Integrand size = 23, antiderivative size = 138 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {a^3 \coth (c+d x)}{d}-\frac {a^3 \coth ^3(c+d x)}{3 d}+\frac {3 a^2 b \log (\tanh (c+d x))}{d}-\frac {3 a^2 b \tanh ^2(c+d x)}{2 d}+\frac {a b^2 \tanh ^3(c+d x)}{d}-\frac {3 a b^2 \tanh ^5(c+d x)}{5 d}+\frac {b^3 \tanh ^6(c+d x)}{6 d}-\frac {b^3 \tanh ^8(c+d x)}{8 d} \] Output:
a^3*coth(d*x+c)/d-1/3*a^3*coth(d*x+c)^3/d+3*a^2*b*ln(tanh(d*x+c))/d-3/2*a^ 2*b*tanh(d*x+c)^2/d+a*b^2*tanh(d*x+c)^3/d-3/5*a*b^2*tanh(d*x+c)^5/d+1/6*b^ 3*tanh(d*x+c)^6/d-1/8*b^3*tanh(d*x+c)^8/d
Time = 1.45 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.54 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {2 a^3 \coth (c+d x)}{3 d}-\frac {a^3 \coth (c+d x) \text {csch}^2(c+d x)}{3 d}-\frac {3 a^2 b \log (\cosh (c+d x))}{d}+\frac {3 a^2 b \log (\sinh (c+d x))}{d}+\frac {3 a^2 b \text {sech}^2(c+d x)}{2 d}-\frac {b^3 \text {sech}^4(c+d x)}{4 d}+\frac {b^3 \text {sech}^6(c+d x)}{3 d}-\frac {b^3 \text {sech}^8(c+d x)}{8 d}+\frac {2 a b^2 \tanh (c+d x)}{5 d}+\frac {a b^2 \text {sech}^2(c+d x) \tanh (c+d x)}{5 d}-\frac {3 a b^2 \text {sech}^4(c+d x) \tanh (c+d x)}{5 d} \] Input:
Integrate[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
(2*a^3*Coth[c + d*x])/(3*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^2)/(3*d) - (3*a^2*b*Log[Cosh[c + d*x]])/d + (3*a^2*b*Log[Sinh[c + d*x]])/d + (3*a^2*b *Sech[c + d*x]^2)/(2*d) - (b^3*Sech[c + d*x]^4)/(4*d) + (b^3*Sech[c + d*x] ^6)/(3*d) - (b^3*Sech[c + d*x]^8)/(8*d) + (2*a*b^2*Tanh[c + d*x])/(5*d) + (a*b^2*Sech[c + d*x]^2*Tanh[c + d*x])/(5*d) - (3*a*b^2*Sech[c + d*x]^4*Tan h[c + d*x])/(5*d)
Time = 0.37 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4146, 2333, 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.
| \(\displaystyle \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+i b \tan (i c+i d x)^3\right )^3}{\sin (i c+i d x)^4}dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^3(c+d x)+a\right )^3d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2333 |
| \(\displaystyle \frac {\int \left (-b^3 \tanh ^7(c+d x)+b^3 \tanh ^5(c+d x)-3 a b^2 \tanh ^4(c+d x)+3 a b^2 \tanh ^2(c+d x)-3 a^2 b \tanh (c+d x)+a^3 \coth ^4(c+d x)-a^3 \coth ^2(c+d x)+3 a^2 b \coth (c+d x)\right )d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{3} a^3 \coth ^3(c+d x)+a^3 \coth (c+d x)-\frac {3}{2} a^2 b \tanh ^2(c+d x)+3 a^2 b \log (\tanh (c+d x))-\frac {3}{5} a b^2 \tanh ^5(c+d x)+a b^2 \tanh ^3(c+d x)-\frac {1}{8} b^3 \tanh ^8(c+d x)+\frac {1}{6} b^3 \tanh ^6(c+d x)}{d}\) |
Input:
Int[Csch[c + d*x]^4*(a + b*Tanh[c + d*x]^3)^3,x]
Output:
(a^3*Coth[c + d*x] - (a^3*Coth[c + d*x]^3)/3 + 3*a^2*b*Log[Tanh[c + d*x]] - (3*a^2*b*Tanh[c + d*x]^2)/2 + a*b^2*Tanh[c + d*x]^3 - (3*a*b^2*Tanh[c + d*x]^5)/5 + (b^3*Tanh[c + d*x]^6)/6 - (b^3*Tanh[c + d*x]^8)/8)/d
