\(\int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx\) [82]

Optimal result

Integrand size = 13, antiderivative size = 83 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x}{b^2}+\frac {2 a \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))} \] Output:

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))
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x+\frac {2 a \left (a^2+2 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{b^2} \] Input:

Integrate[Sinh[x]^2/(a + b*Sinh[x])^2,x]
 

Output:

(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
 

Rubi [A] (verified)

Time = 0.50 (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

\(\displaystyle -\frac {-\frac {2 a \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{b \sqrt {a^2+b^2}}-\frac {x \left (a^2+b^2\right )}{b}}{b \left (a^2+b^2\right )}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}\)

Input:

Int[Sinh[x]^2/(a + b*Sinh[x])^2,x]
 

Output:

-((-(((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]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 219

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])
 

rule 1083

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3139

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]
 

rule 3214

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]
 

rule 3269

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]
 

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.53

method	result	size
default	\(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{b^{2}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{b^{2}}+\frac {2 a \left (\frac {\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a^{2}+b^{2}}+\frac {a b}{a^{2}+b^{2}}}{\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (a^{2}+2 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{b^{2}}\)	\(127\)
risch	\(\frac {x}{b^{2}}+\frac {2 a^{2} \left ({\mathrm e}^{x} a -b \right )}{b^{2} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{2}}-\frac {2 a \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\)	\(286\)

Input:

int(sinh(x)^2/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)
 

Output:

-1/b^2*ln(tanh(1/2*x)-1)+1/b^2*ln(tanh(1/2*x)+1)+2*a/b^2*((b^2/(a^2+b^2)*t 
anh(1/2*x)+a*b/(a^2+b^2))/(tanh(1/2*x)^2*a-2*b*tanh(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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 521 vs. \(2 (79) = 158\).

Time = 0.09 (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 )} \] Input:

integrate(sinh(x)^2/(a+b*sinh(x))^2,x, algorithm="fricas")
 

Output:

(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))
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\text {Timed out} \] Input:

integrate(sinh(x)**2/(a+b*sinh(x))**2,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.80 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (a^{2} + 2 \, b^{2}\right )} a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (a^{3} e^{\left (-x\right )} + a^{2} b\right )}}{a^{2} b^{3} + b^{5} + 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} - {\left (a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{b^{2}} \] Input:

integrate(sinh(x)^2/(a+b*sinh(x))^2,x, algorithm="maxima")
 

Output:

-(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
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.58 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a^{3} e^{x} - a^{2} b\right )}}{{\left (a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac {x}{b^{2}} \] Input:

integrate(sinh(x)^2/(a+b*sinh(x))^2,x, algorithm="giac")
 

Output:

-(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
 

Mupad [B] (verification not implemented)

Time = 2.03 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.75 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x}{b^2}-\frac {\frac {2\,a^2}{a^2\,b+b^3}-\frac {2\,a^3\,{\mathrm {e}}^x}{b\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^3+2\,a\,b^2\right )}{b^3\,\left (a^2+b^2\right )}-\frac {2\,a\,\left (a^2+2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (a^2+2\,b^2\right )}{b^2\,{\left (a^2+b^2\right )}^{3/2}}+\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^3+2\,a\,b^2\right )}{b^3\,\left (a^2+b^2\right )}+\frac {2\,a\,\left (a^2+2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (a^2+2\,b^2\right )}{b^2\,{\left (a^2+b^2\right )}^{3/2}} \] Input:

int(sinh(x)^2/(a + b*sinh(x))^2,x)
 

Output:

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))
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 452, normalized size of antiderivative = 5.45 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {-2 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b i -4 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i -4 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} i -8 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i +2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b i +4 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i +e^{2 x} a^{4} b x -e^{2 x} a^{4} b +2 e^{2 x} a^{2} b^{3} x -e^{2 x} a^{2} b^{3}+e^{2 x} b^{5} x +2 e^{x} a^{5} x +4 e^{x} a^{3} b^{2} x +2 e^{x} a \,b^{4} x -a^{4} b x -a^{4} b -2 a^{2} b^{3} x -a^{2} b^{3}-b^{5} x}{b^{2} \left (e^{2 x} a^{4} b +2 e^{2 x} a^{2} b^{3}+e^{2 x} b^{5}+2 e^{x} a^{5}+4 e^{x} a^{3} b^{2}+2 e^{x} a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}\right )} \] Input:

int(sinh(x)^2/(a+b*sinh(x))^2,x)
 

Output:

( - 2*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))* 
a**3*b*i - 4*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + 
b**2))*a*b**3*i - 4*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 
 + b**2))*a**4*i - 8*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a** 
2 + b**2))*a**2*b**2*i + 2*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a* 
*2 + b**2))*a**3*b*i + 4*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 
 + b**2))*a*b**3*i + e**(2*x)*a**4*b*x - e**(2*x)*a**4*b + 2*e**(2*x)*a**2 
*b**3*x - e**(2*x)*a**2*b**3 + e**(2*x)*b**5*x + 2*e**x*a**5*x + 4*e**x*a* 
*3*b**2*x + 2*e**x*a*b**4*x - a**4*b*x - a**4*b - 2*a**2*b**3*x - a**2*b** 
3 - b**5*x)/(b**2*(e**(2*x)*a**4*b + 2*e**(2*x)*a**2*b**3 + e**(2*x)*b**5 
+ 2*e**x*a**5 + 4*e**x*a**3*b**2 + 2*e**x*a*b**4 - a**4*b - 2*a**2*b**3 - 
b**5))