Integrand size = 23, antiderivative size = 178 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {5 a b \arctan (\sinh (c+d x))}{d}-\frac {a^2 \cosh (c+d x)}{d}-\frac {4 b^2 \cosh (c+d x)}{d}+\frac {a^2 \cosh ^3(c+d x)}{3 d}+\frac {b^2 \cosh ^3(c+d x)}{3 d}-\frac {6 b^2 \text {sech}(c+d x)}{d}+\frac {4 b^2 \text {sech}^3(c+d x)}{3 d}-\frac {b^2 \text {sech}^5(c+d x)}{5 d}-\frac {4 a b \sinh (c+d x)}{d}+\frac {2 a b \sinh ^3(c+d x)}{3 d}-\frac {a b \text {sech}(c+d x) \tanh (c+d x)}{d} \] Output:
5*a*b*arctan(sinh(d*x+c))/d-a^2*cosh(d*x+c)/d-4*b^2*cosh(d*x+c)/d+1/3*a^2* cosh(d*x+c)^3/d+1/3*b^2*cosh(d*x+c)^3/d-6*b^2*sech(d*x+c)/d+4/3*b^2*sech(d *x+c)^3/d-1/5*b^2*sech(d*x+c)^5/d-4*a*b*sinh(d*x+c)/d+2/3*a*b*sinh(d*x+c)^ 3/d-a*b*sech(d*x+c)*tanh(d*x+c)/d
Time = 1.39 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.68 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {-45 \left (a^2+5 b^2\right ) \cosh (c+d x)+5 \left (a^2+b^2\right ) \cosh (3 (c+d x))-2 b \left (-40 b \text {sech}^3(c+d x)+6 b \text {sech}^5(c+d x)-5 a \left (60 \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-27 \sinh (c+d x)+\sinh (3 (c+d x))\right )+30 \text {sech}(c+d x) (6 b+a \tanh (c+d x))\right )}{60 d} \] Input:
Integrate[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^2,x]
Output:
(-45*(a^2 + 5*b^2)*Cosh[c + d*x] + 5*(a^2 + b^2)*Cosh[3*(c + d*x)] - 2*b*( -40*b*Sech[c + d*x]^3 + 6*b*Sech[c + d*x]^5 - 5*a*(60*ArcTan[Tanh[(c + d*x )/2]] - 27*Sinh[c + d*x] + Sinh[3*(c + d*x)]) + 30*Sech[c + d*x]*(6*b + a* Tanh[c + d*x])))/(60*d)
Result contains complex when optimal does not.
Time = 0.50 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 26, 4149, 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.
| \(\displaystyle \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int i \sin (i c+i d x)^3 \left (a+i b \tan (i c+i d x)^3\right )^2dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \sin (i c+i d x)^3 \left (i b \tan (i c+i d x)^3+a\right )^2dx\) |
| \(\Big \downarrow \) 4149 |
| \(\displaystyle i \int \left (-i b^2 \sinh ^3(c+d x) \tanh ^6(c+d x)-2 i a b \sinh ^3(c+d x) \tanh ^3(c+d x)-i a^2 \sinh ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (-\frac {i a^2 \cosh ^3(c+d x)}{3 d}+\frac {i a^2 \cosh (c+d x)}{d}-\frac {5 i a b \arctan (\sinh (c+d x))}{d}-\frac {5 i a b \sinh ^3(c+d x)}{3 d}+\frac {5 i a b \sinh (c+d x)}{d}+\frac {i a b \sinh ^3(c+d x) \tanh ^2(c+d x)}{d}-\frac {i b^2 \cosh ^3(c+d x)}{3 d}+\frac {4 i b^2 \cosh (c+d x)}{d}+\frac {i b^2 \text {sech}^5(c+d x)}{5 d}-\frac {4 i b^2 \text {sech}^3(c+d x)}{3 d}+\frac {6 i b^2 \text {sech}(c+d x)}{d}\right )\) |
Input:
Int[Sinh[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^2,x]
Output:
I*(((-5*I)*a*b*ArcTan[Sinh[c + d*x]])/d + (I*a^2*Cosh[c + d*x])/d + ((4*I) *b^2*Cosh[c + d*x])/d - ((I/3)*a^2*Cosh[c + d*x]^3)/d - ((I/3)*b^2*Cosh[c + d*x]^3)/d + ((6*I)*b^2*Sech[c + d*x])/d - (((4*I)/3)*b^2*Sech[c + d*x]^3 )/d + ((I/5)*b^2*Sech[c + d*x]^5)/d + ((5*I)*a*b*Sinh[c + d*x])/d - (((5*I )/3)*a*b*Sinh[c + d*x]^3)/d + (I*a*b*Sinh[c + d*x]^3*Tanh[c + d*x]^2)/d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0]
