Integrand size = 13, antiderivative size = 83 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x}{b^2}+\frac {2 a \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))} \]
x/b^2+2*a*(a^2+2*b^2)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/b^2/(a^2+ b^2)^(3/2)-a^2*cosh(x)/b/(a^2+b^2)/(a+b*sinh(x))
Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x+\frac {2 a \left (a^2+2 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{b^2} \]
(x + (2*a*(a^2 + 2*b^2)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/(-a^2 - b^2)^(3/2) - (a^2*b*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])))/b^2
Time = 0.48 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.34, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 25, 3269, 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.
\(\displaystyle \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin (i x)^2}{(a-i b \sin (i x))^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\sin (i x)^2}{(a-i b \sin (i x))^2}dx\) |
\(\Big \downarrow \) 3269 |
\(\displaystyle -\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \int -\frac {i \left (a b-\left (a^2+b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\int \frac {a b-\left (a^2+b^2\right ) \sinh (x)}{a+b \sinh (x)}dx}{b \left (a^2+b^2\right )}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\int \frac {a b+i \left (a^2+b^2\right ) \sin (i x)}{a-i b \sin (i x)}dx}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 3214 |
\(\displaystyle -\frac {\frac {a \left (a^2+2 b^2\right ) \int \frac {1}{a+b \sinh (x)}dx}{b}-\frac {x \left (a^2+b^2\right )}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {-\frac {x \left (a^2+b^2\right )}{b}+\frac {a \left (a^2+2 b^2\right ) \int \frac {1}{a-i b \sin (i x)}dx}{b}}{b \left (a^2+b^2\right )}\) |
\(\Big \downarrow \) 3139 |
\(\displaystyle -\frac {\frac {2 a \left (a^2+2 b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{b}-\frac {x \left (a^2+b^2\right )}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle -\frac {-\frac {4 a \left (a^2+2 b^2\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b}-\frac {x \left (a^2+b^2\right )}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {-\frac {2 a \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}-\frac {x \left (a^2+b^2\right )}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}\) |
-((-(((a^2 + b^2)*x)/b) - (2*a*(a^2 + 2*b^2)*ArcTanh[(2*b - 2*a*Tanh[x/2]) /(2*Sqrt[a^2 + b^2])])/(b*Sqrt[a^2 + b^2]))/(b*(a^2 + b^2))) - (a^2*Cosh[x ])/(b*(a^2 + b^2)*(a + b*Sinh[x]))
3.1.82.3.1 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_)])^2, x_Symbol] :> Simp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 - b^2))), x] - Simp[ 1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(m + 1) *(2*b*c*d - a*(c^2 + d^2)) + (a^2*d^2 - 2*a*b*c*d*(m + 2) + b^2*(d^2*(m + 1 ) + c^2*(m + 2)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
Time = 0.53 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.53
method | result | size |
default | \(\frac {2 a \left (\frac {\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a^{2}+b^{2}}+\frac {a b}{a^{2}+b^{2}}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{2}+2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{b^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}}\) | \(127\) |
risch | \(\frac {x}{b^{2}}+\frac {2 a^{2} \left ({\mathrm e}^{x} a -b \right )}{b^{2} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{2}}-\frac {2 a \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\) | \(286\) |
2*a/b^2*((b^2/(a^2+b^2)*tanh(1/2*x)+a*b/(a^2+b^2))/(tanh(1/2*x)^2*a-2*b*ta nh(1/2*x)-a)-(a^2+2*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)))-1/b^2*ln(tanh(1/2*x)-1)+1/b^2*ln(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 521, normalized size of antiderivative = 6.28 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {2 \, a^{4} b + 2 \, a^{2} b^{3} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} x \cosh \left (x\right )^{2} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} x \sinh \left (x\right )^{2} + {\left (a^{3} b + 2 \, a b^{3} - {\left (a^{3} b + 2 \, a b^{3}\right )} \cosh \left (x\right )^{2} - {\left (a^{3} b + 2 \, a