Integrand size = 17, antiderivative size = 138 \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=-\frac {3 \cosh (a+b x)}{8 b}+\frac {\cosh (3 a+3 b x)}{24 b}-\frac {3 \cosh (a-2 c+(b-2 d) x)}{16 (b-2 d)}+\frac {\cosh (3 a-2 c+(3 b-2 d) x)}{16 (3 b-2 d)}-\frac {3 \cosh (a+2 c+(b+2 d) x)}{16 (b+2 d)}+\frac {\cosh (3 a+2 c+(3 b+2 d) x)}{16 (3 b+2 d)} \] Output:
-3/8*cosh(b*x+a)/b+1/24*cosh(3*b*x+3*a)/b-3*cosh(a-2*c+(b-2*d)*x)/(16*b-32 *d)+cosh(3*a-2*c+(3*b-2*d)*x)/(48*b-32*d)-3*cosh(a+2*c+(b+2*d)*x)/(16*b+32 *d)+cosh(3*a+2*c+(3*b+2*d)*x)/(48*b+32*d)
Time = 1.09 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.11 \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=\frac {1}{48} \left (-\frac {18 \cosh (a) \cosh (b x)}{b}+\frac {2 \cosh (3 a) \cosh (3 b x)}{b}-\frac {9 \cosh (a-2 c+b x-2 d x)}{b-2 d}+\frac {3 \cosh (3 a-2 c+3 b x-2 d x)}{3 b-2 d}-\frac {9 \cosh (a+2 c+b x+2 d x)}{b+2 d}+\frac {3 \cosh (3 a+2 c+3 b x+2 d x)}{3 b+2 d}-\frac {18 \sinh (a) \sinh (b x)}{b}+\frac {2 \sinh (3 a) \sinh (3 b x)}{b}\right ) \] Input:
Integrate[Cosh[c + d*x]^2*Sinh[a + b*x]^3,x]
Output:
((-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
Time = 0.34 (sec) , antiderivative size = 138, 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 = {6152, 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.
| \(\displaystyle \int \sinh ^3(a+b x) \cosh ^2(c+d x) \, dx\) |
| \(\Big \downarrow \) 6152 |
| \(\displaystyle \int \left (-\frac {3}{16} \sinh (a+x (b-2 d)-2 c)+\frac {1}{16} \sinh (3 a+x (3 b-2 d)-2 c)-\frac {3}{16} \sinh (a+x (b+2 d)+2 c)+\frac {1}{16} \sinh (3 a+x (3 b+2 d)+2 c)-\frac {3}{8} \sinh (a+b x)+\frac {1}{8} \sinh (3 a+3 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \cosh (a+x (b-2 d)-2 c)}{16 (b-2 d)}+\frac {\cosh (3 a+x (3 b-2 d)-2 c)}{16 (3 b-2 d)}-\frac {3 \cosh (a+x (b+2 d)+2 c)}{16 (b+2 d)}+\frac {\cosh (3 a+x (3 b+2 d)+2 c)}{16 (3 b+2 d)}-\frac {3 \cosh (a+b x)}{8 b}+\frac {\cosh (3 a+3 b x)}{24 b}\) |
Input:
Int[Cosh[c + d*x]^2*Sinh[a + b*x]^3,x]
Output:
(-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))
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 ]))
Time = 8.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.92
| method | result | size |
| default | \(-\frac {3 \cosh \left (b x +a \right )}{8 b}+\frac {\cosh \left (3 b x +3 a \right )}{24 b}-\frac {3 \cosh \left (a -2 c +\left (b -2 d \right ) x \right )}{16 \left (b -2 d \right )}-\frac {3 \cosh \left (a +2 c +\left (b +2 d \right ) x \right )}{16 \left (b +2 d \right )}+\frac {\cosh \left (3 a -2 c +\left (3 b -2 d \right ) x \right )}{48 b -32 d}+\frac {\cosh \left (3 a +2 c +\left (3 b +2 d \right ) x \right )}{48 b +32 d}\) | \(127\) |
