Integrand size = 23, antiderivative size = 81 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {(a+b)^3 \text {csch}^2(c+d x)}{2 d}+\frac {b^2 (3 a+2 b) \log (\cosh (c+d x))}{d}+\frac {(a-2 b) (a+b)^2 \log (\sinh (c+d x))}{d}-\frac {b^3 \text {sech}^2(c+d x)}{2 d} \] Output:
-1/2*(a+b)^3*csch(d*x+c)^2/d+b^2*(3*a+2*b)*ln(cosh(d*x+c))/d+(a-2*b)*(a+b) ^2*ln(sinh(d*x+c))/d-1/2*b^3*sech(d*x+c)^2/d
Time = 0.88 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.36 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=-\frac {4 \cosh ^6(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \left ((a+b)^3 \text {csch}^2(c+d x)-2 b^2 (3 a+2 b) \log (\cosh (c+d x))-2 (a-2 b) (a+b)^2 \log (\sinh (c+d x))+b^3 \text {sech}^2(c+d x)\right )}{d (a+2 b+a \cosh (2 c+2 d x))^3} \] Input:
Integrate[Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]
Output:
(-4*Cosh[c + d*x]^6*(a + b*Sech[c + d*x]^2)^3*((a + b)^3*Csch[c + d*x]^2 - 2*b^2*(3*a + 2*b)*Log[Cosh[c + d*x]] - 2*(a - 2*b)*(a + b)^2*Log[Sinh[c + d*x]] + b^3*Sech[c + d*x]^2))/(d*(a + 2*b + a*Cosh[2*c + 2*d*x])^3)
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 4626, 354, 99, 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.
| \(\displaystyle \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \left (a+b \sec (i c+i d x)^2\right )^3}{\tan (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\left (b \sec (i c+i d x)^2+a\right )^3}{\tan (i c+i d x)^3}dx\) |
| \(\Big \downarrow \) 4626 |
| \(\displaystyle \frac {\int \frac {\left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^3(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\left (a \cosh ^2(c+d x)+b\right )^3 \text {sech}^2(c+d x)}{\left (1-\cosh ^2(c+d x)\right )^2}d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle \frac {\int \left (\text {sech}^2(c+d x) b^3+(3 a+2 b) \text {sech}(c+d x) b^2+\frac {(a-2 b) (a+b)^2}{\cosh ^2(c+d x)-1}+\frac {(a+b)^3}{\left (\cosh ^2(c+d x)-1\right )^2}\right )d\cosh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b^2 (3 a+2 b) \log \left (\cosh ^2(c+d x)\right )+\frac {(a+b)^3}{1-\cosh ^2(c+d x)}+(a-2 b) (a+b)^2 \log \left (1-\cosh ^2(c+d x)\right )+b^3 (-\text {sech}(c+d x))}{2 d}\) |
Input:
Int[Coth[c + d*x]^3*(a + b*Sech[c + d*x]^2)^3,x]
Output:
((a + b)^3/(1 - Cosh[c + d*x]^2) + b^2*(3*a + 2*b)*Log[Cosh[c + d*x]^2] + (a - 2*b)*(a + b)^2*Log[1 - Cosh[c + d*x]^2] - b^3*Sech[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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_ )]^(m_.), x_Symbol] :> Module[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-(f *ff^(m + n*p - 1))^(-1) Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff* x)^n)^p/x^(m + n*p)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, n} , x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p]
Time = 45.08 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )-\frac {3 a^{2} b}{2 \sinh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2}}-\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}-\frac {1}{\cosh \left (d x +c \right )^{2}}-2 \ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(110\) |
| default | \(\frac {a^{3} \left (\ln \left (\sinh \left (d x +c \right )\right )-\frac {\coth \left (d x +c \right )^{2}}{2}\right )-\frac {3 a^{2} b}{2 \sinh \left (d x +c \right )^{2}}+3 a \,b^{2} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2}}-\ln \left (\tanh \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )^{2}}-\frac {1}{\cosh \left (d x +c \right )^{2}}-2 \ln \left (\tanh \left (d x +c \right )\right )\right )}{d}\) | \(110\) |
| risch | \(-a^{3} x -\frac {2 a^{3} c}{d}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}+3 a^{2} b \,{\mathrm e}^{4 d x +4 c}+3 a \,b^{2} {\mathrm e}^{4 d x +4 c}+2 b^{3} {\mathrm e}^{4 d x +4 c}+2 a^{3} {\mathrm e}^{2 d x +2 c}+6 a^{2} b \,{\mathrm e}^{2 d x +2 c}+6 a \,b^{2} {\mathrm e}^{2 d x +2 c}+a^{3}+3 a^{2} b +3 a \,b^{2}+2 b^{3}\right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a^{3}}{d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) a \,b^{2}}{d}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b^{3}}{d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right ) a}{d}+\frac {2 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+1\right )}{d}\) | \(280\) |
Input:
int(coth(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/d*(a^3*(ln(sinh(d*x+c))-1/2*coth(d*x+c)^2)-3/2*a^2*b/sinh(d*x+c)^2+3*a*b ^2*(-1/2/sinh(d*x+c)^2-ln(tanh(d*x+c)))+b^3*(-1/2/sinh(d*x+c)^2/cosh(d*x+c )^2-1/cosh(d*x+c)^2-2*ln(tanh(d*x+c))))
Leaf count of result is larger than twice the leaf count of optimal. 1701 vs. \(2 (77) = 154\).