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c*x)^m*Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Time = 85.43 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{6 \cosh \left (d x +c \right )^{8}}-\frac {1}{24 \cosh \left (d x +c \right )^{8}}\right )}{d}\) | \(156\) |
| default | \(\frac {a^{3} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+3 a^{2} b \left (\frac {1}{2 \cosh \left (d x +c \right )^{2}}+\ln \left (\tanh \left (d x +c \right )\right )\right )+3 b^{2} a \left (-\frac {\sinh \left (d x +c \right )}{4 \cosh \left (d x +c \right )^{5}}+\frac {\left (\frac {8}{15}+\frac {\operatorname {sech}\left (d x +c \right )^{4}}{5}+\frac {4 \operatorname {sech}\left (d x +c \right )^{2}}{15}\right ) \tanh \left (d x +c \right )}{4}\right )+b^{3} \left (-\frac {\sinh \left (d x +c \right )^{4}}{4 \cosh \left (d x +c \right )^{8}}-\frac {\sinh \left (d x +c \right )^{2}}{6 \cosh \left (d x +c \right )^{8}}-\frac {1}{24 \cosh \left (d x +c \right )^{8}}\right )}{d}\) | \(156\) |
| risch | \(-\frac {2 \left (-310 b^{3} {\mathrm e}^{8 d x +8 c}+130 b^{3} {\mathrm e}^{6 d x +6 c}-30 b^{3} {\mathrm e}^{4 d x +4 c}-50 \,{\mathrm e}^{2 d x +2 c} a^{3}-6 b^{2} a -10 a^{3}+270 a^{2} b \,{\mathrm e}^{12 d x +12 c}-270 a^{2} b \,{\mathrm e}^{10 d x +10 c}+180 a \,b^{2} {\mathrm e}^{10 d x +10 c}-360 a^{2} b \,{\mathrm e}^{8 d x +8 c}-108 a \,b^{2} {\mathrm e}^{8 d x +8 c}+135 a^{2} b \,{\mathrm e}^{4 d x +4 c}+48 a \,b^{2} {\mathrm e}^{4 d x +4 c}+45 a^{2} b \,{\mathrm e}^{2 d x +2 c}-30 a \,b^{2} {\mathrm e}^{2 d x +2 c}+280 a^{3} {\mathrm e}^{6 d x +6 c}+490 b^{3} {\mathrm e}^{10 d x +10 c}+980 a^{3} {\mathrm e}^{8 d x +8 c}-490 b^{3} {\mathrm e}^{12 d x +12 c}-45 a^{2} b \,{\mathrm e}^{20 d x +20 c}-135 a^{2} b \,{\mathrm e}^{18 d x +18 c}+30 b^{3} {\mathrm e}^{18 d x +18 c}-130 b^{3} {\mathrm e}^{16 d x +16 c}+760 a^{3} {\mathrm e}^{14 d x +14 c}+360 a^{2} b \,{\mathrm e}^{14 d x +14 c}-240 a \,b^{2} {\mathrm e}^{14 d x +14 c}+96 a \,b^{2} {\mathrm e}^{12 d x +12 c}-30 a \,b^{2} {\mathrm e}^{16 d x +16 c}+30 a^{3} {\mathrm e}^{18 d x +18 c}+310 b^{3} {\mathrm e}^{14 d x +14 c}+1400 a^{3} {\mathrm e}^{12 d x +12 c}+230 a^{3} {\mathrm e}^{16 d x +16 c}+90 a \,b^{2} {\mathrm e}^{18 d x +18 c}-40 \,{\mathrm e}^{4 d x +4 c} a^{3}+1540 \,{\mathrm e}^{10 d x +10 c} a^{3}\right )}{15 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{8} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}+\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d}-\frac {3 a^{2} b \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\) | \(565\) |
Input:
int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(2/3-1/3*csch(d*x+c)^2)*coth(d*x+c)+3*a^2*b*(1/2/cosh(d*x+c)^2+ln (tanh(d*x+c)))+3*b^2*a*(-1/4*sinh(d*x+c)/cosh(d*x+c)^5+1/4*(8/15+1/5*sech( d*x+c)^4+4/15*sech(d*x+c)^2)*tanh(d*x+c))+b^3*(-1/4*sinh(d*x+c)^4/cosh(d*x +c)^8-1/6*sinh(d*x+c)^2/cosh(d*x+c)^8-1/24/cosh(d*x+c)^8))
Leaf count of result is larger than twice the leaf count of optimal. 9459 vs. \(2 (128) = 256\).
