Integrand size = 17, antiderivative size = 144 \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\frac {\sinh (2 a-3 c+(2 b-3 d) x)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a-c+(2 b-d) x)}{16 (2 b-d)}+\frac {3 \sinh (c+d x)}{8 d}+\frac {\sinh (3 c+3 d x)}{24 d}+\frac {3 \sinh (2 a+c+(2 b+d) x)}{16 (2 b+d)}+\frac {\sinh (2 a+3 c+(2 b+3 d) x)}{16 (2 b+3 d)} \] Output:
sinh(2*a-3*c+(2*b-3*d)*x)/(32*b-48*d)+3*sinh(2*a-c+(2*b-d)*x)/(32*b-16*d)+ 3/8*sinh(d*x+c)/d+1/24*sinh(3*d*x+3*c)/d+3*sinh(2*a+c+(2*b+d)*x)/(32*b+16* d)+sinh(2*a+3*c+(2*b+3*d)*x)/(32*b+48*d)
Time = 1.07 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.10 \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\frac {1}{48} \left (\frac {18 \cosh (d x) \sinh (c)}{d}+\frac {2 \cosh (3 d x) \sinh (3 c)}{d}+\frac {18 \cosh (c) \sinh (d x)}{d}+\frac {2 \cosh (3 c) \sinh (3 d x)}{d}+\frac {3 \sinh (2 a-3 c+2 b x-3 d x)}{2 b-3 d}+\frac {9 \sinh (2 a-c+2 b x-d x)}{2 b-d}+\frac {9 \sinh (2 a+c+2 b x+d x)}{2 b+d}+\frac {3 \sinh (2 a+3 c+2 b x+3 d x)}{2 b+3 d}\right ) \] Input:
Integrate[Cosh[a + b*x]^2*Cosh[c + d*x]^3,x]
Output:
((18*Cosh[d*x]*Sinh[c])/d + (2*Cosh[3*d*x]*Sinh[3*c])/d + (18*Cosh[c]*Sinh [d*x])/d + (2*Cosh[3*c]*Sinh[3*d*x])/d + (3*Sinh[2*a - 3*c + 2*b*x - 3*d*x ])/(2*b - 3*d) + (9*Sinh[2*a - c + 2*b*x - d*x])/(2*b - d) + (9*Sinh[2*a + c + 2*b*x + d*x])/(2*b + d) + (3*Sinh[2*a + 3*c + 2*b*x + 3*d*x])/(2*b + 3*d))/48
Time = 0.36 (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 = {6148, 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.
| \(\displaystyle \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx\) |
| \(\Big \downarrow \) 6148 |
| \(\displaystyle \int \left (\frac {1}{16} \cosh (2 a+x (2 b-3 d)-3 c)+\frac {3}{16} \cosh (2 a+x (2 b-d)-c)+\frac {3}{16} \cosh (2 a+x (2 b+d)+c)+\frac {1}{16} \cosh (2 a+x (2 b+3 d)+3 c)+\frac {3}{8} \cosh (c+d x)+\frac {1}{8} \cosh (3 c+3 d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sinh (2 a+x (2 b-3 d)-3 c)}{16 (2 b-3 d)}+\frac {3 \sinh (2 a+x (2 b-d)-c)}{16 (2 b-d)}+\frac {3 \sinh (2 a+x (2 b+d)+c)}{16 (2 b+d)}+\frac {\sinh (2 a+x (2 b+3 d)+3 c)}{16 (2 b+3 d)}+\frac {3 \sinh (c+d x)}{8 d}+\frac {\sinh (3 c+3 d x)}{24 d}\) |
Input:
Int[Cosh[a + b*x]^2*Cosh[c + d*x]^3,x]
Output:
Sinh[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) + (3*Sinh[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Sinh[c + d*x])/(8*d) + Sinh[3*c + 3*d*x]/(24* d) + (3*Sinh[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Sinh[2*a + 3*c + (2* b + 3*d)*x]/(16*(2*b + 3*d))
Int[Cosh[v_]^(p_.)*Cosh[w_]^(q_.), x_Symbol] :> Int[ExpandTrigReduce[Cosh[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 ]))
