Integrand size = 13, antiderivative size = 137 \[ \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx=-\frac {a^3 b \arctan \left (\frac {b \cosh (x)+a \sinh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2}}-\frac {a b^2 \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {a \cosh (x)}{a^2-b^2}+\frac {a \cosh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {a^2 b \sinh (x)}{\left (a^2-b^2\right )^2}-\frac {b \sinh ^3(x)}{3 \left (a^2-b^2\right )} \]
-a^3*b*arctan((b*cosh(x)+a*sinh(x))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)-a*b^2 *cosh(x)/(a^2-b^2)^2-a*cosh(x)/(a^2-b^2)+1/3*a*cosh(x)^3/(a^2-b^2)+a^2*b*s inh(x)/(a^2-b^2)^2-1/3*b*sinh(x)^3/(a^2-b^2)
Time = 1.02 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.31 \[ \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx=\frac {-3 a \sqrt {a-b} \sqrt {a+b} \left (3 a^2+b^2\right ) \cosh (x)+a \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right ) \cosh (3 x)+b \left (-24 a^3 \arctan \left (\frac {b+a \tanh \left (\frac {x}{2}\right )}{\sqrt {a-b} \sqrt {a+b}}\right )+3 \sqrt {a-b} \sqrt {a+b} \left (5 a^2-b^2\right ) \sinh (x)-\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right ) \sinh (3 x)\right )}{12 (a-b)^{5/2} (a+b)^{5/2}} \]
(-3*a*Sqrt[a - b]*Sqrt[a + b]*(3*a^2 + b^2)*Cosh[x] + a*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)*Cosh[3*x] + b*(-24*a^3*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])] + 3*Sqrt[a - b]*Sqrt[a + b]*(5*a^2 - b^2)*Sinh[x] - Sqr t[a - b]*Sqrt[a + b]*(a^2 - b^2)*Sinh[3*x]))/(12*(a - b)^(5/2)*(a + b)^(5/ 2))
Result contains complex when optimal does not.
Time = 0.87 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.12, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.769, Rules used = {3042, 26, 4001, 26, 3042, 26, 3588, 25, 26, 3042, 25, 26, 3044, 15, 3113, 2009, 3578, 26, 3042, 26, 3118, 3553, 219}
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.
| \(\displaystyle \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3}{a-i b \tan (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i x)^3}{a-i b \tan (i x)}dx\) |
| \(\Big \downarrow \) 4001 |
| \(\displaystyle i \int -\frac {i \cosh (x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\sinh ^3(x) \cosh (x)}{a \cosh (x)+b \sinh (x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3 \cos (i x)}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\cos (i x) \sin (i x)^3}{a \cos (i x)-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3588 |
| \(\displaystyle i \left (\frac {a \int -i \sinh ^3(x)dx}{a^2-b^2}-\frac {i b \int -\cosh (x) \sinh ^2(x)dx}{a^2-b^2}+\frac {i a b \int -\frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (\frac {a \int -i \sinh ^3(x)dx}{a^2-b^2}+\frac {i b \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (-\frac {i a \int \sinh ^3(x)dx}{a^2-b^2}+\frac {i b \int \cosh (x) \sinh ^2(x)dx}{a^2-b^2}-\frac {i a b \int \frac {\sinh ^2(x)}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (-\frac {i a \int i \sin (i x)^3dx}{a^2-b^2}+\frac {i b \int -\cos (i x) \sin (i x)^2dx}{a^2-b^2}-\frac {i a b \int -\frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle i \left (-\frac {i a \int i \sin (i x)^3dx}{a^2-b^2}-\frac {i b \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}-\frac {i b \int \cos (i x) \sin (i x)^2dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3044 |
