm e}^{-2 b x +3 d x -2 a +3 c}}{96 \left (2 b +3 d \right ) \left (2 b -3 d \right ) d}+\frac {3 \left (-2 d \,{\mathrm e}^{4 b x +4 a} b +d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}+2 b d +d^{2}\right ) {\mathrm e}^{-2 b x +d x -2 a +c}}{32 \left (2 b +d \right ) \left (2 b -d \right ) d}+\frac {3 \left (2 d \,{\mathrm e}^{4 b x +4 a} b +d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}-2 b d +d^{2}\right ) {\mathrm e}^{-2 b x -d x -2 a -c}}{32 \left (2 b +d \right ) \left (2 b -d \right ) d}-\frac {\left (6 d \,{\mathrm e}^{4 b x +4 a} b +9 d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-18 d^{2} {\mathrm e}^{2 b x +2 a}-6 b d +9 d^{2}\right ) {\mathrm e}^{-2 b x -3 d x -2 a -3 c}}{96 \left (2 b +3 d \right ) \left (2 b -3 d \right ) d}\) | \(405\) |
| orering | \(\text {Expression too large to display}\) | \(3070\) |
Input:
int(sinh(b*x+a)^2*sinh(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
3/8*cosh(d*x+c)/d-1/24*cosh(3*d*x+3*c)/d-1/16*cosh(2*a-3*c+(2*b-3*d)*x)/(2 *b-3*d)+3/16/(2*b-d)*cosh(2*a-c+(2*b-d)*x)-3/16/(2*b+d)*cosh(2*a+c+(2*b+d) *x)+1/16/(2*b+3*d)*cosh(2*a+3*c+(2*b+3*d)*x)
Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (132) = 264\).
Time = 0.10 (sec) , antiderivative size = 414, normalized size of antiderivative = 2.88 \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=\frac {12 \, {\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{3} - {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{3} - 9 \, {\left ({\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right )^{3} - {\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right )^{2} + 36 \, {\left ({\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} - {\left (4 \, b^{3} d - 9 \, b d^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) \sinh \left (d x + c\right ) - 3 \, {\left (9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{2} + {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 9 \, {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} + {\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )}{24 \, {\left ({\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} - {\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{2}\right )}} \] Input:
integrate(sinh(b*x+a)^2*sinh(d*x+c)^3,x, algorithm="fricas")
Output:
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)*cosh (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)
Leaf count of result is larger than twice the leaf count of optimal. 1999 vs. \(2 (116) = 232\).
Time = 5.33 (sec) , antiderivative size = 1999, normalized size of antiderivative = 13.88 \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=\text {Too large to display} \] Input:
integrate(sinh(b*x+a)**2*sinh(d*x+c)**3,x)
Output:
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*sinh(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*c osh(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*sin h(c + d*x)**3/16 - 3*x*sinh(a - d*x/2)**2*sinh(c + d*x)*cosh(c + d*x)**2/1 6 + 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 + sinh(a - d*x/2)**2*sinh(c + d*x)**2*cosh(c + d*x)/d - 31*sin h(a - d*x/2)**2*cosh(c + d*x)**3/(48*d) + 3*sinh(a - d*x/2)*sinh(c + d*x)* *3*cosh(a - d*x/2)/(8*d) - sinh(a - d*x/2)*sinh(c + d*x)*cosh(a - d*x/2)*c osh(c + d*x)**2/(4*d) + 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)*cosh(c + d*x)**2/16 - 3*x*sinh(a + d*x/2)*sinh(c + d*x...
Exception generated. \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sinh(b*x+a)^2*sinh(d*x+c)^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(1-(3*d)/b>0)', see `assume?` for more deta
Time = 0.12 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.81 \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=\frac {e^{\left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} - \frac {3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{32 \, {\left (2 \, b + d\right )}} + \frac {3 \, e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{32 \, {\left (2 \, b - d\right )}} - \frac {e^{\left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {e^{\left (-2 \, b x + 3 \, d x - 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} + \frac {3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{32 \, {\left (2 \, b - d\right )}} - \frac {3 \, e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{32 \, {\left (2 \, b + d\right )}} + \frac {e^{\left (-2 \, b x - 3 \, d x - 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} - \frac {e^{\left (3 \, d x + 3 \, c\right )}}{48 \, d} + \frac {3 \, e^{\left (d x + c\right )}}{16 \, d} + \frac {3 \, e^{\left (-d x - c\right )}}{16 \, d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{48 \, d} \] Input:
integrate(sinh(b*x+a)^2*sinh(d*x+c)^3,x, algorithm="giac")
Output:
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
Time = 1.51 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.34 \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=\frac {\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (8\,b^4-26\,b^2\,d^2+9\,d^4\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}+\frac {1}{3\,d}\right )-\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (7\,b\,d^2-4\,b^3\right )}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (8\,b^4-14\,b^2\,d^2\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {1}{3\,d}\right )+\frac {12\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{16\,b^4-40\,b^2\,d^2+9\,d^4} \] Input:
int(sinh(a + b*x)^2*sinh(c + d*x)^3,x)
Output:
(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)
Time = 0.24 (sec) , antiderivative size = 865, normalized size of antiderivative = 6.01 \[ \int \sinh ^2(a+b x) \sinh ^3(c+d x) \, dx=\frac {36 e^{4 b x +4 d x +4 a +4 c} b^{2} d^{2}+162 e^{4 b x +4 d x +4 a +4 c} b \,d^{3}+72 e^{4 b x +2 d x +4 a +2 c} b^{3} d +36 e^{4 b x +2 d x +4 a +2 c} b^{2} d^{2}-162 e^{4 b x +2 d x +4 a +2 c} b \,d^{3}-24 e^{4 b x +4 a} b^{3} d -36 e^{4 b x +4 a} b^{2} d^{2}+6 e^{4 b x +4 a} b \,d^{3}+80 e^{2 b x +6 d x +2 a +6 c} b^{2} d^{2}-720 e^{2 b x +4 d x +2 a +4 c} b^{2} d^{2}-720 e^{2 b x +2 d x +2 a +2 c} b^{2} d^{2}+80 e^{2 b x +2 a} b^{2} d^{2}-24 e^{6 d x +6 c} b^{3} d -36 e^{6 d x +6 c} b^{2} d^{2}+6 e^{6 d x +6 c} b \,d^{3}+72 e^{4 d x +4 c} b^{3} d +36 e^{4 d x +4 c} b^{2} d^{2}-162 e^{4 d x +4 c} b \,d^{3}-72 e^{2 d x +2 c} b^{3} d +36 e^{2 d x +2 c} b^{2} d^{2}+162 e^{2 d x +2 c} b \,d^{3}+24 e^{4 b x +6 d x +4 a +6 c} b^{3} d -36 e^{4 b x +6 d x +4 a +6 c} b^{2} d^{2}-6 e^{4 b x +6 d x +4 a +6 c} b \,d^{3}-72 e^{4 b x +4 d x +4 a +4 c} b^{3} d +288 e^{2 b x +2 d x +2 a +2 c} b^{4}+162 e^{2 b x +2 d x +2 a +2 c} d^{4}-32 e^{2 b x +2 a} b^{4}-18 e^{2 b x +2 a} d^{4}+9 e^{6 d x +6 c} d^{4}-81 e^{4 d x +4 c} d^{4}-81 e^{2 d x +2 c} d^{4}+24 b^{3} d -6 b \,d^{3}+9 e^{4 b x +6 d x +4 a +6 c} d^{4}-81 e^{4 b x +4 d x +4 a +4 c} d^{4}-81 e^{4 b x +2 d x +4 a +2 c} d^{4}+9 e^{4 b x +4 a} d^{4}-32 e^{2 b x +6 d x +2 a +6 c} b^{4}-18 e^{2 b x +6 d x +2 a +6 c} d^{4}+288 e^{2 b x +4 d x +2 a +4 c} b^{4}+162 e^{2 b x +4 d x +2 a +4 c} d^{4}+9 d^{4}-36 b^{2} d^{2}}{96 e^{2 b x +3 d x +2 a +3 c} d \left (16 b^{4}-40 b^{2} d^{2}+9 d^{4}\right )} \] Input:
int(sinh(b*x+a)^2*sinh(d*x+c)^3,x)
Output:
(24*e**(4*a + 4*b*x + 6*c + 6*d*x)*b**3*d - 36*e**(4*a + 4*b*x + 6*c + 6*d *x)*b**2*d**2 - 6*e**(4*a + 4*b*x + 6*c + 6*d*x)*b*d**3 + 9*e**(4*a + 4*b* x + 6*c + 6*d*x)*d**4 - 72*e**(4*a + 4*b*x + 4*c + 4*d*x)*b**3*d + 36*e**( 4*a + 4*b*x + 4*c + 4*d*x)*b**2*d**2 + 162*e**(4*a + 4*b*x + 4*c + 4*d*x)* b*d**3 - 81*e**(4*a + 4*b*x + 4*c + 4*d*x)*d**4 + 72*e**(4*a + 4*b*x + 2*c + 2*d*x)*b**3*d + 36*e**(4*a + 4*b*x + 2*c + 2*d*x)*b**2*d**2 - 162*e**(4 *a + 4*b*x + 2*c + 2*d*x)*b*d**3 - 81*e**(4*a + 4*b*x + 2*c + 2*d*x)*d**4 - 24*e**(4*a + 4*b*x)*b**3*d - 36*e**(4*a + 4*b*x)*b**2*d**2 + 6*e**(4*a + 4*b*x)*b*d**3 + 9*e**(4*a + 4*b*x)*d**4 - 32*e**(2*a + 2*b*x + 6*c + 6*d* x)*b**4 + 80*e**(2*a + 2*b*x + 6*c + 6*d*x)*b**2*d**2 - 18*e**(2*a + 2*b*x + 6*c + 6*d*x)*d**4 + 288*e**(2*a + 2*b*x + 4*c + 4*d*x)*b**4 - 720*e**(2 *a + 2*b*x + 4*c + 4*d*x)*b**2*d**2 + 162*e**(2*a + 2*b*x + 4*c + 4*d*x)*d **4 + 288*e**(2*a + 2*b*x + 2*c + 2*d*x)*b**4 - 720*e**(2*a + 2*b*x + 2*c + 2*d*x)*b**2*d**2 + 162*e**(2*a + 2*b*x + 2*c + 2*d*x)*d**4 - 32*e**(2*a + 2*b*x)*b**4 + 80*e**(2*a + 2*b*x)*b**2*d**2 - 18*e**(2*a + 2*b*x)*d**4 - 24*e**(6*c + 6*d*x)*b**3*d - 36*e**(6*c + 6*d*x)*b**2*d**2 + 6*e**(6*c + 6*d*x)*b*d**3 + 9*e**(6*c + 6*d*x)*d**4 + 72*e**(4*c + 4*d*x)*b**3*d + 36* e**(4*c + 4*d*x)*b**2*d**2 - 162*e**(4*c + 4*d*x)*b*d**3 - 81*e**(4*c + 4* d*x)*d**4 - 72*e**(2*c + 2*d*x)*b**3*d + 36*e**(2*c + 2*d*x)*b**2*d**2 + 1 62*e**(2*c + 2*d*x)*b*d**3 - 81*e**(2*c + 2*d*x)*d**4 + 24*b**3*d - 36*...