Time = 8.21 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.10
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{8}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8 \sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{5}}-\frac {16 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {64 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {128}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(195\) |
| default | \(\frac {a^{2} \left (-\frac {2}{3}+\frac {\sinh \left (d x +c \right )^{2}}{3}\right ) \cosh \left (d x +c \right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{2}}-\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {5 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+5 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{8}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8 \sinh \left (d x +c \right )^{6}}{3 \cosh \left (d x +c \right )^{5}}-\frac {16 \sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {64 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {128}{15 \cosh \left (d x +c \right )^{5}}\right )}{d}\) | \(195\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c} a^{2}}{24 d}+\frac {{\mathrm e}^{3 d x +3 c} a b}{12 d}+\frac {{\mathrm e}^{3 d x +3 c} b^{2}}{24 d}-\frac {3 \,{\mathrm e}^{d x +c} a^{2}}{8 d}-\frac {9 \,{\mathrm e}^{d x +c} a b}{4 d}-\frac {15 \,{\mathrm e}^{d x +c} b^{2}}{8 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2}}{8 d}+\frac {9 \,{\mathrm e}^{-d x -c} a b}{4 d}-\frac {15 \,{\mathrm e}^{-d x -c} b^{2}}{8 d}+\frac {{\mathrm e}^{-3 d x -3 c} a^{2}}{24 d}-\frac {{\mathrm e}^{-3 d x -3 c} a b}{12 d}+\frac {{\mathrm e}^{-3 d x -3 c} b^{2}}{24 d}-\frac {2 b \,{\mathrm e}^{d x +c} \left (15 \,{\mathrm e}^{8 d x +8 c} a +90 \,{\mathrm e}^{8 d x +8 c} b +30 \,{\mathrm e}^{6 d x +6 c} a +280 \,{\mathrm e}^{6 d x +6 c} b +428 b \,{\mathrm e}^{4 d x +4 c}-30 \,{\mathrm e}^{2 d x +2 c} a +280 \,{\mathrm e}^{2 d x +2 c} b -15 a +90 b \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{5}}+\frac {5 i b a \ln \left ({\mathrm e}^{d x +c}+i\right )}{d}-\frac {5 i b a \ln \left ({\mathrm e}^{d x +c}-i\right )}{d}\) | \(345\) |
Input:
int(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c)+2*a*b*(1/3*sinh(d*x+c)^5/cos h(d*x+c)^2-5/3*sinh(d*x+c)^3/cosh(d*x+c)^2-5*sinh(d*x+c)/cosh(d*x+c)^2+5/2 *sech(d*x+c)*tanh(d*x+c)+5*arctan(exp(d*x+c)))+b^2*(1/3*sinh(d*x+c)^8/cosh (d*x+c)^5-8/3*sinh(d*x+c)^6/cosh(d*x+c)^5-16*sinh(d*x+c)^4/cosh(d*x+c)^5-6 4/3*sinh(d*x+c)^2/cosh(d*x+c)^5-128/15/cosh(d*x+c)^5))
Leaf count of result is larger than twice the leaf count of optimal. 3341 vs. \(2 (168) = 336\).
Time = 0.15 (sec) , antiderivative size = 3341, normalized size of antiderivative = 18.77 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\text {Too large to display} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{2} \sinh ^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(sinh(d*x+c)**3*(a+b*tanh(d*x+c)**3)**2,x)
Output:
Integral((a + b*tanh(c + d*x)**3)**2*sinh(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (168) = 336\).
