Integrand size = 13, antiderivative size = 115 \[ \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {2 a x}{b^3}-\frac {2 a^2 \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^3 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b^2 \left (a^2+b^2\right )}-\frac {a^2 \cosh (x) \sinh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))} \]
-2*a*x/b^3-2*a^2*(2*a^2+3*b^2)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/ b^3/(a^2+b^2)^(3/2)+(2*a^2+b^2)*cosh(x)/b^2/(a^2+b^2)-a^2*cosh(x)*sinh(x)/ b/(a^2+b^2)/(a+b*sinh(x))
Time = 0.43 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.83 \[ \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {-2 a x-\frac {2 a^2 \left (2 a^2+3 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+\cosh (x) \left (b+\frac {a^3 b}{\left (a^2+b^2\right ) (a+b \sinh (x))}\right )}{b^3} \]
(-2*a*x - (2*a^2*(2*a^2 + 3*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2] ])/(-a^2 - b^2)^(3/2) + Cosh[x]*(b + (a^3*b)/((a^2 + b^2)*(a + b*Sinh[x])) ))/b^3
Result contains complex when optimal does not.
Time = 0.67 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.27, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.923, Rules used = {3042, 26, 3271, 3042, 3502, 26, 3042, 3214, 3042, 3139, 1083, 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.
| \(\displaystyle \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \sin (i x)^3}{(a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\sin (i x)^3}{(a-i b \sin (i x))^2}dx\) |
| \(\Big \downarrow \) 3271 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \int \frac {a^2-b \sinh (x) a+\left (2 a^2+b^2\right ) \sinh ^2(x)}{a+b \sinh (x)}dx}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \int \frac {a^2+i b \sin (i x) a-\left (2 a^2+b^2\right ) \sin (i x)^2}{a-i b \sin (i x)}dx}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}+\frac {i \int -\frac {i \left (a^2 b-2 a \left (a^2+b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {\int \frac {a^2 b-2 a \left (a^2+b^2\right ) \sinh (x)}{a+b \sinh (x)}dx}{b}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}+\frac {\int \frac {b a^2+2 i \left (a^2+b^2\right ) \sin (i x) a}{a-i b \sin (i x)}dx}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {\frac {a^2 \left (2 a^2+3 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{b}-\frac {2 a x \left (a^2+b^2\right )}{b}}{b}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}+\frac {-\frac {2 a x \left (a^2+b^2\right )}{b}+\frac {a^2 \left (2 a^2+3 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{b}}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {\frac {2 a^2 \left (2 a^2+3 b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{b}-\frac {2 a x \left (a^2+b^2\right )}{b}}{b}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {-\frac {4 a^2 \left (2 a^2+3 b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b}-\frac {2 a x \left (a^2+b^2\right )}{b}}{b}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle i \left (\frac {i a^2 \sinh (x) \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \left (\frac {-\frac {2 a^2 \left (2 a^2+3 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}-\frac {2 a x \left (a^2+b^2\right )}{b}}{b}+\frac {\left (2 a^2+b^2\right ) \cosh (x)}{b}\right )}{b \left (a^2+b^2\right )}\right )\) |
I*(((-I)*(((-2*a*(a^2 + b^2)*x)/b - (2*a^2*(2*a^2 + 3*b^2)*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]))/b + ((2*a^2 + b^ 2)*Cosh[x])/b))/(b*(a^2 + b^2)) + (I*a^2*Cosh[x]*Sinh[x])/(b*(a^2 + b^2)*( a + b*Sinh[x])))
3.1.81.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Co s[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f* (n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin [e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^ 2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 + c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 + d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x] , x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 0.68 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.40
| method | result | size |
| default | \(-\frac {1}{b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {2 a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{3}}+\frac {1}{b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {2 a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{3}}-\frac {4 a^{2} \left (\frac {\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{2 a^{2}+2 b^{2}}+\frac {a b}{2 a^{2}+2 b^{2}}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (2 a^{2}+3 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{b^{3}}\) | \(161\) |
| risch | \(-\frac {2 a x}{b^{3}}+\frac {{\mathrm e}^{x}}{2 b^{2}}+\frac {{\mathrm e}^{-x}}{2 b^{2}}-\frac {2 a^{3} \left ({\mathrm e}^{x} a -b \right )}{b^{3} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {2 a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{3}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b}-\frac {2 a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{3}}-\frac {3 a^{2} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b}\) | \(315\) |
-1/b^2/(tanh(1/2*x)-1)+2/b^3*a*ln(tanh(1/2*x)-1)+1/b^2/(tanh(1/2*x)+1)-2/b ^3*a*ln(tanh(1/2*x)+1)-4/b^3*a^2*((1/2*b^2/(a^2+b^2)*tanh(1/2*x)+1/2*a*b/( a^2+b^2))/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)-1/2*(2*a^2+3*b^2)/(a^2+b^2)^ (3/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1053 vs. \(2 (111) = 222\).