Time = 0.29 (sec) , antiderivative size = 1701, normalized size of antiderivative = 21.00 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Too large to display} \] Input:
integrate(coth(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x, algorithm="fricas")
Output:
-(a^3*d*x*cosh(d*x + c)^8 + 8*a^3*d*x*cosh(d*x + c)*sinh(d*x + c)^7 + a^3* d*x*sinh(d*x + c)^8 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^6 + 2*(14*a^3*d*x*cosh(d*x + c)^2 + a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*sinh(d* x + c)^6 + 4*(14*a^3*d*x*cosh(d*x + c)^3 + 3*(a^3 + 3*a^2*b + 3*a*b^2 + 2* b^3)*cosh(d*x + c))*sinh(d*x + c)^5 + a^3*d*x - 2*(a^3*d*x - 2*a^3 - 6*a^2 *b - 6*a*b^2)*cosh(d*x + c)^4 + 2*(35*a^3*d*x*cosh(d*x + c)^4 - a^3*d*x + 2*a^3 + 6*a^2*b + 6*a*b^2 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 8*(7*a^3*d*x*cosh(d*x + c)^5 + 5*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^3 - (a^3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2) *cosh(d*x + c))*sinh(d*x + c)^3 + 2*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh (d*x + c)^2 + 2*(14*a^3*d*x*cosh(d*x + c)^6 + 15*(a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3)*cosh(d*x + c)^4 + a^3 + 3*a^2*b + 3*a*b^2 + 2*b^3 - 6*(a^3*d*x - 2*a^3 - 6*a^2*b - 6*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - ((3*a*b^2 + 2*b^3)*cosh(d*x + c)^8 + 56*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^3*sinh(d*x + c )^5 + 28*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*(3*a*b^2 + 2*b^3)*cosh(d*x + c)*sinh(d*x + c)^7 + (3*a*b^2 + 2*b^3)*sinh(d*x + c)^8 - 2*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^4 + 2*(35*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^4 - 3*a*b^2 - 2*b^3)*sinh(d*x + c)^4 + 8*(7*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^5 - (3*a*b^2 + 2*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*a*b^2 + 2*b^ 3 + 4*(7*(3*a*b^2 + 2*b^3)*cosh(d*x + c)^6 - 3*(3*a*b^2 + 2*b^3)*cosh(d...
Timed out. \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\text {Timed out} \] Input:
integrate(coth(d*x+c)**3*(a+b*sech(d*x+c)**2)**3,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (77) = 154\).
Time = 0.12 (sec) , antiderivative size = 314, normalized size of antiderivative = 3.88 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=a^{3} {\left (x + \frac {c}{d} + \frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - 2 \, b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - 3 \, a b^{2} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} + \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{d} - \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {6 \, a^{2} b}{d {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2}} \] Input:
integrate(coth(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x, algorithm="maxima")
Output:
a^3*(x + c/d + log(e^(-d*x - c) + 1)/d + log(e^(-d*x - 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))) - 2*b^3*(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) + e^(-6*d*x - 6*c))/(d*(2*e^(-4*d*x - 4*c) - e^(-8 *d*x - 8*c) - 1))) - 3*a*b^2*(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))) - 6*a^2*b/(d*(e^(d*x + c) - e^(-d*x - c))^ 2)
Leaf count of result is larger than twice the leaf count of optimal. 247 vs. \(2 (77) = 154\).