| \(\displaystyle i \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}-\frac {b \int -\sinh ^2(x)d(i \sinh (x))}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle i \left (\frac {a \int \sin (i x)^3dx}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle i \left (\frac {i a \int \left (1-\cosh ^2(x)\right )d\cosh (x)}{a^2-b^2}+\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle i \left (\frac {i a b \int \frac {\sin (i x)^2}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3578 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {i b \int i \sinh (x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {b \int \sinh (x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cosh (x)+b \sinh (x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {b \int -i \sin (i x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a b \left (-\frac {i b \int \sin (i x)dx}{a^2-b^2}+\frac {a^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3118 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {a^2 \int \frac {1}{a \cos (i x)-i b \sin (i x)}dx}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 3553 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a^2 \int \frac {1}{a^2-b^2-(-i b \cosh (x)-i a \sinh (x))^2}d(-i b \cosh (x)-i a \sinh (x))}{a^2-b^2}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (\frac {i a b \left (\frac {i a^2 \text {arctanh}\left (\frac {-i a \sinh (x)-i b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2}}-\frac {a \sinh (x)}{a^2-b^2}+\frac {b \cosh (x)}{a^2-b^2}\right )}{a^2-b^2}+\frac {i b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac {i a \left (\cosh (x)-\frac {\cosh ^3(x)}{3}\right )}{a^2-b^2}\right )\) |
I*((I*a*(Cosh[x] - Cosh[x]^3/3))/(a^2 - b^2) + ((I/3)*b*Sinh[x]^3)/(a^2 - b^2) + (I*a*b*((I*a^2*ArcTanh[((-I)*b*Cosh[x] - I*a*Sinh[x])/Sqrt[a^2 - b^ 2]])/(a^2 - b^2)^(3/2) + (b*Cosh[x])/(a^2 - b^2) - (a*Sinh[x])/(a^2 - b^2) ))/(a^2 - b^2))
3.1.81.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x _Symbol] :> Simp[-d^(-1) Subst[Int[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]
Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin [(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[(-a)*(Sin[c + d*x]^(m - 1)/(d*(a^2 + b^2)*(m - 1))), x] + (Simp[a^2/(a^2 + b^2) Int[Sin[c + d*x]^(m - 2)/(a *Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Simp[b/(a^2 + b^2) Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ [m, 1]
Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_. ) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp[b /(a^2 + b^2) Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Simp[a/(a ^2 + b^2) Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Simp[a*(b/(a^ 2 + b^2)) Int[Cos[c + d*x]^(m - 1)*(Sin[c + d*x]^(n - 1)/(a*Cos[c + d*x] + b*Sin[c + d*x])), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n _.), x_Symbol] :> Int[Sin[e + f*x]^m*((a*Cos[e + f*x] + b*Sin[e + f*x])^n/C os[e + f*x]^n), x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && ILtQ [n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))
Time = 2.07 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.21
| method | result | size |
| default | \(-\frac {2 a^{3} b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{2} \left (a +b \right )^{2} \sqrt {a^{2}-b^{2}}}-\frac {8}{\left (16 a -16 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {16}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3} \left (16 a -16 b \right )}-\frac {a}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {16}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {a}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(166\) |
| risch | \(\frac {{\mathrm e}^{3 x}}{24 a +24 b}-\frac {3 \,{\mathrm e}^{x} a}{8 \left (a +b \right )^{2}}-\frac {{\mathrm e}^{x} b}{8 \left (a +b \right )^{2}}-\frac {3 \,{\mathrm e}^{-x} a}{8 \left (a -b \right )^{2}}+\frac {{\mathrm e}^{-x} b}{8 \left (a -b \right )^{2}}+\frac {{\mathrm e}^{-3 x}}{24 a -24 b}-\frac {b \,a^{3} \ln \left ({\mathrm e}^{x}+\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {b \,a^{3} \ln \left ({\mathrm e}^{x}-\frac {a -b}{\sqrt {-a^{2}+b^{2}}}\right )}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(174\) |
-2*a^3*b/(a-b)^2/(a+b)^2/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/ (a^2-b^2)^(1/2))-8/(16*a-16*b)/(tanh(1/2*x)+1)^2+16/3/(tanh(1/2*x)+1)^3/(1 6*a-16*b)-1/2*a/(a-b)^2/(tanh(1/2*x)+1)-16/3/(tanh(1/2*x)-1)^3/(16*a+16*b) -8/(16*a+16*b)/(tanh(1/2*x)-1)^2+1/2*a/(a+b)^2/(tanh(1/2*x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (129) = 258\).