Time = 0.28 (sec) , antiderivative size = 1053, normalized size of antiderivative = 9.16 \[ \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]
-1/2*(a^4*b^2 + 2*a^2*b^4 + b^6 - (a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^4 - (a^4*b^2 + 2*a^2*b^4 + b^6)*sinh(x)^4 - 2*(a^5*b + 2*a^3*b^3 + a*b^5 - 2*( a^5*b + 2*a^3*b^3 + a*b^5)*x)*cosh(x)^3 - 2*(a^5*b + 2*a^3*b^3 + a*b^5 - 2 *(a^5*b + 2*a^3*b^3 + a*b^5)*x + 2*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x))*si nh(x)^3 + 4*(a^6 + a^4*b^2 + 2*(a^6 + 2*a^4*b^2 + a^2*b^4)*x)*cosh(x)^2 + 2*(2*a^6 + 2*a^4*b^2 - 3*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^2 + 4*(a^6 + 2*a^4*b^2 + a^2*b^4)*x - 3*(a^5*b + 2*a^3*b^3 + a*b^5 - 2*(a^5*b + 2*a^3*b ^3 + a*b^5)*x)*cosh(x))*sinh(x)^2 - 2*((2*a^4*b + 3*a^2*b^3)*cosh(x)^3 + ( 2*a^4*b + 3*a^2*b^3)*sinh(x)^3 + 2*(2*a^5 + 3*a^3*b^2)*cosh(x)^2 + (4*a^5 + 6*a^3*b^2 + 3*(2*a^4*b + 3*a^2*b^3)*cosh(x))*sinh(x)^2 - (2*a^4*b + 3*a^ 2*b^3)*cosh(x) - (2*a^4*b + 3*a^2*b^3 - 3*(2*a^4*b + 3*a^2*b^3)*cosh(x)^2 - 4*(2*a^5 + 3*a^3*b^2)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh(x) ^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*s inh(x) - 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*s inh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 2*(3*a^5*b + 4* a^3*b^3 + a*b^5 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*x)*cosh(x) - 2*(3*a^5*b + 4*a^3*b^3 + a*b^5 + 2*(a^4*b^2 + 2*a^2*b^4 + b^6)*cosh(x)^3 + 3*(a^5*b + 2 *a^3*b^3 + a*b^5 - 2*(a^5*b + 2*a^3*b^3 + a*b^5)*x)*cosh(x)^2 + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*x - 4*(a^6 + a^4*b^2 + 2*(a^6 + 2*a^4*b^2 + a^2*b^4)*x )*cosh(x))*sinh(x))/((a^4*b^4 + 2*a^2*b^6 + b^8)*cosh(x)^3 + (a^4*b^4 +...