b^{3}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{4} + 2 \, a^{2} b^{2}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + {\left (a^{3} b + 2 \, a b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {a^{2} + b^{2}} \log \left (\frac {b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right ) + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (x\right ) + b \sinh \left (x\right ) + a\right )}}{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + a\right )} \sinh \left (x\right ) - b}\right ) + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} x - 2 \, {\left (a^{5} + a^{3} b^{2} + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} x\right )} \cosh \left (x\right ) - 2 \, {\left (a^{5} + a^{3} b^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} x \cosh \left (x\right ) + {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} x\right )} \sinh \left (x\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} - {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )^{2} - {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \sinh \left (x\right )^{2} - 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \cosh \left (x\right ) - 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]
(2*a^4*b + 2*a^2*b^3 - (a^4*b + 2*a^2*b^3 + b^5)*x*cosh(x)^2 - (a^4*b + 2* a^2*b^3 + b^5)*x*sinh(x)^2 + (a^3*b + 2*a*b^3 - (a^3*b + 2*a*b^3)*cosh(x)^ 2 - (a^3*b + 2*a*b^3)*sinh(x)^2 - 2*(a^4 + 2*a^2*b^2)*cosh(x) - 2*(a^4 + 2 *a^2*b^2 + (a^3*b + 2*a*b^3)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*co sh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a *b)*sinh(x) + 2*sqrt(a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) + (a^4*b + 2 *a^2*b^3 + b^5)*x - 2*(a^5 + a^3*b^2 + (a^5 + 2*a^3*b^2 + a*b^4)*x)*cosh(x ) - 2*(a^5 + a^3*b^2 + (a^4*b + 2*a^2*b^3 + b^5)*x*cosh(x) + (a^5 + 2*a^3* b^2 + a*b^4)*x)*sinh(x))/(a^4*b^3 + 2*a^2*b^5 + b^7 - (a^4*b^3 + 2*a^2*b^5 + b^7)*cosh(x)^2 - (a^4*b^3 + 2*a^2*b^5 + b^7)*sinh(x)^2 - 2*(a^5*b^2 + 2 *a^3*b^4 + a*b^6)*cosh(x) - 2*(a^5*b^2 + 2*a^3*b^4 + a*b^6 + (a^4*b^3 + 2* a^2*b^5 + b^7)*cosh(x))*sinh(x))
Timed out. \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.80 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (a^{2} + 2 \, b^{2}\right )} a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (a^{3} e^{\left (-x\right )} + a^{2} b\right )}}{a^{2} b^{3} + b^{5} + 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} - {\left (a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{b^{2}} \]
-(a^2 + 2*b^2)*a*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt (a^2 + b^2)))/((a^2*b^2 + b^4)*sqrt(a^2 + b^2)) - 2*(a^3*e^(-x) + a^2*b)/( a^2*b^3 + b^5 + 2*(a^3*b^2 + a*b^4)*e^(-x) - (a^2*b^3 + b^5)*e^(-2*x)) + x /b^2
Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.58 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a^{3} e^{x} - a^{2} b\right )}}{{\left (a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac {x}{b^{2}} \]
-(a^3 + 2*a*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^2 + b^4)*sqrt(a^2 + b^2)) + 2*(a^3*e^x - a^2*b)/((a^2*b^2 + b^4)*(b*e^(2*x) + 2*a*e^x - b)) + x/b^2
Time = 1.55 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.75 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x}{b^2}-\frac {\frac {2\,a^2}{a^2\,b+b^3}-\frac {2\,a^3\,{\mathrm {e}}^x}{b\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^3+2\,a\,b^2\right )}{b^3\,\left (a^2+b^2\right )}-\frac {2\,a\,\left (a^2+2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (a^2+2\,b^2\right )}{b^2\,{\left (a^2+b^2\right )}^{3/2}}+\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^3+2\,a\,b^2\right )}{b^3\,\left (a^2+b^2\right )}+\frac {2\,a\,\left (a^2+2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (a^2+2\,b^2\right )}{b^2\,{\left (a^2+b^2\right )}^{3/2}} \]
x/b^2 - ((2*a^2)/(a^2*b + b^3) - (2*a^3*exp(x))/(b*(a^2*b + b^3)))/(2*a*ex p(x) - b + b*exp(2*x)) - (a*log((2*exp(x)*(2*a*b^2 + a^3))/(b^3*(a^2 + b^2 )) - (2*a*(a^2 + 2*b^2)*(b - a*exp(x)))/(b^3*(a^2 + b^2)^(3/2)))*(a^2 + 2* b^2))/(b^2*(a^2 + b^2)^(3/2)) + (a*log((2*exp(x)*(2*a*b^2 + a^3))/(b^3*(a^ 2 + b^2)) + (2*a*(a^2 + 2*b^2)*(b - a*exp(x)))/(b^3*(a^2 + b^2)^(3/2)))*(a ^2 + 2*b^2))/(b^2*(a^2 + b^2)^(3/2))