Integrand size = 21, antiderivative size = 120 \[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \left (3 a^2+b^2\right ) \arctan (\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}+\frac {a^3 \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac {a^3 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) d} \] Output:
1/2*b*(3*a^2+b^2)*arctan(sinh(d*x+c))/(a^2+b^2)^2/d+a^3*ln(cosh(d*x+c))/(a ^2+b^2)^2/d-a^3*ln(a+b*sinh(d*x+c))/(a^2+b^2)^2/d+1/2*sech(d*x+c)^2*(a-b*s inh(d*x+c))/(a^2+b^2)/d
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.27 \[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {b \left (a^2+b^2\right ) \arctan (\sinh (c+d x))-\left (a^3-i \left (2 a^2 b+b^3\right )\right ) \log (i-\sinh (c+d x))-\left (a^3+i \left (2 a^2 b+b^3\right )\right ) \log (i+\sinh (c+d x))+2 a^3 \log (a+b \sinh (c+d x))-a \left (a^2+b^2\right ) \text {sech}^2(c+d x)+b \left (a^2+b^2\right ) \text {sech}(c+d x) \tanh (c+d x)}{2 \left (a^2+b^2\right )^2 d} \] Input:
Integrate[Tanh[c + d*x]^3/(a + b*Sinh[c + d*x]),x]
Output:
-1/2*(b*(a^2 + b^2)*ArcTan[Sinh[c + d*x]] - (a^3 - I*(2*a^2*b + b^3))*Log[ I - Sinh[c + d*x]] - (a^3 + I*(2*a^2*b + b^3))*Log[I + Sinh[c + d*x]] + 2* a^3*Log[a + b*Sinh[c + d*x]] - a*(a^2 + b^2)*Sech[c + d*x]^2 + b*(a^2 + b^ 2)*Sech[c + d*x]*Tanh[c + d*x])/((a^2 + b^2)^2*d)
Time = 0.41 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.22, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3042, 26, 3200, 601, 25, 27, 657, 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.
| \(\displaystyle \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \tan (i c+i d x)^3}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\tan (i c+i d x)^3}{a-i b \sin (i c+i d x)}dx\) |
| \(\Big \downarrow \) 3200 |
| \(\displaystyle \frac {\int \frac {b^3 \sinh ^3(c+d x)}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )^2}d(b \sinh (c+d x))}{d}\) |
| \(\Big \downarrow \) 601 |
| \(\displaystyle \frac {\frac {b^2 (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}-\frac {\int -\frac {b^2 \left (a b^2+\left (2 a^2+b^2\right ) \sinh (c+d x) b\right )}{\left (a^2+b^2\right ) (a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )}d(b \sinh (c+d x))}{2 b^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {b^2 \left (a b^2+\left (2 a^2+b^2\right ) \sinh (c+d x) b\right )}{\left (a^2+b^2\right ) (a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )}d(b \sinh (c+d x))}{2 b^2}+\frac {b^2 (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {a b^2+\left (2 a^2+b^2\right ) \sinh (c+d x) b}{(a+b \sinh (c+d x)) \left (\sinh ^2(c+d x) b^2+b^2\right )}d(b \sinh (c+d x))}{2 \left (a^2+b^2\right )}+\frac {b^2 (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}}{d}\) |
| \(\Big \downarrow \) 657 |
| \(\displaystyle \frac {\frac {\int \left (\frac {b^4+3 a^2 b^2+2 a^3 \sinh (c+d x) b}{\left (a^2+b^2\right ) \left (\sinh ^2(c+d x) b^2+b^2\right )}-\frac {2 a^3}{\left (a^2+b^2\right ) (a+b \sinh (c+d x))}\right )d(b \sinh (c+d x))}{2 \left (a^2+b^2\right )}+\frac {b^2 (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^2 (a-b \sinh (c+d x))}{2 \left (a^2+b^2\right ) \left (b^2 \sinh ^2(c+d x)+b^2\right )}+\frac {\frac {b \left (3 a^2+b^2\right ) \arctan (\sinh (c+d x))}{a^2+b^2}+\frac {a^3 \log \left (b^2 \sinh ^2(c+d x)+b^2\right )}{a^2+b^2}-\frac {2 a^3 \log (a+b \sinh (c+d x))}{a^2+b^2}}{2 \left (a^2+b^2\right )}}{d}\) |
Input:
Int[Tanh[c + d*x]^3/(a + b*Sinh[c + d*x]),x]
Output:
(((b*(3*a^2 + b^2)*ArcTan[Sinh[c + d*x]])/(a^2 + b^2) - (2*a^3*Log[a + b*S inh[c + d*x]])/(a^2 + b^2) + (a^3*Log[b^2 + b^2*Sinh[c + d*x]^2])/(a^2 + b ^2))/(2*(a^2 + b^2)) + (b^2*(a - b*Sinh[c + d*x]))/(2*(a^2 + b^2)*(b^2 + b ^2*Sinh[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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[(c + d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* (2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
Int[(((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.))/((a_) + (c_.)*( x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*((f + g*x)^n/(a + c*x^ 2)), x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && IntegersQ[n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p _.), x_Symbol] :> Simp[1/f Subst[Int[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b ^2, 0] && IntegerQ[(p + 1)/2]