Time = 9.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(\frac {3 \sinh \left (d x +c \right )}{8 d}+\frac {\sinh \left (3 d x +3 c \right )}{24 d}+\frac {\sinh \left (2 a -3 c +\left (2 b -3 d \right ) x \right )}{32 b -48 d}+\frac {3 \sinh \left (2 a -c +\left (2 b -d \right ) x \right )}{16 \left (2 b -d \right )}+\frac {3 \sinh \left (2 a +c +\left (2 b +d \right ) x \right )}{16 \left (2 b +d \right )}+\frac {\sinh \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{32 b +48 d}\) | \(133\) |
| parallelrisch | \(\frac {\left (24 b^{3} d +36 b^{2} d^{2}-6 b \,d^{3}-9 d^{4}\right ) \sinh \left (2 a -3 c +\left (2 b -3 d \right ) x \right )+72 \left (b -\frac {3 d}{2}\right ) \left (\left (b +\frac {3 d}{2}\right ) d \left (b +\frac {d}{2}\right ) \sinh \left (2 a -c +\left (2 b -d \right ) x \right )+\left (b -\frac {d}{2}\right ) \left (\frac {d \left (b +\frac {d}{2}\right ) \sinh \left (2 a +3 c +\left (2 b +3 d \right ) x \right )}{3}+\left (\sinh \left (2 a +c +\left (2 b +d \right ) x \right ) d +4 \left (\sinh \left (d x +c \right )+\frac {\sinh \left (3 d x +3 c \right )}{9}\right ) \left (b +\frac {d}{2}\right )\right ) \left (b +\frac {3 d}{2}\right )\right )\right )}{768 b^{4} d -1920 b^{2} d^{3}+432 d^{5}}\) | \(185\) |
| risch | \(\frac {\left (6 d \,{\mathrm e}^{4 b x +4 a} b -9 d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-18 d^{2} {\mathrm e}^{2 b x +2 a}-6 b d -9 d^{2}\right ) {\mathrm e}^{-2 b x +3 d x -2 a +3 c}}{96 \left (2 b +3 d \right ) \left (2 b -3 d \right ) d}+\frac {3 \left (2 d \,{\mathrm e}^{4 b x +4 a} b -d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}-2 b d -d^{2}\right ) {\mathrm e}^{-2 b x +d x -2 a +c}}{32 \left (2 b +d \right ) \left (2 b -d \right ) d}-\frac {3 \left (-2 d \,{\mathrm e}^{4 b x +4 a} b -d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-2 d^{2} {\mathrm e}^{2 b x +2 a}+2 b d -d^{2}\right ) {\mathrm e}^{-2 b x -d x -2 a -c}}{32 \left (2 b +d \right ) \left (2 b -d \right ) d}-\frac {\left (-6 d \,{\mathrm e}^{4 b x +4 a} b -9 d^{2} {\mathrm e}^{4 b x +4 a}+8 b^{2} {\mathrm e}^{2 b x +2 a}-18 d^{2} {\mathrm e}^{2 b x +2 a}+6 b d -9 d^{2}\right ) {\mathrm e}^{-2 b x -3 d x -2 a -3 c}}{96 \left (2 b +3 d \right ) \left (2 b -3 d \right ) d}\) | \(411\) |
| orering | \(\text {Expression too large to display}\) | \(3070\) |
Input:
int(cosh(b*x+a)^2*cosh(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
3/8*sinh(d*x+c)/d+1/24*sinh(3*d*x+3*c)/d+1/16*sinh(2*a-3*c+(2*b-3*d)*x)/(2 *b-3*d)+3/16*sinh(2*a-c+(2*b-d)*x)/(2*b-d)+3/16*sinh(2*a+c+(2*b+d)*x)/(2*b +d)+1/16*sinh(2*a+3*c+(2*b+3*d)*x)/(2*b+3*d)
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (132) = 264\).