Time = 0.12 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.96 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=-\frac {1}{120} \, b^{2} {\left (\frac {5 \, {\left (45 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d} + \frac {200 \, e^{\left (-2 \, d x - 2 \, c\right )} + 2515 \, e^{\left (-4 \, d x - 4 \, c\right )} + 6680 \, e^{\left (-6 \, d x - 6 \, c\right )} + 9073 \, e^{\left (-8 \, d x - 8 \, c\right )} + 5600 \, e^{\left (-10 \, d x - 10 \, c\right )} + 1665 \, e^{\left (-12 \, d x - 12 \, c\right )} - 5}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 5 \, e^{\left (-5 \, d x - 5 \, c\right )} + 10 \, e^{\left (-7 \, d x - 7 \, c\right )} + 10 \, e^{\left (-9 \, d x - 9 \, c\right )} + 5 \, e^{\left (-11 \, d x - 11 \, c\right )} + e^{\left (-13 \, d x - 13 \, c\right )}\right )}}\right )} + \frac {1}{12} \, a b {\left (\frac {27 \, e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d} - \frac {120 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {25 \, e^{\left (-2 \, d x - 2 \, c\right )} + 77 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} - 1}{d {\left (e^{\left (-3 \, d x - 3 \, c\right )} + 2 \, e^{\left (-5 \, d x - 5 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )}\right )}}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^2,x, algorithm="maxima")
Output:
-1/120*b^2*(5*(45*e^(-d*x - c) - e^(-3*d*x - 3*c))/d + (200*e^(-2*d*x - 2* c) + 2515*e^(-4*d*x - 4*c) + 6680*e^(-6*d*x - 6*c) + 9073*e^(-8*d*x - 8*c) + 5600*e^(-10*d*x - 10*c) + 1665*e^(-12*d*x - 12*c) - 5)/(d*(e^(-3*d*x - 3*c) + 5*e^(-5*d*x - 5*c) + 10*e^(-7*d*x - 7*c) + 10*e^(-9*d*x - 9*c) + 5* e^(-11*d*x - 11*c) + e^(-13*d*x - 13*c)))) + 1/12*a*b*((27*e^(-d*x - c) - e^(-3*d*x - 3*c))/d - 120*arctan(e^(-d*x - c))/d - (25*e^(-2*d*x - 2*c) + 77*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) - 1)/(d*(e^(-3*d*x - 3*c) + 2*e^( -5*d*x - 5*c) + e^(-7*d*x - 7*c)))) + 1/24*a^2*(e^(3*d*x + 3*c)/d - 9*e^(d *x + c)/d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d)
Time = 0.25 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.62 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {1200 \, a b \arctan \left (e^{\left (d x + c\right )}\right ) + 5 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 10 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 5 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 45 \, a^{2} e^{\left (d x + c\right )} - 270 \, a b e^{\left (d x + c\right )} - 225 \, b^{2} e^{\left (d x + c\right )} - 5 \, {\left (9 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 54 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} + 2 \, a b - b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )} - \frac {16 \, {\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} + 90 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} + 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 280 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 428 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 280 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 15 \, a b e^{\left (d x + c\right )} + 90 \, b^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{5}}}{120 \, d} \] Input:
integrate(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^2,x, algorithm="giac")
Output:
1/120*(1200*a*b*arctan(e^(d*x + c)) + 5*a^2*e^(3*d*x + 3*c) + 10*a*b*e^(3* d*x + 3*c) + 5*b^2*e^(3*d*x + 3*c) - 45*a^2*e^(d*x + c) - 270*a*b*e^(d*x + c) - 225*b^2*e^(d*x + c) - 5*(9*a^2*e^(2*d*x + 2*c) - 54*a*b*e^(2*d*x + 2 *c) + 45*b^2*e^(2*d*x + 2*c) - a^2 + 2*a*b - b^2)*e^(-3*d*x - 3*c) - 16*(1 5*a*b*e^(9*d*x + 9*c) + 90*b^2*e^(9*d*x + 9*c) + 30*a*b*e^(7*d*x + 7*c) + 280*b^2*e^(7*d*x + 7*c) + 428*b^2*e^(5*d*x + 5*c) - 30*a*b*e^(3*d*x + 3*c) + 280*b^2*e^(3*d*x + 3*c) - 15*a*b*e^(d*x + c) + 90*b^2*e^(d*x + c))/(e^( 2*d*x + 2*c) + 1)^5)/d