Time = 0.20 (sec) , antiderivative size = 9459, normalized size of antiderivative = 68.54 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")
Output:
Too large to include
\[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \] Input:
integrate(csch(d*x+c)**4*(a+b*tanh(d*x+c)**3)**3,x)
Output:
Integral((a + b*tanh(c + d*x)**3)**3*csch(c + d*x)**4, x)
Leaf count of result is larger than twice the leaf count of optimal. 997 vs. \(2 (128) = 256\).
Time = 0.15 (sec) , antiderivative size = 997, normalized size of antiderivative = 7.22 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="maxima")
Output:
3*a^2*b*(log(e^(-d*x - c) + 1)/d + log(e^(-d*x - c) - 1)/d - log(e^(-2*d*x - 2*c) + 1)/d + 2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4 *c) + 1))) + 4/5*a*b^2*(5*e^(-2*d*x - 2*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^(-4*d*x - 4*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^(-6*d*x - 6*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)) + 1/(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))) + 4/3*a^3*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2 *d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2* d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1))) - 4/3*b^3*(3*e^( -4*d*x - 4*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)) - 4*e^(-6*d*x - 6*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)) + 10*e^(-8*d*x - 8*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...
Leaf count of result is larger than twice the leaf count of optimal. 437 vs. \(2 (128) = 256\).
Time = 0.36 (sec) , antiderivative size = 437, normalized size of antiderivative = 3.17 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=-\frac {2520 \, a^{2} b \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right ) - 2520 \, a^{2} b \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right ) + \frac {140 \, {\left (33 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 99 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 99 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a^{3} - 33 \, a^{2} b\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}} - \frac {6849 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 59832 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 222012 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 10080 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 3360 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 459144 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 26880 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 4480 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 580230 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 23520 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 11200 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 459144 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 10752 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 4480 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 222012 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 8736 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 3360 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 59832 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 5376 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6849 \, a^{2} b - 672 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{840 \, d} \] Input:
integrate(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x, algorithm="giac")
Output:
-1/840*(2520*a^2*b*log(e^(2*d*x + 2*c) + 1) - 2520*a^2*b*log(abs(e^(2*d*x + 2*c) - 1)) + 140*(33*a^2*b*e^(6*d*x + 6*c) - 99*a^2*b*e^(4*d*x + 4*c) + 24*a^3*e^(2*d*x + 2*c) + 99*a^2*b*e^(2*d*x + 2*c) - 8*a^3 - 33*a^2*b)/(e^( 2*d*x + 2*c) - 1)^3 - (6849*a^2*b*e^(16*d*x + 16*c) + 59832*a^2*b*e^(14*d* x + 14*c) + 222012*a^2*b*e^(12*d*x + 12*c) - 10080*a*b^2*e^(12*d*x + 12*c) - 3360*b^3*e^(12*d*x + 12*c) + 459144*a^2*b*e^(10*d*x + 10*c) - 26880*a*b ^2*e^(10*d*x + 10*c) + 4480*b^3*e^(10*d*x + 10*c) + 580230*a^2*b*e^(8*d*x + 8*c) - 23520*a*b^2*e^(8*d*x + 8*c) - 11200*b^3*e^(8*d*x + 8*c) + 459144* a^2*b*e^(6*d*x + 6*c) - 10752*a*b^2*e^(6*d*x + 6*c) + 4480*b^3*e^(6*d*x + 6*c) + 222012*a^2*b*e^(4*d*x + 4*c) - 8736*a*b^2*e^(4*d*x + 4*c) - 3360*b^ 3*e^(4*d*x + 4*c) + 59832*a^2*b*e^(2*d*x + 2*c) - 5376*a*b^2*e^(2*d*x + 2* c) + 6849*a^2*b - 672*a*b^2)/(e^(2*d*x + 2*c) + 1)^8)/d