| parallelrisch | \(\frac {9 \left (b +\frac {2 d}{3}\right ) b \left (b -2 d \right ) \left (b +2 d \right ) \cosh \left (3 a -2 c +\left (3 b -2 d \right ) x \right )+9 b \left (b -2 d \right ) \left (b +2 d \right ) \left (b -\frac {2 d}{3}\right ) \cosh \left (3 a +2 c +\left (3 b +2 d \right ) x \right )-81 \left (b +\frac {2 d}{3}\right ) b \left (b +2 d \right ) \left (b -\frac {2 d}{3}\right ) \cosh \left (a -2 c +\left (b -2 d \right ) x \right )-81 \left (b +\frac {2 d}{3}\right ) b \left (b -2 d \right ) \left (b -\frac {2 d}{3}\right ) \cosh \left (a +2 c +\left (b +2 d \right ) x \right )+\left (18 b^{4}-80 b^{2} d^{2}+32 d^{4}\right ) \cosh \left (3 b x +3 a \right )+\left (-162 b^{4}+720 b^{2} d^{2}-288 d^{4}\right ) \cosh \left (b x +a \right )-288 b^{4}+640 b^{2} d^{2}-256 d^{4}}{432 b^{5}-1920 b^{3} d^{2}+768 b \,d^{4}}\) | \(230\) |
| risch | \(\frac {{\mathrm e}^{3 b x +3 a}}{48 b}-\frac {3 \,{\mathrm e}^{b x +a}}{16 b}-\frac {3 \,{\mathrm e}^{-b x -a}}{16 b}+\frac {{\mathrm e}^{-3 b x -3 a}}{48 b}+\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}-2 b^{2} d \,{\mathrm e}^{6 b x +6 a}-12 b \,d^{2} {\mathrm e}^{6 b x +6 a}+8 d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}+54 b^{2} d \,{\mathrm e}^{4 b x +4 a}+12 b \,d^{2} {\mathrm e}^{4 b x +4 a}-24 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}-54 b^{2} d \,{\mathrm e}^{2 b x +2 a}+12 b \,d^{2} {\mathrm e}^{2 b x +2 a}+24 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}+2 b^{2} d -12 b \,d^{2}-8 d^{3}\right ) {\mathrm e}^{-3 b x +2 d x -3 a +2 c}}{32 \left (3 b +2 d \right ) \left (b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}+\frac {\left (3 b^{3} {\mathrm e}^{6 b x +6 a}+2 b^{2} d \,{\mathrm e}^{6 b x +6 a}-12 b \,d^{2} {\mathrm e}^{6 b x +6 a}-8 d^{3} {\mathrm e}^{6 b x +6 a}-27 b^{3} {\mathrm e}^{4 b x +4 a}-54 b^{2} d \,{\mathrm e}^{4 b x +4 a}+12 b \,d^{2} {\mathrm e}^{4 b x +4 a}+24 d^{3} {\mathrm e}^{4 b x +4 a}-27 b^{3} {\mathrm e}^{2 b x +2 a}+54 b^{2} d \,{\mathrm e}^{2 b x +2 a}+12 b \,d^{2} {\mathrm e}^{2 b x +2 a}-24 d^{3} {\mathrm e}^{2 b x +2 a}+3 b^{3}-2 b^{2} d -12 b \,d^{2}+8 d^{3}\right ) {\mathrm e}^{-3 b x -2 d x -3 a -2 c}}{32 \left (3 b +2 d \right ) \left (b +2 d \right ) \left (3 b -2 d \right ) \left (b -2 d \right )}\) | \(549\) |
| orering | \(\text {Expression too large to display}\) | \(3070\) |
Input:
int(cosh(d*x+c)^2*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
-3/8*cosh(b*x+a)/b+1/24*cosh(3*b*x+3*a)/b-3/16/(b-2*d)*cosh(a-2*c+(b-2*d)* x)-3/16/(b+2*d)*cosh(a+2*c+(b+2*d)*x)+1/16/(3*b-2*d)*cosh(3*a-2*c+(3*b-2*d )*x)+1/16/(3*b+2*d)*cosh(3*a+2*c+(3*b+2*d)*x)
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (126) = 252\).