Time = 0.10 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.76 \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\frac {36 \, {\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right ) \sinh \left (b x + a\right ) \sinh \left (d x + c\right )^{2} + {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} - 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2} - 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 12 \, {\left ({\left (4 \, b^{3} d - b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{3} + 3 \, {\left (4 \, b^{3} d - 9 \, b d^{3}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )\right )} \sinh \left (b x + a\right ) + 3 \, {\left (48 \, b^{4} - 120 \, b^{2} d^{2} + 27 \, d^{4} - 3 \, {\left (4 \, b^{2} d^{2} - 9 \, d^{4}\right )} \cosh \left (b x + a\right )^{2} + {\left (16 \, b^{4} - 40 \, b^{2} d^{2} + 9 \, d^{4} - 9 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right )^{2} - 3 \, {\left (4 \, b^{2} d^{2} - 9 \, d^{4} + 3 \, {\left (4 \, b^{2} d^{2} - d^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left ({\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \cosh \left (b x + a\right )^{2} - {\left (16 \, b^{4} d - 40 \, b^{2} d^{3} + 9 \, d^{5}\right )} \sinh \left (b x + a\right )^{2}\right )}} \] Input:
integrate(cosh(b*x+a)^2*cosh(d*x+c)^3,x, algorithm="fricas")
Output:
1/24*(36*(4*b^3*d - b*d^3)*cosh(b*x + a)*cosh(d*x + c)*sinh(b*x + a)*sinh( d*x + c)^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 - 9*(4*b^2*d^2 - d^4)*sinh(b*x + a)^2)*sinh(d*x + c)^3 + 12*((4*b^3* d - b*d^3)*cosh(b*x + a)*cosh(d*x + c)^3 + 3*(4*b^3*d - 9*b*d^3)*cosh(b*x + a)*cosh(d*x + c))*sinh(b*x + a) + 3*(48*b^4 - 120*b^2*d^2 + 27*d^4 - 3*( 4*b^2*d^2 - 9*d^4)*cosh(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)^2 - 3*(4*b^2*d^2 - 9*d^4 + 3* (4*b^2*d^2 - d^4)*cosh(d*x + c)^2)*sinh(b*x + a)^2)*sinh(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. 2009 vs. \(2 (116) = 232\).
Time = 5.32 (sec) , antiderivative size = 2009, normalized size of antiderivative = 13.95 \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\text {Too large to display} \] Input:
integrate(cosh(b*x+a)**2*cosh(d*x+c)**3,x)
Output:
Piecewise((x*cosh(a)**2*cosh(c)**3, Eq(b, 0) & Eq(d, 0)), (3*x*sinh(a - 3* d*x/2)**2*sinh(c + d*x)**2*cosh(c + d*x)/16 + x*sinh(a - 3*d*x/2)**2*cosh( c + d*x)**3/16 + x*sinh(a - 3*d*x/2)*sinh(c + d*x)**3*cosh(a - 3*d*x/2)/8 + 3*x*sinh(a - 3*d*x/2)*sinh(c + d*x)*cosh(a - 3*d*x/2)*cosh(c + d*x)**2/8 + 3*x*sinh(c + d*x)**2*cosh(a - 3*d*x/2)**2*cosh(c + d*x)/16 + x*cosh(a - 3*d*x/2)**2*cosh(c + d*x)**3/16 + 11*sinh(a - 3*d*x/2)**2*sinh(c + d*x)** 3/(48*d) - sinh(a - 3*d*x/2)**2*sinh(c + d*x)*cosh(c + d*x)**2/d - 3*sinh( a - 3*d*x/2)*sinh(c + d*x)**2*cosh(a - 3*d*x/2)*cosh(c + d*x)/(4*d) - 5*si nh(a - 3*d*x/2)*cosh(a - 3*d*x/2)*cosh(c + d*x)**3/(8*d) - 7*sinh(c + d*x) **3*cosh(a - 3*d*x/2)**2/(16*d), Eq(b, -3*d/2)), (-3*x*sinh(a - d*x/2)**2* sinh(c + d*x)**2*cosh(c + d*x)/16 + 3*x*sinh(a - d*x/2)**2*cosh(c + d*x)** 3/16 - 3*x*sinh(a - d*x/2)*sinh(c + d*x)**3*cosh(a - d*x/2)/8 + 3*x*sinh(a - d*x/2)*sinh(c + d*x)*cosh(a - d*x/2)*cosh(c + d*x)**2/8 - 3*x*sinh(c + d*x)**2*cosh(a - d*x/2)**2*cosh(c + d*x)/16 + 3*x*cosh(a - d*x/2)**2*cosh( c + d*x)**3/16 + 49*sinh(a - d*x/2)**2*sinh(c + d*x)**3/(48*d) - sinh(a - d*x/2)**2*sinh(c + d*x)*cosh(c + d*x)**2/d + 7*sinh(a - d*x/2)*sinh(c + d* x)**2*cosh(a - d*x/2)*cosh(c + d*x)/(4*d) - 13*sinh(a - d*x/2)*cosh(a - d* x/2)*cosh(c + d*x)**3/(8*d) + 17*sinh(c + d*x)**3*cosh(a - d*x/2)**2/(48*d ), Eq(b, -d/2)), (-3*x*sinh(a + d*x/2)**2*sinh(c + d*x)**2*cosh(c + d*x)/1 6 + 3*x*sinh(a + d*x/2)**2*cosh(c + d*x)**3/16 + 3*x*sinh(a + d*x/2)*si...
Exception generated. \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(b*x+a)^2*cosh(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 = 0.13 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.81 \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\frac {e^{\left (2 \, b x + 3 \, d x + 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} + \frac {3 \, e^{\left (2 \, b x + d x + 2 \, a + c\right )}}{32 \, {\left (2 \, b + d\right )}} + \frac {3 \, e^{\left (2 \, b x - d x + 2 \, a - c\right )}}{32 \, {\left (2 \, b - d\right )}} + \frac {e^{\left (2 \, b x - 3 \, d x + 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {e^{\left (-2 \, b x + 3 \, d x - 2 \, a + 3 \, c\right )}}{32 \, {\left (2 \, b - 3 \, d\right )}} - \frac {3 \, e^{\left (-2 \, b x + d x - 2 \, a + c\right )}}{32 \, {\left (2 \, b - d\right )}} - \frac {3 \, e^{\left (-2 \, b x - d x - 2 \, a - c\right )}}{32 \, {\left (2 \, b + d\right )}} - \frac {e^{\left (-2 \, b x - 3 \, d x - 2 \, a - 3 \, c\right )}}{32 \, {\left (2 \, b + 3 \, d\right )}} + \frac {e^{\left (3 \, d x + 3 \, c\right )}}{48 \, d} + \frac {3 \, e^{\left (d x + c\right )}}{16 \, d} - \frac {3 \, e^{\left (-d x - c\right )}}{16 \, d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{48 \, d} \] Input:
integrate(cosh(b*x+a)^2*cosh(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 = 1.54 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.34 \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\frac {{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (8\,b^4-26\,b^2\,d^2+9\,d^4\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )}-{\mathrm {sinh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {1}{3\,d}\right )-\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^3\,\mathrm {sinh}\left (a+b\,x\right )\,\left (7\,b\,d^2-4\,b^3\right )}{16\,b^4-40\,b^2\,d^2+9\,d^4}-{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3\,\left (\frac {3\,d^3}{16\,b^4-40\,b^2\,d^2+9\,d^4}+\frac {1}{3\,d}\right )+\frac {12\,b\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (c+d\,x\right )}^2}{16\,b^4-40\,b^2\,d^2+9\,d^4}-\frac {2\,b^2\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (c+d\,x\right )\,\left (4\,b^2-7\,d^2\right )}{d\,\left (16\,b^4-40\,b^2\,d^2+9\,d^4\right )} \] Input:
int(cosh(a + b*x)^2*cosh(c + d*x)^3,x)
Output:
(cosh(a + b*x)^2*cosh(c + d*x)^2*sinh(c + d*x)*(8*b^4 + 9*d^4 - 26*b^2*d^2 ))/(d*(16*b^4 + 9*d^4 - 40*b^2*d^2)) - sinh(a + b*x)^2*sinh(c + d*x)^3*((3 *d^3)/(16*b^4 + 9*d^4 - 40*b^2*d^2) - 1/(3*d)) - (2*cosh(a + b*x)*cosh(c + d*x)^3*sinh(a + b*x)*(7*b*d^2 - 4*b^3))/(16*b^4 + 9*d^4 - 40*b^2*d^2) - c osh(a + b*x)^2*sinh(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)*sinh(a + b*x)*sinh(c + d*x) ^2)/(16*b^4 + 9*d^4 - 40*b^2*d^2) - (2*b^2*cosh(c + d*x)^2*sinh(a + b*x)^2 *sinh(c + d*x)*(4*b^2 - 7*d^2))/(d*(16*b^4 + 9*d^4 - 40*b^2*d^2))
Time = 0.26 (sec) , antiderivative size = 865, normalized size of antiderivative = 6.01 \[ \int \cosh ^2(a+b x) \cosh ^3(c+d x) \, dx=\frac {-36 e^{4 b x +4 d x +4 a +4 c} b^{2} d^{2}-162 e^{4 b x +4 d x +4 a +4 c} b \,d^{3}+72 e^{4 b x +2 d x +4 a +2 c} b^{3} d +36 e^{4 b x +2 d x +4 a +2 c} b^{2} d^{2}-162 e^{4 b x +2 d x +4 a +2 c} b \,d^{3}+24 e^{4 b x +4 a} b^{3} d +36 e^{4 b x +4 a} b^{2} d^{2}-6 e^{4 b x +4 a} b \,d^{3}-80 e^{2 b x +6 d x +2 a +6 c} b^{2} d^{2}-720 e^{2 b x +4 d x +2 a +4 c} b^{2} d^{2}+720 e^{2 b x +2 d x +2 a +2 c} b^{2} d^{2}+80 e^{2 b x +2 a} b^{2} d^{2}-24 e^{6 d x +6 c} b^{3} d -36 e^{6 d x +6 c} b^{2} d^{2}+6 e^{6 d x +6 c} b \,d^{3}-72 e^{4 d x +4 c} b^{3} d -36 e^{4 d x +4 c} b^{2} d^{2}+162 e^{4 d x +4 c} b \,d^{3}-72 e^{2 d x +2 c} b^{3} d +36 e^{2 d x +2 c} b^{2} d^{2}+162 e^{2 d x +2 c} b \,d^{3}+24 e^{4 b x +6 d x +4 a +6 c} b^{3} d -36 e^{4 b x +6 d x +4 a +6 c} b^{2} d^{2}-6 e^{4 b x +6 d x +4 a +6 c} b \,d^{3}+72 e^{4 b x +4 d x +4 a +4 c} b^{3} d -288 e^{2 b x +2 d x +2 a +2 c} b^{4}-162 e^{2 b x +2 d x +2 a +2 c} d^{4}-32 e^{2 b x +2 a} b^{4}-18 e^{2 b x +2 a} d^{4}+9 e^{6 d x +6 c} d^{4}+81 e^{4 d x +4 c} d^{4}-81 e^{2 d x +2 c} d^{4}-24 b^{3} d +6 b \,d^{3}+9 e^{4 b x +6 d x +4 a +6 c} d^{4}+81 e^{4 b x +4 d x +4 a +4 c} d^{4}-81 e^{4 b x +2 d x +4 a +2 c} d^{4}-9 e^{4 b x +4 a} d^{4}+32 e^{2 b x +6 d x +2 a +6 c} b^{4}+18 e^{2 b x +6 d x +2 a +6 c} d^{4}+288 e^{2 b x +4 d x +2 a +4 c} b^{4}+162 e^{2 b x +4 d x +2 a +4 c} d^{4}-9 d^{4}+36 b^{2} d^{2}}{96 e^{2 b x +3 d x +2 a +3 c} d \left (16 b^{4}-40 b^{2} d^{2}+9 d^{4}\right )} \] Input:
int(cosh(b*x+a)^2*cosh(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*...