Time = 0.20 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.05 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {2 \, {\left (3 \, a b^{2} + 2 \, b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} + 2\right ) + 2 \, {\left (a^{3} - 3 \, a b^{2} - 2 \, b^{3}\right )} \log \left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )} - 2\right ) - \frac {a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} + 8 \, a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 24 \, a^{2} b {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 24 \, a b^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 16 \, b^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )} + 12 \, a^{3} + 48 \, a^{2} b + 48 \, a b^{2}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + e^{\left (-2 \, d x - 2 \, c\right )}\right )}^{2} - 4}}{4 \, d} \] Input:
integrate(coth(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x, algorithm="giac")
Output:
1/4*(2*(3*a*b^2 + 2*b^3)*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) + 2) + 2*( a^3 - 3*a*b^2 - 2*b^3)*log(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c) - 2) - (a^3* (e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^2 + 8*a^3*(e^(2*d*x + 2*c) + e^(-2*d* x - 2*c)) + 24*a^2*b*(e^(2*d*x + 2*c) + e^(-2*d*x - 2*c)) + 24*a*b^2*(e^(2 *d*x + 2*c) + e^(-2*d*x - 2*c)) + 16*b^3*(e^(2*d*x + 2*c) + e^(-2*d*x - 2* c)) + 12*a^3 + 48*a^2*b + 48*a*b^2)/((e^(2*d*x + 2*c) + e^(-2*d*x - 2*c))^ 2 - 4))/d
Time = 2.85 (sec) , antiderivative size = 324, normalized size of antiderivative = 4.00 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (4\,b^3\,\sqrt {-d^2}-a^3\,\sqrt {-d^2}+6\,a\,b^2\,\sqrt {-d^2}\right )}{d\,\sqrt {a^6-12\,a^4\,b^2-8\,a^3\,b^3+36\,a^2\,b^4+48\,a\,b^5+16\,b^6}}\right )\,\sqrt {a^6-12\,a^4\,b^2-8\,a^3\,b^3+36\,a^2\,b^4+48\,a\,b^5+16\,b^6}}{\sqrt {-d^2}}-\frac {\frac {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2\right )}{d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+2\,b^3\right )}{d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-1}-\frac {\frac {4\,\left (a^3+3\,a^2\,b+3\,a\,b^2\right )}{d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+2\,b^3\right )}{d}}{{\mathrm {e}}^{8\,c+8\,d\,x}-2\,{\mathrm {e}}^{4\,c+4\,d\,x}+1}-a^3\,x+\frac {a^3\,\ln \left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}{2\,d} \] Input:
int(coth(c + d*x)^3*(a + b/cosh(c + d*x)^2)^3,x)
Output:
(atan((exp(2*c)*exp(2*d*x)*(4*b^3*(-d^2)^(1/2) - a^3*(-d^2)^(1/2) + 6*a*b^ 2*(-d^2)^(1/2)))/(d*(48*a*b^5 + a^6 + 16*b^6 + 36*a^2*b^4 - 8*a^3*b^3 - 12 *a^4*b^2)^(1/2)))*(48*a*b^5 + a^6 + 16*b^6 + 36*a^2*b^4 - 8*a^3*b^3 - 12*a ^4*b^2)^(1/2))/(-d^2)^(1/2) - ((4*(3*a*b^2 + 3*a^2*b + a^3))/d + (2*exp(2* c + 2*d*x)*(3*a*b^2 + 3*a^2*b + a^3 + 2*b^3))/d)/(exp(4*c + 4*d*x) - 1) - ((4*(3*a*b^2 + 3*a^2*b + a^3))/d + (4*exp(2*c + 2*d*x)*(3*a*b^2 + 3*a^2*b + a^3 + 2*b^3))/d)/(exp(8*c + 8*d*x) - 2*exp(4*c + 4*d*x) + 1) - a^3*x + ( a^3*log(exp(4*c + 4*d*x) - 1))/(2*d)