Time = 0.97 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.75
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3}-a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+a^{3} \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (3 a^{2} b +b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {8 a^{3} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}}{d}\) | \(210\) |
| default | \(\frac {\frac {\frac {2 \left (\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+\left (-a^{3}-a \,b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+a^{3} \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (3 a^{2} b +b^{3}\right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {8 a^{3} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-a \right )}{8 a^{4}+16 a^{2} b^{2}+8 b^{4}}}{d}\) | \(210\) |
| risch | \(-\frac {2 a^{3} d^{2} x}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}-\frac {2 a^{3} d c}{a^{4} d^{2}+2 a^{2} b^{2} d^{2}+b^{4} d^{2}}+\frac {2 a^{3} x}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 a^{3} c}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {{\mathrm e}^{d x +c} \left (-b \,{\mathrm e}^{2 d x +2 c}+2 a \,{\mathrm e}^{d x +c}+b \right )}{d \left (a^{2}+b^{2}\right ) \left (1+{\mathrm e}^{2 d x +2 c}\right )^{2}}+\frac {3 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{d x +c}+i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}+\frac {\ln \left ({\mathrm e}^{d x +c}-i\right ) a^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) d}-\frac {a^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \,{\mathrm e}^{d x +c}}{b}-1\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(441\) |
Input:
int(tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(2/(a^4+2*a^2*b^2+b^4)*(((1/2*a^2*b+1/2*b^3)*tanh(1/2*d*x+1/2*c)^3+(-a ^3-a*b^2)*tanh(1/2*d*x+1/2*c)^2+(-1/2*a^2*b-1/2*b^3)*tanh(1/2*d*x+1/2*c))/ (1+tanh(1/2*d*x+1/2*c)^2)^2+1/2*a^3*ln(1+tanh(1/2*d*x+1/2*c)^2)+1/2*(3*a^2 *b+b^3)*arctan(tanh(1/2*d*x+1/2*c)))-8*a^3/(8*a^4+16*a^2*b^2+8*b^4)*ln(tan h(1/2*d*x+1/2*c)^2*a-2*b*tanh(1/2*d*x+1/2*c)-a))
Leaf count of result is larger than twice the leaf count of optimal. 896 vs. \(2 (117) = 234\).
Time = 0.13 (sec) , antiderivative size = 896, normalized size of antiderivative = 7.47 \[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \] Input:
integrate(tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
-((a^2*b + b^3)*cosh(d*x + c)^3 + (a^2*b + b^3)*sinh(d*x + c)^3 - 2*(a^3 + a*b^2)*cosh(d*x + c)^2 - (2*a^3 + 2*a*b^2 - 3*(a^2*b + b^3)*cosh(d*x + c) )*sinh(d*x + c)^2 - ((3*a^2*b + b^3)*cosh(d*x + c)^4 + 4*(3*a^2*b + b^3)*c osh(d*x + c)*sinh(d*x + c)^3 + (3*a^2*b + b^3)*sinh(d*x + c)^4 + 3*a^2*b + b^3 + 2*(3*a^2*b + b^3)*cosh(d*x + c)^2 + 2*(3*a^2*b + b^3 + 3*(3*a^2*b + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 4*((3*a^2*b + b^3)*cosh(d*x + c)^ 3 + (3*a^2*b + b^3)*cosh(d*x + c))*sinh(d*x + c))*arctan(cosh(d*x + c) + s inh(d*x + c)) - (a^2*b + b^3)*cosh(d*x + c) + (a^3*cosh(d*x + c)^4 + 4*a^3 *cosh(d*x + c)*sinh(d*x + c)^3 + a^3*sinh(d*x + c)^4 + 2*a^3*cosh(d*x + c) ^2 + a^3 + 2*(3*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^2 + 4*(a^3*cosh(d *x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c))*log(2*(b*sinh(d*x + c) + a)/ (cosh(d*x + c) - sinh(d*x + c))) - (a^3*cosh(d*x + c)^4 + 4*a^3*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*sinh(d*x + c)^4 + 2*a^3*cosh(d*x + c)^2 + a^3 + 2*(3*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^2 + 4*(a^3*cosh(d*x + c)^3 + a^3*cosh(d*x + c))*sinh(d*x + c))*log(2*cosh(d*x + c)/(cosh(d*x + c) - si nh(d*x + c))) - (a^2*b + b^3 - 3*(a^2*b + b^3)*cosh(d*x + c)^2 + 4*(a^3 + a*b^2)*cosh(d*x + c))*sinh(d*x + c))/((a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^4 + 4*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^4 + 2*a^2*b^2 + b^4)*d*sinh(d*x + c)^4 + 2*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + 2*(3*(a^4 + 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2 + (a^4 + 2*a^2...
