Integrand size = 23, antiderivative size = 232 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {3 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {5 b^3 \arctan (\sinh (c+d x))}{128 d}+\frac {a^3 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {5 b^3 \text {sech}(c+d x) \tanh (c+d x)}{128 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh ^5(c+d x)}{8 d} \] Output:
3/2*a^2*b*arctan(sinh(d*x+c))/d+5/128*b^3*arctan(sinh(d*x+c))/d+1/2*a^3*ar ctanh(cosh(d*x+c))/d-1/2*a^3*coth(d*x+c)*csch(d*x+c)/d-a*b^2*sech(d*x+c)^3 /d+3/5*a*b^2*sech(d*x+c)^5/d+3/2*a^2*b*sech(d*x+c)*tanh(d*x+c)/d+5/128*b^3 *sech(d*x+c)*tanh(d*x+c)/d-5/64*b^3*sech(d*x+c)^3*tanh(d*x+c)/d-5/48*b^3*s ech(d*x+c)^3*tanh(d*x+c)^3/d-1/8*b^3*sech(d*x+c)^3*tanh(d*x+c)^5/d
Time = 12.70 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.14 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {\left (192 a^2 b+5 b^3\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a^3 \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^3 \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {\text {sech}^2(c+d x) \left (192 a^2 b \sinh (c+d x)+5 b^3 \sinh (c+d x)\right )}{128 d}-\frac {59 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {17 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{48 d}-\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \] Input:
Integrate[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
((192*a^2*b + 5*b^3)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) - (a^3*Csch[(c + d* x)/2]^2)/(8*d) + (a^3*Log[Cosh[(c + d*x)/2]])/(2*d) - (a^3*Log[Sinh[(c + d *x)/2]])/(2*d) - (a^3*Sech[(c + d*x)/2]^2)/(8*d) - (a*b^2*Sech[c + d*x]^3) /d + (3*a*b^2*Sech[c + d*x]^5)/(5*d) + (Sech[c + d*x]^2*(192*a^2*b*Sinh[c + d*x] + 5*b^3*Sinh[c + d*x]))/(128*d) - (59*b^3*Sech[c + d*x]^3*Tanh[c + d*x])/(192*d) + (17*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d) - (b^3*Sech[ c + d*x]^7*Tanh[c + d*x])/(8*d)
Result contains complex when optimal does not.
Time = 0.73 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.11, 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.
| \(\displaystyle \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+i b \tan (i c+i d x)^3\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (i b \tan (i c+i d x)^3+a\right )^3}{\sin (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4149 |
| \(\displaystyle -i \int \left (i b^3 \text {sech}^3(c+d x) \tanh ^6(c+d x)+3 i a b^2 \text {sech}^3(c+d x) \tanh ^3(c+d x)+i a^3 \text {csch}^3(c+d x)+3 i a^2 b \text {sech}^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -i \left (\frac {i a^3 \text {arctanh}(\cosh (c+d x))}{2 d}-\frac {i a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {3 i a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {3 i a^2 b \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {3 i a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {i a b^2 \text {sech}^3(c+d x)}{d}+\frac {5 i b^3 \arctan (\sinh (c+d x))}{128 d}-\frac {i b^3 \tanh ^5(c+d x) \text {sech}^3(c+d x)}{8 d}-\frac {5 i b^3 \tanh ^3(c+d x) \text {sech}^3(c+d x)}{48 d}-\frac {5 i b^3 \tanh (c+d x) \text {sech}^3(c+d x)}{64 d}+\frac {5 i b^3 \tanh (c+d x) \text {sech}(c+d x)}{128 d}\right )\) |
Input:
Int[Csch[c + d*x]^3*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
(-I)*((((3*I)/2)*a^2*b*ArcTan[Sinh[c + d*x]])/d + (((5*I)/128)*b^3*ArcTan[ Sinh[c + d*x]])/d + ((I/2)*a^3*ArcTanh[Cosh[c + d*x]])/d - ((I/2)*a^3*Coth [c + d*x]*Csch[c + d*x])/d - (I*a*b^2*Sech[c + d*x]^3)/d + (((3*I)/5)*a*b^ 2*Sech[c + d*x]^5)/d + (((3*I)/2)*a^2*b*Sech[c + d*x]*Tanh[c + d*x])/d + ( ((5*I)/128)*b^3*Sech[c + d*x]*Tanh[c + d*x])/d - (((5*I)/64)*b^3*Sech[c + d*x]^3*Tanh[c + d*x])/d - (((5*I)/48)*b^3*Sech[c + d*x]^3*Tanh[c + d*x]^3) /d - ((I/8)*b^3*Sech[c + d*x]^3*Tanh[c + d*x]^5)/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 = 55.35 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {2}{15 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )}{7 \cosh \left (d x +c \right )^{8}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )}{7}+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(208\) |
| default | \(\frac {a^{3} \left (-\frac {\operatorname {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {2}{15 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )}{7 \cosh \left (d x +c \right )^{8}}+\frac {\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )}{7}+\frac {5 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(208\) |
| risch | \(-\frac {{\mathrm e}^{d x +c} \left (30950 b^{3} {\mathrm e}^{8 d x +8 c}-19760 b^{3} {\mathrm e}^{6 d x +6 c}+8520 b^{3} {\mathrm e}^{4 d x +4 c}+8640 \,{\mathrm e}^{2 d x +2 c} a^{3}+960 a^{3}+2880 a^{2} b +23040 a^{2} b \,{\mathrm e}^{12 d x +12 c}+17280 a^{2} b \,{\mathrm e}^{10 d x +10 c}-1536 a \,b^{2} {\mathrm e}^{10 d x +10 c}-17280 a^{2} b \,{\mathrm e}^{8 d x +8 c}-1536 a \,b^{2} {\mathrm e}^{8 d x +8 c}-23040 a^{2} b \,{\mathrm e}^{6 d x +6 c}-10752 a \,b^{2} {\mathrm e}^{6 d x +6 c}+4608 a \,b^{2} {\mathrm e}^{4 d x +4 c}+8640 a^{2} b \,{\mathrm e}^{2 d x +2 c}+7680 a \,b^{2} {\mathrm e}^{2 d x +2 c}+75 b^{3}+80640 a^{3} {\mathrm e}^{6 d x +6 c}-30950 b^{3} {\mathrm e}^{10 d x +10 c}+120960 a^{3} {\mathrm e}^{8 d x +8 c}+19760 b^{3} {\mathrm e}^{12 d x +12 c}-2135 b^{3} {\mathrm e}^{2 d x +2 c}-2880 a^{2} b \,{\mathrm e}^{18 d x +18 c}-75 b^{3} {\mathrm e}^{18 d x +18 c}+2135 b^{3} {\mathrm e}^{16 d x +16 c}+34560 a^{3} {\mathrm e}^{14 d x +14 c}+4608 a \,b^{2} {\mathrm e}^{14 d x +14 c}-10752 a \,b^{2} {\mathrm e}^{12 d x +12 c}-8640 a^{2} b \,{\mathrm e}^{16 d x +16 c}+7680 a \,b^{2} {\mathrm e}^{16 d x +16 c}+960 a^{3} {\mathrm e}^{18 d x +18 c}-8520 b^{3} {\mathrm e}^{14 d x +14 c}+80640 a^{3} {\mathrm e}^{12 d x +12 c}+8640 a^{3} {\mathrm e}^{16 d x +16 c}+34560 \,{\mathrm e}^{4 d x +4 c} a^{3}+120960 \,{\mathrm e}^{10 d x +10 c} a^{3}\right )}{960 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}+\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}\) | \(610\) |
Input:
int(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+3*a^2*b*(1/2*s ech(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+3*b^2*a*(-1/3*sinh(d*x+c)^2/cos h(d*x+c)^5-2/15/cosh(d*x+c)^5)+b^3*(-1/3*sinh(d*x+c)^5/cosh(d*x+c)^8-1/3*s inh(d*x+c)^3/cosh(d*x+c)^8-1/7*sinh(d*x+c)/cosh(d*x+c)^8+1/7*(1/8*sech(d*x +c)^7+7/48*sech(d*x+c)^5+35/192*sech(d*x+c)^3+35/128*sech(d*x+c))*tanh(d*x +c)+5/64*arctan(exp(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 10985 vs. \(2 (212) = 424\).