Time = 0.54 (sec) , antiderivative size = 646, normalized size of antiderivative = 4.68 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx=\frac {96\,\left (10\,b^3+a\,b^2\right )}{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 {640\,b^3}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {4\,\left (25\,b^3+12\,a\,b^2\right )}{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 {128\,b^3}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1\right )}-\frac {2\,\left (3\,a^2\,b+6\,a\,b^2+2\,b^3\right )}{d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {6\,\mathrm {atan}\left (\frac {a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\sqrt {-d^2}}{d\,\sqrt {a^4\,b^2}}\right )\,\sqrt {a^4\,b^2}}{\sqrt {-d^2}}-\frac {32\,b^3}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {4\,a^3}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {8\,a^3}{3\,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 {8\,\left (11\,b^3+15\,a\,b^2\right )}{3\,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 {6\,a^2\,b}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \] Input:
int((a + b*tanh(c + d*x)^3)^3/sinh(c + d*x)^4,x)
Output:
(96*(a*b^2 + 10*b^3))/(5*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(10*c + 10*d*x) + 1)) - (640*b^ 3)/(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)) - (4 *(12*a*b^2 + 25*b^3))/(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)) + (128*b^3)/(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)) - (2*(6*a *b^2 + 3*a^2*b + 2*b^3))/(d*(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)) - (6*atan((a^2*b*exp(2*c)*exp(2*d*x)*(-d^2)^(1/2))/(d*(a^4*b^2)^(1/2)))*(a^ 4*b^2)^(1/2))/(-d^2)^(1/2) - (32*b^3)/(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 )) - (4*a^3)/(d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (8*a^3)/(3* d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) + (8*( 15*a*b^2 + 11*b^3))/(3*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6* c + 6*d*x) + 1)) + (6*a^2*b)/(d*(exp(2*c + 2*d*x) + 1))
Time = 0.31 (sec) , antiderivative size = 1626, normalized size of antiderivative = 11.78 \[ \int \text {csch}^4(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx =\text {Too large to display} \] Input:
int(csch(d*x+c)^4*(a+b*tanh(d*x+c)^3)^3,x)
Output:
( - 45*e**(22*c + 22*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2*b + 45*e**(22*c + 22*d*x)*log(e**(c + d*x) - 1)*a**2*b + 45*e**(22*c + 22*d*x)*log(e**(c + d*x) + 1)*a**2*b - 18*e**(22*c + 22*d*x)*a**2*b - 225*e**(20*c + 20*d*x)*l og(e**(2*c + 2*d*x) + 1)*a**2*b + 225*e**(20*c + 20*d*x)*log(e**(c + d*x) - 1)*a**2*b + 225*e**(20*c + 20*d*x)*log(e**(c + d*x) + 1)*a**2*b - 315*e* *(18*c + 18*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2*b + 315*e**(18*c + 18*d*x) *log(e**(c + d*x) - 1)*a**2*b + 315*e**(18*c + 18*d*x)*log(e**(c + d*x) + 1)*a**2*b - 60*e**(18*c + 18*d*x)*a**3 + 144*e**(18*c + 18*d*x)*a**2*b - 1 80*e**(18*c + 18*d*x)*a*b**2 - 60*e**(18*c + 18*d*x)*b**3 + 225*e**(16*c + 16*d*x)*log(e**(2*c + 2*d*x) + 1)*a**2*b - 225*e**(16*c + 16*d*x)*log(e** (c + d*x) - 1)*a**2*b - 225*e**(16*c + 16*d*x)*log(e**(c + d*x) + 1)*a**2* b - 460*e**(16*c + 16*d*x)*a**3 + 90*e**(16*c + 16*d*x)*a**2*b + 60*e**(16 *c + 16*d*x)*a*b**2 + 260*e**(16*c + 16*d*x)*b**3 + 990*e**(14*c + 14*d*x) *log(e**(2*c + 2*d*x) + 1)*a**2*b - 990*e**(14*c + 14*d*x)*log(e**(c + d*x ) - 1)*a**2*b - 990*e**(14*c + 14*d*x)*log(e**(c + d*x) + 1)*a**2*b - 1520 *e**(14*c + 14*d*x)*a**3 - 324*e**(14*c + 14*d*x)*a**2*b + 480*e**(14*c + 14*d*x)*a*b**2 - 620*e**(14*c + 14*d*x)*b**3 + 630*e**(12*c + 12*d*x)*log( e**(2*c + 2*d*x) + 1)*a**2*b - 630*e**(12*c + 12*d*x)*log(e**(c + d*x) - 1 )*a**2*b - 630*e**(12*c + 12*d*x)*log(e**(c + d*x) + 1)*a**2*b - 2800*e**( 12*c + 12*d*x)*a**3 - 288*e**(12*c + 12*d*x)*a**2*b - 192*e**(12*c + 12...