Time = 0.12 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.21 \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=\frac {{\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right )^{3} + 9 \, {\left ({\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )^{3} - {\left (9 \, b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )\right )} \cosh \left (d x + c\right )^{2} + 3 \, {\left (9 \, {\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right ) \cosh \left (d x + c\right )^{2} + {\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 9 \, {\left ({\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )^{3} + 3 \, {\left (b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - {\left (9 \, b^{4} - 4 \, b^{2} d^{2}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (d x + c\right )^{2} - 9 \, {\left (9 \, b^{4} - 40 \, b^{2} d^{2} + 16 \, d^{4}\right )} \cosh \left (b x + a\right ) - 12 \, {\left ({\left (b^{3} d - 4 \, b d^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )^{3} - 3 \, {\left (9 \, b^{3} d - 4 \, b d^{3} - {\left (b^{3} d - 4 \, b d^{3}\right )} \cosh \left (b x + a\right )^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (b x + a\right )\right )} \sinh \left (d x + c\right )}{24 \, {\left ({\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \cosh \left (b x + a\right )^{4} - 2 \, {\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + {\left (9 \, b^{5} - 40 \, b^{3} d^{2} + 16 \, b d^{4}\right )} \sinh \left (b x + a\right )^{4}\right )}} \] Input:
integrate(cosh(d*x+c)^2*sinh(b*x+a)^3,x, algorithm="fricas")
Output:
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)
Leaf count of result is larger than twice the leaf count of optimal. 2030 vs. \(2 (116) = 232\).
Time = 5.34 (sec) , antiderivative size = 2030, normalized size of antiderivative = 14.71 \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate(cosh(d*x+c)**2*sinh(b*x+a)**3,x)
Output:
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 + sinh(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)*cos h(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)/(16*d) + 3*sinh(a - 2*d*x/3)**2*sinh(c + d*x)**2*c osh(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)), (x...
Exception generated. \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=\text {Exception raised: ValueError} \] Input:
integrate(cosh(d*x+c)^2*sinh(b*x+a)^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(-(2*d)/b>0)', see `assume?` for more detai
Leaf count of result is larger than twice the leaf count of optimal. 256 vs. \(2 (126) = 252\).
