Integrand size = 21, antiderivative size = 219 \[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {3 a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {35 b^3 \arctan (\sinh (c+d x))}{128 d}-\frac {a^3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 a b^2 \text {sech}(c+d x)}{d}+\frac {2 a b^2 \text {sech}^3(c+d x)}{d}-\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {35 b^3 \text {sech}(c+d x) \tanh (c+d x)}{128 d}-\frac {35 b^3 \text {sech}(c+d x) \tanh ^3(c+d x)}{192 d}-\frac {7 b^3 \text {sech}(c+d x) \tanh ^5(c+d x)}{48 d}-\frac {b^3 \text {sech}(c+d x) \tanh ^7(c+d x)}{8 d} \] Output:
3/2*a^2*b*arctan(sinh(d*x+c))/d+35/128*b^3*arctan(sinh(d*x+c))/d-a^3*arcta nh(cosh(d*x+c))/d-3*a*b^2*sech(d*x+c)/d+2*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-35/128*b^3*sech(d*x+c) *tanh(d*x+c)/d-35/192*b^3*sech(d*x+c)*tanh(d*x+c)^3/d-7/48*b^3*sech(d*x+c) *tanh(d*x+c)^5/d-1/8*b^3*sech(d*x+c)*tanh(d*x+c)^7/d
Time = 5.64 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.77 \[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {30 \left (b \left (192 a^2+35 b^2\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+64 a^3 \left (-\log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )+240 b^3 \text {sech}^7(c+d x) \tanh (c+d x)-8 b^2 \text {sech}^5(c+d x) (144 a+125 b \tanh (c+d x))+10 b^2 \text {sech}^3(c+d x) (384 a+163 b \tanh (c+d x))-45 b \text {sech}(c+d x) \left (128 a b+\left (64 a^2+31 b^2\right ) \tanh (c+d x)\right )}{1920 d} \] Input:
Integrate[Csch[c + d*x]*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
(30*(b*(192*a^2 + 35*b^2)*ArcTan[Tanh[(c + d*x)/2]] + 64*a^3*(-Log[Cosh[(c + d*x)/2]] + Log[Sinh[(c + d*x)/2]])) + 240*b^3*Sech[c + d*x]^7*Tanh[c + d*x] - 8*b^2*Sech[c + d*x]^5*(144*a + 125*b*Tanh[c + d*x]) + 10*b^2*Sech[c + d*x]^3*(384*a + 163*b*Tanh[c + d*x]) - 45*b*Sech[c + d*x]*(128*a*b + (6 4*a^2 + 31*b^2)*Tanh[c + d*x]))/(1920*d)
Result contains complex when optimal does not.
Time = 0.90 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, 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.
| \(\displaystyle \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \left (a+i b \tan (i c+i d x)^3\right )^3}{\sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\left (i b \tan (i c+i d x)^3+a\right )^3}{\sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 4149 |
| \(\displaystyle i \int \left (-i b^3 \text {sech}(c+d x) \tanh ^8(c+d x)-3 i a b^2 \text {sech}(c+d x) \tanh ^5(c+d x)-3 i a^2 b \text {sech}(c+d x) \tanh ^2(c+d x)-i a^3 \text {csch}(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {i a^3 \text {arctanh}(\cosh (c+d x))}{d}-\frac {3 i a^2 b \arctan (\sinh (c+d x))}{2 d}+\frac {3 i a^2 b \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {3 i a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {2 i a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 i a b^2 \text {sech}(c+d x)}{d}-\frac {35 i b^3 \arctan (\sinh (c+d x))}{128 d}+\frac {i b^3 \tanh ^7(c+d x) \text {sech}(c+d x)}{8 d}+\frac {7 i b^3 \tanh ^5(c+d x) \text {sech}(c+d x)}{48 d}+\frac {35 i b^3 \tanh ^3(c+d x) \text {sech}(c+d x)}{192 d}+\frac {35 i b^3 \tanh (c+d x) \text {sech}(c+d x)}{128 d}\right )\) |
Input:
Int[Csch[c + d*x]*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
I*((((-3*I)/2)*a^2*b*ArcTan[Sinh[c + d*x]])/d - (((35*I)/128)*b^3*ArcTan[S inh[c + d*x]])/d + (I*a^3*ArcTanh[Cosh[c + d*x]])/d + ((3*I)*a*b^2*Sech[c + d*x])/d - ((2*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 + (((35*I)/128)* b^3*Sech[c + d*x]*Tanh[c + d*x])/d + (((35*I)/192)*b^3*Sech[c + d*x]*Tanh[ c + d*x]^3)/d + (((7*I)/48)*b^3*Sech[c + d*x]*Tanh[c + d*x]^5)/d + ((I/8)* b^3*Sech[c + d*x]*Tanh[c + d*x]^7)/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 = 7.94 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.12