\[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\tanh ^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate(tanh(d*x+c)**3/(a+b*sinh(d*x+c)),x)
Output:
Integral(tanh(c + d*x)**3/(a + b*sinh(c + d*x)), x)
Time = 0.12 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.81 \[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {a^{3} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {{\left (3 \, a^{2} b + b^{3}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {b e^{\left (-d x - c\right )} - 2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - b e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} \] Input:
integrate(tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
-a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^4 + 2*a^2*b^2 + b ^4)*d) + a^3*log(e^(-2*d*x - 2*c) + 1)/((a^4 + 2*a^2*b^2 + b^4)*d) - (3*a^ 2*b + b^3)*arctan(e^(-d*x - c))/((a^4 + 2*a^2*b^2 + b^4)*d) - (b*e^(-d*x - c) - 2*a*e^(-2*d*x - 2*c) - b*e^(-3*d*x - 3*c))/((a^2 + b^2 + 2*(a^2 + b^ 2)*e^(-2*d*x - 2*c) + (a^2 + b^2)*e^(-4*d*x - 4*c))*d)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (117) = 234\).
Time = 0.18 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.32 \[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {4 \, a^{3} b \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {2 \, a^{3} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (3 \, a^{2} b + b^{3}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 2 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 4 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}}}{4 \, d} \] Input:
integrate(tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
-1/4*(4*a^3*b*log(abs(b*(e^(d*x + c) - e^(-d*x - c)) + 2*a))/(a^4*b + 2*a^ 2*b^3 + b^5) - 2*a^3*log((e^(d*x + c) - e^(-d*x - c))^2 + 4)/(a^4 + 2*a^2* b^2 + b^4) - (pi + 2*arctan(1/2*(e^(2*d*x + 2*c) - 1)*e^(-d*x - c)))*(3*a^ 2*b + b^3)/(a^4 + 2*a^2*b^2 + b^4) + 2*(a^3*(e^(d*x + c) - e^(-d*x - c))^2 + 2*a^2*b*(e^(d*x + c) - e^(-d*x - c)) + 2*b^3*(e^(d*x + c) - e^(-d*x - c )) - 4*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*((e^(d*x + c) - e^(-d*x - c))^2 + 4 )))/d
Time = 3.54 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.18 \[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {2\,\left (a^3+a\,b^2\right )}{d\,{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^2\,b+b^3\right )}{d\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {2\,a}{d\,\left (a^2+b^2\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,\left (2\,a+b\,1{}\mathrm {i}\right )}{2\,\left (d\,a^2+2{}\mathrm {i}\,d\,a\,b-d\,b^2\right )}+\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (b+a\,2{}\mathrm {i}\right )}{2\,\left (1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b-1{}\mathrm {i}\,d\,b^2\right )}-\frac {a^3\,\ln \left (32\,a^7\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-b^7-6\,a^2\,b^5-9\,a^4\,b^3-16\,a^6\,b+b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+16\,a^6\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+12\,a^3\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+18\,a^5\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+6\,a^2\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+9\,a^4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+2\,a\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d\,a^4+2\,d\,a^2\,b^2+d\,b^4} \] Input:
int(tanh(c + d*x)^3/(a + b*sinh(c + d*x)),x)
Output:
((2*(a*b^2 + a^3))/(d*(a^2 + b^2)^2) - (exp(c + d*x)*(a^2*b + b^3))/(d*(a^ 2 + b^2)^2))/(exp(2*c + 2*d*x) + 1) - ((2*a)/(d*(a^2 + b^2)) - (2*b*exp(c + d*x))/(d*(a^2 + b^2)))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) + (lo g(exp(c + d*x)*1i + 1)*(2*a + b*1i))/(2*(a^2*d - b^2*d + a*b*d*2i)) + (log (exp(c + d*x) + 1i)*(a*2i + b))/(2*(a^2*d*1i - b^2*d*1i + 2*a*b*d)) - (a^3 *log(32*a^7*exp(d*x)*exp(c) - b^7 - 6*a^2*b^5 - 9*a^4*b^3 - 16*a^6*b + b^7 *exp(2*c)*exp(2*d*x) + 16*a^6*b*exp(2*c)*exp(2*d*x) + 12*a^3*b^4*exp(d*x)* exp(c) + 18*a^5*b^2*exp(d*x)*exp(c) + 6*a^2*b^5*exp(2*c)*exp(2*d*x) + 9*a^ 4*b^3*exp(2*c)*exp(2*d*x) + 2*a*b^6*exp(d*x)*exp(c)))/(a^4*d + b^4*d + 2*a ^2*b^2*d)
Time = 0.21 (sec) , antiderivative size = 519, normalized size of antiderivative = 4.32 \[ \int \frac {\tanh ^3(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {3 e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +e^{4 d x +4 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+6 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +2 e^{2 d x +2 c} \mathit {atan} \left (e^{d x +c}\right ) b^{3}+3 \mathit {atan} \left (e^{d x +c}\right ) a^{2} b +\mathit {atan} \left (e^{d x +c}\right ) b^{3}+e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a^{3}-e^{4 d x +4 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) a^{3}-e^{4 d x +4 c} a^{3}-e^{4 d x +4 c} a \,b^{2}-e^{3 d x +3 c} a^{2} b -e^{3 d x +3 c} b^{3}+2 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c}+1\right ) a^{3}-2 e^{2 d x +2 c} \mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) a^{3}+e^{d x +c} a^{2} b +e^{d x +c} b^{3}+\mathrm {log}\left (e^{2 d x +2 c}+1\right ) a^{3}-\mathrm {log}\left (e^{2 d x +2 c} b +2 e^{d x +c} a -b \right ) a^{3}-a^{3}-a \,b^{2}}{d \left (e^{4 d x +4 c} a^{4}+2 e^{4 d x +4 c} a^{2} b^{2}+e^{4 d x +4 c} b^{4}+2 e^{2 d x +2 c} a^{4}+4 e^{2 d x +2 c} a^{2} b^{2}+2 e^{2 d x +2 c} b^{4}+a^{4}+2 a^{2} b^{2}+b^{4}\right )} \] Input:
int(tanh(d*x+c)^3/(a+b*sinh(d*x+c)),x)
Output:
(3*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b + e**(4*c + 4*d*x)*atan(e**( c + d*x))*b**3 + 6*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**2*b + 2*e**(2*c + 2*d*x)*atan(e**(c + d*x))*b**3 + 3*atan(e**(c + d*x))*a**2*b + atan(e**( c + d*x))*b**3 + e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x) + 1)*a**3 - e**(4*c + 4*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**3 - e**(4*c + 4*d*x)*a**3 - e**(4*c + 4*d*x)*a*b**2 - e**(3*c + 3*d*x)*a**2*b - e**(3*c + 3*d*x)*b**3 + 2*e**(2*c + 2*d*x)*log(e**(2*c + 2*d*x) + 1)*a**3 - 2*e**( 2*c + 2*d*x)*log(e**(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**3 + e**(c + d*x)*a**2*b + e**(c + d*x)*b**3 + log(e**(2*c + 2*d*x) + 1)*a**3 - log(e* *(2*c + 2*d*x)*b + 2*e**(c + d*x)*a - b)*a**3 - a**3 - a*b**2)/(d*(e**(4*c + 4*d*x)*a**4 + 2*e**(4*c + 4*d*x)*a**2*b**2 + e**(4*c + 4*d*x)*b**4 + 2* e**(2*c + 2*d*x)*a**4 + 4*e**(2*c + 2*d*x)*a**2*b**2 + 2*e**(2*c + 2*d*x)* b**4 + a**4 + 2*a**2*b**2 + b**4))