Time = 0.13 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.86 \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=\frac {e^{\left (3 \, b x + 2 \, d x + 3 \, a + 2 \, c\right )}}{32 \, {\left (3 \, b + 2 \, d\right )}} + \frac {e^{\left (3 \, b x - 2 \, d x + 3 \, a - 2 \, c\right )}}{32 \, {\left (3 \, b - 2 \, d\right )}} + \frac {e^{\left (3 \, b x + 3 \, a\right )}}{48 \, b} - \frac {3 \, e^{\left (b x + 2 \, d x + a + 2 \, c\right )}}{32 \, {\left (b + 2 \, d\right )}} - \frac {3 \, e^{\left (b x - 2 \, d x + a - 2 \, c\right )}}{32 \, {\left (b - 2 \, d\right )}} - \frac {3 \, e^{\left (b x + a\right )}}{16 \, b} - \frac {3 \, e^{\left (-b x + 2 \, d x - a + 2 \, c\right )}}{32 \, {\left (b - 2 \, d\right )}} - \frac {3 \, e^{\left (-b x - 2 \, d x - a - 2 \, c\right )}}{32 \, {\left (b + 2 \, d\right )}} - \frac {3 \, e^{\left (-b x - a\right )}}{16 \, b} + \frac {e^{\left (-3 \, b x + 2 \, d x - 3 \, a + 2 \, c\right )}}{32 \, {\left (3 \, b - 2 \, d\right )}} + \frac {e^{\left (-3 \, b x - 2 \, d x - 3 \, a - 2 \, c\right )}}{32 \, {\left (3 \, b + 2 \, d\right )}} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{48 \, b} \] Input:
integrate(cosh(d*x+c)^2*sinh(b*x+a)^3,x, algorithm="giac")
Output:
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
Time = 1.53 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.44 \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=\frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,\left (9\,b^4-26\,b^2\,d^2+8\,d^4\right )}{b\,\left (9\,b^4-40\,b^2\,d^2+16\,d^4\right )}-{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (\frac {3\,b^3}{9\,b^4-40\,b^2\,d^2+16\,d^4}-\frac {1}{3\,b}\right )-{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (\frac {3\,b^3}{9\,b^4-40\,b^2\,d^2+16\,d^4}+\frac {1}{3\,b}\right )-\frac {2\,d\,\mathrm {cosh}\left (c+d\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^3\,\mathrm {sinh}\left (c+d\,x\right )\,\left (7\,b^2-4\,d^2\right )}{9\,b^4-40\,b^2\,d^2+16\,d^4}+\frac {12\,b^2\,d\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {cosh}\left (c+d\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )\,\mathrm {sinh}\left (c+d\,x\right )}{9\,b^4-40\,b^2\,d^2+16\,d^4}+\frac {2\,d^2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (c+d\,x\right )}^2\,\left (7\,b^2-4\,d^2\right )}{b\,\left (9\,b^4-40\,b^2\,d^2+16\,d^4\right )} \] Input:
int(cosh(c + d*x)^2*sinh(a + b*x)^3,x)
Output:
(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*s inh(c + d*x)^2*(7*b^2 - 4*d^2))/(b*(9*b^4 + 16*d^4 - 40*b^2*d^2))