| method | result | size |
| derivativedivides | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {4 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8}{15 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}-\frac {7 \sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{8}}-\frac {7 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}+\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )+\frac {35 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(245\) |
| default | \(\frac {-2 a^{3} \operatorname {arctanh}\left ({\mathrm e}^{d x +c}\right )+3 a^{2} b \left (-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}+\frac {\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}+\arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )^{4}}{\cosh \left (d x +c \right )^{5}}-\frac {4 \sinh \left (d x +c \right )^{2}}{3 \cosh \left (d x +c \right )^{5}}-\frac {8}{15 \cosh \left (d x +c \right )^{5}}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{7}}{\cosh \left (d x +c \right )^{8}}-\frac {7 \sinh \left (d x +c \right )^{5}}{3 \cosh \left (d x +c \right )^{8}}-\frac {7 \sinh \left (d x +c \right )^{3}}{3 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{8}}+\left (\frac {\operatorname {sech}\left (d x +c \right )^{7}}{8}+\frac {7 \operatorname {sech}\left (d x +c \right )^{5}}{48}+\frac {35 \operatorname {sech}\left (d x +c \right )^{3}}{192}+\frac {35 \,\operatorname {sech}\left (d x +c \right )}{128}\right ) \tanh \left (d x +c \right )+\frac {35 \arctan \left ({\mathrm e}^{d x +c}\right )}{64}\right )}{d}\) | \(245\) |
| risch | \(-\frac {b \,{\mathrm e}^{d x +c} \left (2880 a^{2} {\mathrm e}^{14 d x +14 c}+5760 a b \,{\mathrm e}^{14 d x +14 c}+1395 b^{2} {\mathrm e}^{14 d x +14 c}+14400 \,{\mathrm e}^{12 d x +12 c} a^{2}+24960 \,{\mathrm e}^{12 d x +12 c} a b +455 \,{\mathrm e}^{12 d x +12 c} b^{2}+25920 \,{\mathrm e}^{10 d x +10 c} a^{2}+62592 \,{\mathrm e}^{10 d x +10 c} a b +8995 \,{\mathrm e}^{10 d x +10 c} b^{2}+14400 \,{\mathrm e}^{8 d x +8 c} a^{2}+103296 \,{\mathrm e}^{8 d x +8 c} a b -5425 \,{\mathrm e}^{8 d x +8 c} b^{2}-14400 \,{\mathrm e}^{6 d x +6 c} a^{2}+103296 \,{\mathrm e}^{6 d x +6 c} a b +5425 \,{\mathrm e}^{6 d x +6 c} b^{2}-25920 \,{\mathrm e}^{4 d x +4 c} a^{2}+62592 \,{\mathrm e}^{4 d x +4 c} a b -8995 \,{\mathrm e}^{4 d x +4 c} b^{2}-14400 \,{\mathrm e}^{2 d x +2 c} a^{2}+24960 \,{\mathrm e}^{2 d x +2 c} b a -455 b^{2} {\mathrm e}^{2 d x +2 c}-2880 a^{2}+5760 a b -1395 b^{2}\right )}{960 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}+\frac {35 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}-\frac {35 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\) | \(418\) |
Input:
int(csch(d*x+c)*(a+b*tanh(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-2*a^3*arctanh(exp(d*x+c))+3*a^2*b*(-sinh(d*x+c)/cosh(d*x+c)^2+1/2*se ch(d*x+c)*tanh(d*x+c)+arctan(exp(d*x+c)))+3*b^2*a*(-sinh(d*x+c)^4/cosh(d*x +c)^5-4/3*sinh(d*x+c)^2/cosh(d*x+c)^5-8/15/cosh(d*x+c)^5)+b^3*(-sinh(d*x+c )^7/cosh(d*x+c)^8-7/3*sinh(d*x+c)^5/cosh(d*x+c)^8-7/3*sinh(d*x+c)^3/cosh(d *x+c)^8-sinh(d*x+c)/cosh(d*x+c)^8+(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)+35/64*arctan(exp(d*x+c) )))
Leaf count of result is larger than twice the leaf count of optimal. 7127 vs. \(2 (203) = 406\).