Time = 0.22 (sec) , antiderivative size = 803, normalized size of antiderivative = 9.91 \[ \int \coth ^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^3 \, dx=\frac {e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3}-2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{3}+e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3}-2 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{3}-2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a^{3}+4 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) b^{3}-2 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a^{3}+4 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) b^{3}-3 \,\mathrm {log}\left (e^{d x +c}-1\right ) a \,b^{2}-3 \,\mathrm {log}\left (e^{d x +c}+1\right ) a \,b^{2}+2 \,\mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}-6 e^{8 d x +8 c} a^{2} b -6 e^{8 d x +8 c} a \,b^{2}-6 e^{6 d x +6 c} a^{2} b -6 e^{6 d x +6 c} a \,b^{2}-6 a \,b^{2}-3 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}-1\right ) a \,b^{2}-3 e^{8 d x +8 c} \mathrm {log}\left (e^{d x +c}+1\right ) a \,b^{2}+6 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}-1\right ) a \,b^{2}+6 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+1\right ) a \,b^{2}-e^{8 d x +8 c} a^{3} d x -6 e^{2 d x +2 c} a \,b^{2}-2 a^{3}-4 e^{2 d x +2 c} b^{3}+2 e^{8 d x +8 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}-4 e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) b^{3}+3 \,\mathrm {log}\left (e^{2 d x +2 c}+1\right ) a \,b^{2}-6 e^{2 d x +2 c} a^{2} b +3 e^{8 d x +8 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a \,b^{2}-6 e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a \,b^{2}-6 a^{2} b -2 e^{8 d x +8 c} a^{3}-2 e^{6 d x +6 c} a^{3}-4 e^{6 d x +6 c} b^{3}-2 e^{2 d x +2 c} a^{3}+\mathrm {log}\left (e^{d x +c}-1\right ) a^{3}-2 \,\mathrm {log}\left (e^{d x +c}-1\right ) b^{3}+\mathrm {log}\left (e^{d x +c}+1\right ) a^{3}-2 \,\mathrm {log}\left (e^{d x +c}+1\right ) b^{3}+2 e^{4 d x +4 c} a^{3} d x -a^{3} d x}{d \left (e^{8 d x +8 c}-2 e^{4 d x +4 c}+1\right )} \] Input:
int(coth(d*x+c)^3*(a+b*sech(d*x+c)^2)^3,x)
Output:
(3*e**(8*c + 8*d*x)*log(e**(2*c + 2*d*x) + 1)*a*b**2 + 2*e**(8*c + 8*d*x)* log(e**(2*c + 2*d*x) + 1)*b**3 + e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a* *3 - 3*e**(8*c + 8*d*x)*log(e**(c + d*x) - 1)*a*b**2 - 2*e**(8*c + 8*d*x)* log(e**(c + d*x) - 1)*b**3 + e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a**3 - 3*e**(8*c + 8*d*x)*log(e**(c + d*x) + 1)*a*b**2 - 2*e**(8*c + 8*d*x)*log( e**(c + d*x) + 1)*b**3 - e**(8*c + 8*d*x)*a**3*d*x - 2*e**(8*c + 8*d*x)*a* *3 - 6*e**(8*c + 8*d*x)*a**2*b - 6*e**(8*c + 8*d*x)*a*b**2 - 2*e**(6*c + 6 *d*x)*a**3 - 6*e**(6*c + 6*d*x)*a**2*b - 6*e**(6*c + 6*d*x)*a*b**2 - 4*e** (6*c + 6*d*x)*b**3 - 6*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a*b**2 - 4*e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*b**3 - 2*e**(4*c + 4*d*x)*lo g(e**(c + d*x) - 1)*a**3 + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*a*b**2 + 4*e**(4*c + 4*d*x)*log(e**(c + d*x) - 1)*b**3 - 2*e**(4*c + 4*d*x)*log( e**(c + d*x) + 1)*a**3 + 6*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*a*b**2 + 4*e**(4*c + 4*d*x)*log(e**(c + d*x) + 1)*b**3 + 2*e**(4*c + 4*d*x)*a**3*d *x - 2*e**(2*c + 2*d*x)*a**3 - 6*e**(2*c + 2*d*x)*a**2*b - 6*e**(2*c + 2*d *x)*a*b**2 - 4*e**(2*c + 2*d*x)*b**3 + 3*log(e**(2*c + 2*d*x) + 1)*a*b**2 + 2*log(e**(2*c + 2*d*x) + 1)*b**3 + log(e**(c + d*x) - 1)*a**3 - 3*log(e* *(c + d*x) - 1)*a*b**2 - 2*log(e**(c + d*x) - 1)*b**3 + log(e**(c + d*x) + 1)*a**3 - 3*log(e**(c + d*x) + 1)*a*b**2 - 2*log(e**(c + d*x) + 1)*b**3 - a**3*d*x - 2*a**3 - 6*a**2*b - 6*a*b**2)/(d*(e**(8*c + 8*d*x) - 2*e**(...