Timed out. \[ \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.81 \[ \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (2 \, a^{2} + 3 \, b^{2}\right )} a^{2} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b^{3} + b^{5}\right )} \sqrt {a^{2} + b^{2}}} + \frac {a^{2} b^{2} + b^{4} + 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} e^{\left (-x\right )} + {\left (4 \, a^{4} - a^{2} b^{2} - b^{4}\right )} e^{\left (-2 \, x\right )}}{2 \, {\left ({\left (a^{2} b^{4} + b^{6}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{3} b^{3} + a b^{5}\right )} e^{\left (-2 \, x\right )} - {\left (a^{2} b^{4} + b^{6}\right )} e^{\left (-3 \, x\right )}\right )}} - \frac {2 \, a x}{b^{3}} + \frac {e^{\left (-x\right )}}{2 \, b^{2}} \]
(2*a^2 + 3*b^2)*a^2*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + s qrt(a^2 + b^2)))/((a^2*b^3 + b^5)*sqrt(a^2 + b^2)) + 1/2*(a^2*b^2 + b^4 + 2*(3*a^3*b + a*b^3)*e^(-x) + (4*a^4 - a^2*b^2 - b^4)*e^(-2*x))/((a^2*b^4 + b^6)*e^(-x) + 2*(a^3*b^3 + a*b^5)*e^(-2*x) - (a^2*b^4 + b^6)*e^(-3*x)) - 2*a*x/b^3 + 1/2*e^(-x)/b^2
Time = 0.28 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.60 \[ \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\left (2 \, a^{4} + 3 \, a^{2} b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} b^{3} + b^{5}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, a x}{b^{3}} + \frac {e^{x}}{2 \, b^{2}} - \frac {{\left (a^{2} b^{2} + b^{4} + {\left (4 \, a^{4} - a^{2} b^{2} - b^{4}\right )} e^{\left (2 \, x\right )} - 2 \, {\left (3 \, a^{3} b + a b^{3}\right )} e^{x}\right )} e^{\left (-x\right )}}{2 \, {\left (a^{2} + b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )} b^{3}} \]
(2*a^4 + 3*a^2*b^2)*log(abs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/((a^2*b^3 + b^5)*sqrt(a^2 + b^2)) - 2*a*x/b^3 + 1/2*e^x/b^2 - 1/2*(a^2*b^2 + b^4 + (4*a^4 - a^2*b^2 - b^4)*e^(2*x) - 2* (3*a^3*b + a*b^3)*e^x)*e^(-x)/((a^2 + b^2)*(b*e^(2*x) + 2*a*e^x - b)*b^3)
Time = 1.54 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.38 \[ \int \frac {\sinh ^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {{\mathrm {e}}^{-x}}{2\,b^2}+\frac {\frac {2\,a^3}{b\,\left (a^2\,b+b^3\right )}-\frac {2\,a^4\,{\mathrm {e}}^x}{b^2\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}+\frac {{\mathrm {e}}^x}{2\,b^2}-\frac {2\,a\,x}{b^3}-\frac {a^2\,\ln \left (-\frac {2\,{\mathrm {e}}^x\,\left (2\,a^4+3\,a^2\,b^2\right )}{a^2\,b^4+b^6}-\frac {2\,a^2\,\left (b-a\,{\mathrm {e}}^x\right )\,\left (2\,a^2+3\,b^2\right )}{b^4\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (2\,a^2+3\,b^2\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}+\frac {a^2\,\ln \left (\frac {2\,a^2\,\left (b-a\,{\mathrm {e}}^x\right )\,\left (2\,a^2+3\,b^2\right )}{b^4\,{\left (a^2+b^2\right )}^{3/2}}-\frac {2\,{\mathrm {e}}^x\,\left (2\,a^4+3\,a^2\,b^2\right )}{a^2\,b^4+b^6}\right )\,\left (2\,a^2+3\,b^2\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}} \]
exp(-x)/(2*b^2) + ((2*a^3)/(b*(a^2*b + b^3)) - (2*a^4*exp(x))/(b^2*(a^2*b + b^3)))/(2*a*exp(x) - b + b*exp(2*x)) + exp(x)/(2*b^2) - (2*a*x)/b^3 - (a ^2*log(- (2*exp(x)*(2*a^4 + 3*a^2*b^2))/(b^6 + a^2*b^4) - (2*a^2*(b - a*ex p(x))*(2*a^2 + 3*b^2))/(b^4*(a^2 + b^2)^(3/2)))*(2*a^2 + 3*b^2))/(b^3*(a^2 + b^2)^(3/2)) + (a^2*log((2*a^2*(b - a*exp(x))*(2*a^2 + 3*b^2))/(b^4*(a^2 + b^2)^(3/2)) - (2*exp(x)*(2*a^4 + 3*a^2*b^2))/(b^6 + a^2*b^4))*(2*a^2 + 3*b^2))/(b^3*(a^2 + b^2)^(3/2))