Time = 0.20 (sec) , antiderivative size = 10985, normalized size of antiderivative = 47.35 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)**3*(a+b*tanh(d*x+c)**3)**3,x)
Output:
Integral((a + b*tanh(c + d*x)**3)**3*csch(c + d*x)**3, x)
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (212) = 424\).
Time = 0.15 (sec) , antiderivative size = 586, normalized size of antiderivative = 2.53 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")
Output:
-1/192*b^3*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) - 397*e^(-3*d*x - 3*c) + 895*e^(-5*d*x - 5*c) - 1765*e^(-7*d*x - 7*c) + 1765*e^(-9*d*x - 9* c) - 895*e^(-11*d*x - 11*c) + 397*e^(-13*d*x - 13*c) - 15*e^(-15*d*x - 15* c))/(d*(8*e^(-2*d*x - 2*c) + 28*e^(-4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) + 7 0*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) + 28*e^(-12*d*x - 12*c) + 8*e^( -14*d*x - 14*c) + e^(-16*d*x - 16*c) + 1))) - 3*a^2*b*(arctan(e^(-d*x - c) )/d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 1/2*a^3*(log(e^(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/ d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) - 1))) - 8/5*a*b^2*(5*e^(-3*d*x - 3*c)/(d*(5*e^(-2*d*x - 2*c) + 10 *e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)) - 2*e^(-5*d*x - 5*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1) ) + 5*e^(-7*d*x - 7*c)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e ^(-6*d*x - 6*c) + 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) + 1)))
Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (212) = 424\).
Time = 0.36 (sec) , antiderivative size = 426, normalized size of antiderivative = 1.84 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {480 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 480 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + 15 \, {\left (192 \, a^{2} b + 5 \, b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) - \frac {960 \, {\left (a^{3} e^{\left (3 \, d x + 3 \, c\right )} + a^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} + \frac {2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 75 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} - 7680 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} - 1985 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} - 19968 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 4475 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} - 21504 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 8825 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} - 21504 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 8825 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 19968 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 4475 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 7680 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 1985 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} - 75 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{960 \, d} \] Input:
integrate(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="giac")
Output:
1/960*(480*a^3*log(e^(d*x + c) + 1) - 480*a^3*log(abs(e^(d*x + c) - 1)) + 15*(192*a^2*b + 5*b^3)*arctan(e^(d*x + c)) - 960*(a^3*e^(3*d*x + 3*c) + a^ 3*e^(d*x + c))/(e^(2*d*x + 2*c) - 1)^2 + (2880*a^2*b*e^(15*d*x + 15*c) + 7 5*b^3*e^(15*d*x + 15*c) + 14400*a^2*b*e^(13*d*x + 13*c) - 7680*a*b^2*e^(13 *d*x + 13*c) - 1985*b^3*e^(13*d*x + 13*c) + 25920*a^2*b*e^(11*d*x + 11*c) - 19968*a*b^2*e^(11*d*x + 11*c) + 4475*b^3*e^(11*d*x + 11*c) + 14400*a^2*b *e^(9*d*x + 9*c) - 21504*a*b^2*e^(9*d*x + 9*c) - 8825*b^3*e^(9*d*x + 9*c) - 14400*a^2*b*e^(7*d*x + 7*c) - 21504*a*b^2*e^(7*d*x + 7*c) + 8825*b^3*e^( 7*d*x + 7*c) - 25920*a^2*b*e^(5*d*x + 5*c) - 19968*a*b^2*e^(5*d*x + 5*c) - 4475*b^3*e^(5*d*x + 5*c) - 14400*a^2*b*e^(3*d*x + 3*c) - 7680*a*b^2*e^(3* d*x + 3*c) + 1985*b^3*e^(3*d*x + 3*c) - 2880*a^2*b*e^(d*x + c) - 75*b^3*e^ (d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d