Time = 0.24 (sec) , antiderivative size = 865, normalized size of antiderivative = 6.27 \[ \int \cosh ^2(c+d x) \sinh ^3(a+b x) \, dx=\frac {9 b^{4}+36 e^{4 b x +4 d x +4 a +4 c} b^{2} d^{2}-72 e^{4 b x +4 d x +4 a +4 c} b \,d^{3}+720 e^{4 b x +2 d x +4 a +2 c} b^{2} d^{2}-162 e^{4 b x +4 a} b^{3} d +36 e^{4 b x +4 a} b^{2} d^{2}+72 e^{4 b x +4 a} b \,d^{3}+36 e^{2 b x +4 d x +2 a +4 c} b^{2} d^{2}+720 e^{2 b x +2 d x +2 a +2 c} b^{2} d^{2}+36 e^{2 b x +2 a} b^{2} d^{2}+6 e^{4 d x +4 c} b^{3} d -36 e^{4 d x +4 c} b^{2} d^{2}-24 e^{4 d x +4 c} b \,d^{3}-80 e^{2 d x +2 c} b^{2} d^{2}+162 e^{4 b x +4 d x +4 a +4 c} b^{3} d -162 e^{2 b x +2 d x +2 a +2 c} b^{4}-288 e^{2 b x +2 d x +2 a +2 c} d^{4}-81 e^{2 b x +2 a} b^{4}+32 e^{2 d x +2 c} d^{4}-6 b^{3} d +24 b \,d^{3}-288 e^{4 b x +2 d x +4 a +2 c} d^{4}-81 e^{2 b x +4 d x +2 a +4 c} b^{4}+9 e^{4 d x +4 c} b^{4}+18 e^{2 d x +2 c} b^{4}-6 e^{6 b x +4 d x +6 a +4 c} b^{3} d -36 e^{6 b x +4 d x +6 a +4 c} b^{2} d^{2}+24 e^{6 b x +4 d x +6 a +4 c} b \,d^{3}-80 e^{6 b x +2 d x +6 a +2 c} b^{2} d^{2}+6 e^{6 b x +6 a} b^{3} d -36 e^{6 b x +6 a} b^{2} d^{2}-24 e^{6 b x +6 a} b \,d^{3}-162 e^{2 b x +4 d x +2 a +4 c} b^{3} d +72 e^{2 b x +4 d x +2 a +4 c} b \,d^{3}+162 e^{2 b x +2 a} b^{3} d -72 e^{2 b x +2 a} b \,d^{3}-36 b^{2} d^{2}+9 e^{6 b x +4 d x +6 a +4 c} b^{4}+18 e^{6 b x +2 d x +6 a +2 c} b^{4}+32 e^{6 b x +2 d x +6 a +2 c} d^{4}+9 e^{6 b x +6 a} b^{4}-81 e^{4 b x +4 d x +4 a +4 c} b^{4}-162 e^{4 b x +2 d x +4 a +2 c} b^{4}-81 e^{4 b x +4 a} b^{4}}{96 e^{3 b x +2 d x +3 a +2 c} b \left (9 b^{4}-40 b^{2} d^{2}+16 d^{4}\right )} \] Input:
int(cosh(d*x+c)^2*sinh(b*x+a)^3,x)
Output:
(9*e**(6*a + 6*b*x + 4*c + 4*d*x)*b**4 - 6*e**(6*a + 6*b*x + 4*c + 4*d*x)* b**3*d - 36*e**(6*a + 6*b*x + 4*c + 4*d*x)*b**2*d**2 + 24*e**(6*a + 6*b*x + 4*c + 4*d*x)*b*d**3 + 18*e**(6*a + 6*b*x + 2*c + 2*d*x)*b**4 - 80*e**(6* a + 6*b*x + 2*c + 2*d*x)*b**2*d**2 + 32*e**(6*a + 6*b*x + 2*c + 2*d*x)*d** 4 + 9*e**(6*a + 6*b*x)*b**4 + 6*e**(6*a + 6*b*x)*b**3*d - 36*e**(6*a + 6*b *x)*b**2*d**2 - 24*e**(6*a + 6*b*x)*b*d**3 - 81*e**(4*a + 4*b*x + 4*c + 4* d*x)*b**4 + 162*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 - 72*e**(4*a + 4*b*x + 4*c + 4*d*x)*b*d**3 - 162 *e**(4*a + 4*b*x + 2*c + 2*d*x)*b**4 + 720*e**(4*a + 4*b*x + 2*c + 2*d*x)* b**2*d**2 - 288*e**(4*a + 4*b*x + 2*c + 2*d*x)*d**4 - 81*e**(4*a + 4*b*x)* b**4 - 162*e**(4*a + 4*b*x)*b**3*d + 36*e**(4*a + 4*b*x)*b**2*d**2 + 72*e* *(4*a + 4*b*x)*b*d**3 - 81*e**(2*a + 2*b*x + 4*c + 4*d*x)*b**4 - 162*e**(2 *a + 2*b*x + 4*c + 4*d*x)*b**3*d + 36*e**(2*a + 2*b*x + 4*c + 4*d*x)*b**2* d**2 + 72*e**(2*a + 2*b*x + 4*c + 4*d*x)*b*d**3 - 162*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 - 288*e**(2 *a + 2*b*x + 2*c + 2*d*x)*d**4 - 81*e**(2*a + 2*b*x)*b**4 + 162*e**(2*a + 2*b*x)*b**3*d + 36*e**(2*a + 2*b*x)*b**2*d**2 - 72*e**(2*a + 2*b*x)*b*d**3 + 9*e**(4*c + 4*d*x)*b**4 + 6*e**(4*c + 4*d*x)*b**3*d - 36*e**(4*c + 4*d* x)*b**2*d**2 - 24*e**(4*c + 4*d*x)*b*d**3 + 18*e**(2*c + 2*d*x)*b**4 - 80* e**(2*c + 2*d*x)*b**2*d**2 + 32*e**(2*c + 2*d*x)*d**4 + 9*b**4 - 6*b**3...