Time = 0.29 (sec) , antiderivative size = 1861, normalized size of antiderivative = 13.58 \[ \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \]
[1/24*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^6 + 6*( a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)*sinh(x)^5 + (a^ 5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*sinh(x)^6 + a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 - 3*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6* a^2*b^3 - a*b^4 - b^5)*cosh(x)^4 - 3*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2* b^3 - a*b^4 - b^5 - 5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)* cosh(x)^2)*sinh(x)^4 + 4*(5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^3 - 3*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2*b^3 - a*b^4 - b^5 )*cosh(x))*sinh(x)^3 - 3*(3*a^5 + 5*a^4*b - 2*a^3*b^2 - 6*a^2*b^3 - a*b^4 + b^5)*cosh(x)^2 - 3*(3*a^5 + 5*a^4*b - 2*a^3*b^2 - 6*a^2*b^3 - a*b^4 + b^ 5 - 5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^4 + 6*(3 *a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2*b^3 - a*b^4 - b^5)*cosh(x)^2)*sinh(x)^2 - 24*(a^3*b*cosh(x)^3 + 3*a^3*b*cosh(x)^2*sinh(x) + 3*a^3*b*cosh(x)*sinh( x)^2 + a^3*b*sinh(x)^3)*sqrt(-a^2 + b^2)*log(((a + b)*cosh(x)^2 + 2*(a + b )*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh (x)) - a + b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sin h(x)^2 + a - b)) + 6*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)* cosh(x)^5 - 2*(3*a^5 - 5*a^4*b - 2*a^3*b^2 + 6*a^2*b^3 - a*b^4 - b^5)*cosh (x)^3 - (3*a^5 + 5*a^4*b - 2*a^3*b^2 - 6*a^2*b^3 - a*b^4 + b^5)*cosh(x))*s inh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 3*(a^6 - 3*a^4...
\[ \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx=\int \frac {\sinh ^{3}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \]
Exception generated. \[ \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Time = 0.27 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.19 \[ \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx=-\frac {2 \, a^{3} b \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {{\left (9 \, a e^{\left (2 \, x\right )} - 3 \, b e^{\left (2 \, x\right )} - a + b\right )} e^{\left (-3 \, x\right )}}{24 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} + \frac {a^{2} e^{\left (3 \, x\right )} + 2 \, a b e^{\left (3 \, x\right )} + b^{2} e^{\left (3 \, x\right )} - 9 \, a^{2} e^{x} - 12 \, a b e^{x} - 3 \, b^{2} e^{x}}{24 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} \]
-2*a^3*b*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/((a^4 - 2*a^2*b^2 + b^4)* sqrt(a^2 - b^2)) - 1/24*(9*a*e^(2*x) - 3*b*e^(2*x) - a + b)*e^(-3*x)/(a^2 - 2*a*b + b^2) + 1/24*(a^2*e^(3*x) + 2*a*b*e^(3*x) + b^2*e^(3*x) - 9*a^2*e ^x - 12*a*b*e^x - 3*b^2*e^x)/(a^3 + 3*a^2*b + 3*a*b^2 + b^3)
Time = 1.93 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.91 \[ \int \frac {\sinh ^3(x)}{a+b \tanh (x)} \, dx=\frac {{\mathrm {e}}^{-3\,x}}{24\,a-24\,b}+\frac {{\mathrm {e}}^{3\,x}}{24\,a+24\,b}-\frac {{\mathrm {e}}^x\,\left (3\,a+b\right )}{8\,{\left (a+b\right )}^2}-\frac {{\mathrm {e}}^{-x}\,\left (3\,a-b\right )}{8\,{\left (a-b\right )}^2}-\frac {2\,\mathrm {atan}\left (\frac {a^3\,b\,{\mathrm {e}}^x\,\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}}{a^5\,\sqrt {a^6\,b^2}-b^5\,\sqrt {a^6\,b^2}+2\,a^2\,b^3\,\sqrt {a^6\,b^2}-2\,a^3\,b^2\,\sqrt {a^6\,b^2}+a\,b^4\,\sqrt {a^6\,b^2}-a^4\,b\,\sqrt {a^6\,b^2}}\right )\,\sqrt {a^6\,b^2}}{\sqrt {a^{10}-5\,a^8\,b^2+10\,a^6\,b^4-10\,a^4\,b^6+5\,a^2\,b^8-b^{10}}} \]
exp(-3*x)/(24*a - 24*b) + exp(3*x)/(24*a + 24*b) - (exp(x)*(3*a + b))/(8*( a + b)^2) - (exp(-x)*(3*a - b))/(8*(a - b)^2) - (2*atan((a^3*b*exp(x)*(a^1 0 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2)^(1/2))/(a^5*(a ^6*b^2)^(1/2) - b^5*(a^6*b^2)^(1/2) + 2*a^2*b^3*(a^6*b^2)^(1/2) - 2*a^3*b^ 2*(a^6*b^2)^(1/2) + a*b^4*(a^6*b^2)^(1/2) - a^4*b*(a^6*b^2)^(1/2)))*(a^6*b ^2)^(1/2))/(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2) ^(1/2)