Time = 9.49 (sec) , antiderivative size = 731, normalized size of antiderivative = 3.15 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int((a + b*tanh(c + d*x)^3)^3/sinh(c + d*x)^3,x)
Output:
(a^3*log(exp(c + d*x) + 1))/(2*d) - (a^3*log(exp(c + d*x) - 1))/(2*d) + (e xp(c + d*x)*(192*a^2*b + 5*b^3))/(64*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(3264*a*b^2 + 2245*b^3))/(120*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d *x) + exp(6*c + 6*d*x) + 1)) - (b*log(exp(c + d*x) - 1i)*(192*a^2 + 5*b^2) *1i)/(128*d) + (b*log(exp(c + d*x) + 1i)*(192*a^2 + 5*b^2)*1i)/(128*d) - ( exp(c + d*x)*(768*a*b^2 + 1325*b^3))/(20*d*(4*exp(2*c + 2*d*x) + 6*exp(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (500*b^3*exp(c + d*x))/(3*d*(6*exp(2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x ) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1)) + (2*exp(c + d*x)*(144*a*b^2 + 1025*b^3))/(15*d*(5*exp(2*c + 2*d*x) + 10*e xp(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)) + (112*b^3*exp(c + d*x))/(d*(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 35*exp(8*c + 8*d*x) + 21*exp(10*c + 10*d*x ) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1)) - (exp(c + d*x)*(768*a *b^2 + 576*a^2*b + 251*b^3))/(96*d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (32*b^3*exp(c + d*x))/(d*(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) + 56*exp(10*c + 10*d*x) + 28* exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1)) - (a^ 3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (2*a^3*exp(c + d*x))/(d*(exp( 4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1))
Time = 0.24 (sec) , antiderivative size = 1702, normalized size of antiderivative = 7.34 \[ \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^3*(a+b*tanh(d*x+c)^3)^3,x)
Output:
(2880*e**(20*c + 20*d*x)*atan(e**(c + d*x))*a**2*b + 75*e**(20*c + 20*d*x) *atan(e**(c + d*x))*b**3 + 17280*e**(18*c + 18*d*x)*atan(e**(c + d*x))*a** 2*b + 450*e**(18*c + 18*d*x)*atan(e**(c + d*x))*b**3 + 37440*e**(16*c + 16 *d*x)*atan(e**(c + d*x))*a**2*b + 975*e**(16*c + 16*d*x)*atan(e**(c + d*x) )*b**3 + 23040*e**(14*c + 14*d*x)*atan(e**(c + d*x))*a**2*b + 600*e**(14*c + 14*d*x)*atan(e**(c + d*x))*b**3 - 40320*e**(12*c + 12*d*x)*atan(e**(c + d*x))*a**2*b - 1050*e**(12*c + 12*d*x)*atan(e**(c + d*x))*b**3 - 80640*e* *(10*c + 10*d*x)*atan(e**(c + d*x))*a**2*b - 2100*e**(10*c + 10*d*x)*atan( e**(c + d*x))*b**3 - 40320*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**2*b - 10 50*e**(8*c + 8*d*x)*atan(e**(c + d*x))*b**3 + 23040*e**(6*c + 6*d*x)*atan( e**(c + d*x))*a**2*b + 600*e**(6*c + 6*d*x)*atan(e**(c + d*x))*b**3 + 3744 0*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b + 975*e**(4*c + 4*d*x)*atan(e **(c + d*x))*b**3 + 17280*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b + 450 *e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**3 + 2880*atan(e**(c + d*x))*a**2*b + 75*atan(e**(c + d*x))*b**3 - 480*e**(20*c + 20*d*x)*log(e**(c + d*x) - 1)*a**3 + 480*e**(20*c + 20*d*x)*log(e**(c + d*x) + 1)*a**3 - 960*e**(19*c + 19*d*x)*a**3 + 2880*e**(19*c + 19*d*x)*a**2*b + 75*e**(19*c + 19*d*x)*b **3 - 2880*e**(18*c + 18*d*x)*log(e**(c + d*x) - 1)*a**3 + 2880*e**(18*c + 18*d*x)*log(e**(c + d*x) + 1)*a**3 - 8640*e**(17*c + 17*d*x)*a**3 + 8640* e**(17*c + 17*d*x)*a**2*b - 7680*e**(17*c + 17*d*x)*a*b**2 - 2135*e**(1...