Time = 2.54 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.23 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {{\mathrm {e}}^{3\,c+3\,d\,x}\,{\left (a+b\right )}^2}{24\,d}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (3\,a^2+18\,a\,b+15\,b^2\right )}{8\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,\left (3\,a^2-18\,a\,b+15\,b^2\right )}{8\,d}+\frac {{\mathrm {e}}^{-3\,c-3\,d\,x}\,{\left (a-b\right )}^2}{24\,d}+\frac {10\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {d^2}}-\frac {256\,b^2\,{\mathrm {e}}^{c+d\,x}}{15\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}+\frac {64\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {32\,b^2\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (6\,b^2+a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (8\,b^2+3\,a\,b\right )}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )} \] Input:
int(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^3)^2,x)
Output:
(exp(3*c + 3*d*x)*(a + b)^2)/(24*d) - (exp(c + d*x)*(18*a*b + 3*a^2 + 15*b ^2))/(8*d) - (exp(- c - d*x)*(3*a^2 - 18*a*b + 15*b^2))/(8*d) + (exp(- 3*c - 3*d*x)*(a - b)^2)/(24*d) + (10*atan((a*b*exp(d*x)*exp(c)*(d^2)^(1/2))/( d*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(d^2)^(1/2) - (256*b^2*exp(c + d*x))/ (15*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) + 1)) + (64*b^2*exp(c + d*x))/(5*d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*ex p(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (32*b^2*exp(c + d*x))/(5*d*(5*ex p(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8 *d*x) + exp(10*c + 10*d*x) + 1)) - (2*exp(c + d*x)*(a*b + 6*b^2))/(d*(exp( 2*c + 2*d*x) + 1)) + (4*exp(c + d*x)*(3*a*b + 8*b^2))/(3*d*(2*exp(2*c + 2* d*x) + exp(4*c + 4*d*x) + 1))
Time = 0.24 (sec) , antiderivative size = 564, normalized size of antiderivative = 3.17 \[ \int \sinh ^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^2 \, dx=\frac {10 e^{16 d x +16 c} a b -20 e^{14 d x +14 c} a^{2}-200 e^{14 d x +14 c} b^{2}-220 e^{12 d x +12 c} a^{2}-2740 e^{12 d x +12 c} b^{2}-620 e^{10 d x +10 c} a^{2}-7800 e^{10 d x +10 c} b^{2}+1200 e^{13 d x +13 c} \mathit {atan} \left (e^{d x +c}\right ) a b +6000 e^{11 d x +11 c} \mathit {atan} \left (e^{d x +c}\right ) a b +12000 e^{9 d x +9 c} \mathit {atan} \left (e^{d x +c}\right ) a b +12000 e^{7 d x +7 c} \mathit {atan} \left (e^{d x +c}\right ) a b +6000 e^{5 d x +5 c} \mathit {atan} \left (e^{d x +c}\right ) a b +1200 e^{3 d x +3 c} \mathit {atan} \left (e^{d x +c}\right ) a b +5 a^{2}+5 e^{16 d x +16 c} a^{2}+5 e^{16 d x +16 c} b^{2}-200 e^{2 d x +2 c} b^{2}+5 b^{2}-220 e^{14 d x +14 c} a b -1220 e^{12 d x +12 c} a b -1740 e^{10 d x +10 c} a b +1740 e^{6 d x +6 c} a b +1220 e^{4 d x +4 c} a b +220 e^{2 d x +2 c} a b -10 a b -850 e^{8 d x +8 c} a^{2}-11298 e^{8 d x +8 c} b^{2}-620 e^{6 d x +6 c} a^{2}-7800 e^{6 d x +6 c} b^{2}-220 e^{4 d x +4 c} a^{2}-2740 e^{4 d x +4 c} b^{2}-20 e^{2 d x +2 c} a^{2}}{120 e^{3 d x +3 c} d \left (e^{10 d x +10 c}+5 e^{8 d x +8 c}+10 e^{6 d x +6 c}+10 e^{4 d x +4 c}+5 e^{2 d x +2 c}+1\right )} \] Input:
int(sinh(d*x+c)^3*(a+b*tanh(d*x+c)^3)^2,x)
Output:
(1200*e**(13*c + 13*d*x)*atan(e**(c + d*x))*a*b + 6000*e**(11*c + 11*d*x)* atan(e**(c + d*x))*a*b + 12000*e**(9*c + 9*d*x)*atan(e**(c + d*x))*a*b + 1 2000*e**(7*c + 7*d*x)*atan(e**(c + d*x))*a*b + 6000*e**(5*c + 5*d*x)*atan( e**(c + d*x))*a*b + 1200*e**(3*c + 3*d*x)*atan(e**(c + d*x))*a*b + 5*e**(1 6*c + 16*d*x)*a**2 + 10*e**(16*c + 16*d*x)*a*b + 5*e**(16*c + 16*d*x)*b**2 - 20*e**(14*c + 14*d*x)*a**2 - 220*e**(14*c + 14*d*x)*a*b - 200*e**(14*c + 14*d*x)*b**2 - 220*e**(12*c + 12*d*x)*a**2 - 1220*e**(12*c + 12*d*x)*a*b - 2740*e**(12*c + 12*d*x)*b**2 - 620*e**(10*c + 10*d*x)*a**2 - 1740*e**(1 0*c + 10*d*x)*a*b - 7800*e**(10*c + 10*d*x)*b**2 - 850*e**(8*c + 8*d*x)*a* *2 - 11298*e**(8*c + 8*d*x)*b**2 - 620*e**(6*c + 6*d*x)*a**2 + 1740*e**(6* c + 6*d*x)*a*b - 7800*e**(6*c + 6*d*x)*b**2 - 220*e**(4*c + 4*d*x)*a**2 + 1220*e**(4*c + 4*d*x)*a*b - 2740*e**(4*c + 4*d*x)*b**2 - 20*e**(2*c + 2*d* x)*a**2 + 220*e**(2*c + 2*d*x)*a*b - 200*e**(2*c + 2*d*x)*b**2 + 5*a**2 - 10*a*b + 5*b**2)/(120*e**(3*c + 3*d*x)*d*(e**(10*c + 10*d*x) + 5*e**(8*c + 8*d*x) + 10*e**(6*c + 6*d*x) + 10*e**(4*c + 4*d*x) + 5*e**(2*c + 2*d*x) + 1))