Time = 0.16 (sec) , antiderivative size = 7127, normalized size of antiderivative = 32.54 \[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)*(a+b*tanh(d*x+c)**3)**3,x)
Output:
Integral((a + b*tanh(c + d*x)**3)**3*csch(c + d*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (203) = 406\).
Time = 0.12 (sec) , antiderivative size = 654, normalized size of antiderivative = 2.99 \[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
integrate(csch(d*x+c)*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")
Output:
-1/192*b^3*(105*arctan(e^(-d*x - c))/d + (279*e^(-d*x - c) + 91*e^(-3*d*x - 3*c) + 1799*e^(-5*d*x - 5*c) - 1085*e^(-7*d*x - 7*c) + 1085*e^(-9*d*x - 9*c) - 1799*e^(-11*d*x - 11*c) - 91*e^(-13*d*x - 13*c) - 279*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) + 70*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))) - 2/5*a*b^2*(15*e^(-d*x - 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)) + 20*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 )) + 58*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)) + 20*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)) + 15*e^(-9*d*x - 9* 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))) + a^3*log(tanh(1/2*d*x + 1/2* c))/d
Leaf count of result is larger than twice the leaf count of optimal. 414 vs. \(2 (203) = 406\).
Time = 0.32 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.89 \[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {960 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 960 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - 15 \, {\left (192 \, a^{2} b + 35 \, b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) + \frac {2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 5760 \, a b^{2} e^{\left (15 \, d x + 15 \, c\right )} + 1395 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} + 24960 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} + 455 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} + 62592 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 8995 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} + 103296 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 5425 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} + 103296 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 5425 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} + 62592 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 8995 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 24960 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 455 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} + 5760 \, a b^{2} e^{\left (d x + c\right )} - 1395 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{960 \, d} \] Input:
integrate(csch(d*x+c)*(a+b*tanh(d*x+c)^3)^3,x, algorithm="giac")
Output:
-1/960*(960*a^3*log(e^(d*x + c) + 1) - 960*a^3*log(abs(e^(d*x + c) - 1)) - 15*(192*a^2*b + 35*b^3)*arctan(e^(d*x + c)) + (2880*a^2*b*e^(15*d*x + 15* c) + 5760*a*b^2*e^(15*d*x + 15*c) + 1395*b^3*e^(15*d*x + 15*c) + 14400*a^2 *b*e^(13*d*x + 13*c) + 24960*a*b^2*e^(13*d*x + 13*c) + 455*b^3*e^(13*d*x + 13*c) + 25920*a^2*b*e^(11*d*x + 11*c) + 62592*a*b^2*e^(11*d*x + 11*c) + 8 995*b^3*e^(11*d*x + 11*c) + 14400*a^2*b*e^(9*d*x + 9*c) + 103296*a*b^2*e^( 9*d*x + 9*c) - 5425*b^3*e^(9*d*x + 9*c) - 14400*a^2*b*e^(7*d*x + 7*c) + 10 3296*a*b^2*e^(7*d*x + 7*c) + 5425*b^3*e^(7*d*x + 7*c) - 25920*a^2*b*e^(5*d *x + 5*c) + 62592*a*b^2*e^(5*d*x + 5*c) - 8995*b^3*e^(5*d*x + 5*c) - 14400 *a^2*b*e^(3*d*x + 3*c) + 24960*a*b^2*e^(3*d*x + 3*c) - 455*b^3*e^(3*d*x + 3*c) - 2880*a^2*b*e^(d*x + c) + 5760*a*b^2*e^(d*x + c) - 1395*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d
Time = 7.74 (sec) , antiderivative size = 671, normalized size of antiderivative = 3.06 \[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int((a + b*tanh(c + d*x)^3)^3/sinh(c + d*x),x)
Output:
(a^3*log(exp(c + d*x) - 1))/d - (a^3*log(exp(c + d*x) + 1))/d - (exp(c + d *x)*(4224*a*b^2 + 4445*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 + 35*b^2) *1i)/(128*d) + (b*log(exp(c + d*x) + 1i)*(192*a^2 + 35*b^2)*1i)/(128*d) + (exp(c + d*x)*(768*a*b^2 + 1925*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)) + (532*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)) - (3*exp(c + d*x)*(128*a*b^2 + 64*a^2*b + 31*b^3))/(64*d*(exp(2*c + 2*d*x ) + 1)) - (2*exp(c + d*x)*(144*a*b^2 + 1225*b^3))/(15*d*(5*exp(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(1 0*c + 10*d*x) + 1)) - (112*b^3*exp(c + d*x))/(d*(7*exp(2*c + 2*d*x) + 21*e xp(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 )*(1536*a*b^2 + 576*a^2*b + 931*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))
Time = 0.25 (sec) , antiderivative size = 1304, normalized size of antiderivative = 5.95 \[ \int \text {csch}(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)*(a+b*tanh(d*x+c)^3)^3,x)
Output:
(2880*e**(16*c + 16*d*x)*atan(e**(c + d*x))*a**2*b + 525*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 + 4200*e**(14*c + 14*d*x)*atan(e**(c + d*x))*b**3 + 80640*e**(12*c + 12*d*x)*atan(e**(c + d*x))*a**2*b + 14700*e**(12*c + 12*d*x)*atan(e**(c + d*x))*b**3 + 161280*e**(10*c + 10*d*x)*atan(e**(c + d*x))*a**2*b + 29400*e **(10*c + 10*d*x)*atan(e**(c + d*x))*b**3 + 201600*e**(8*c + 8*d*x)*atan(e **(c + d*x))*a**2*b + 36750*e**(8*c + 8*d*x)*atan(e**(c + d*x))*b**3 + 161 280*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a**2*b + 29400*e**(6*c + 6*d*x)*at an(e**(c + d*x))*b**3 + 80640*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b + 14700*e**(4*c + 4*d*x)*atan(e**(c + d*x))*b**3 + 23040*e**(2*c + 2*d*x)*a tan(e**(c + d*x))*a**2*b + 4200*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**3 + 2880*atan(e**(c + d*x))*a**2*b + 525*atan(e**(c + d*x))*b**3 + 960*e**(16 *c + 16*d*x)*log(e**(c + d*x) - 1)*a**3 - 960*e**(16*c + 16*d*x)*log(e**(c + d*x) + 1)*a**3 - 2880*e**(15*c + 15*d*x)*a**2*b - 5760*e**(15*c + 15*d* x)*a*b**2 - 1395*e**(15*c + 15*d*x)*b**3 + 7680*e**(14*c + 14*d*x)*log(e** (c + d*x) - 1)*a**3 - 7680*e**(14*c + 14*d*x)*log(e**(c + d*x) + 1)*a**3 - 14400*e**(13*c + 13*d*x)*a**2*b - 24960*e**(13*c + 13*d*x)*a*b**2 - 455*e **(13*c + 13*d*x)*b**3 + 26880*e**(12*c + 12*d*x)*log(e**(c + d*x) - 1)*a* *3 - 26880*e**(12*c + 12*d*x)*log(e**(c + d*x) + 1)*a**3 - 25920*e**(11*c + 11*d*x)*a**2*b - 62592*e**(11*c + 11*d*x)*a*b**2 - 8995*e**(11*c + 11...