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

Optimal result

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

[Out]

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))

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2869, 2814, 2739, 632, 212} \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {2 a \left (a^2+2 b^2\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right )^{3/2}}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {x}{b^2} \]

[In]

Int[Sinh[x]^2/(a + b*Sinh[x])^2,x]

[Out]

x/b^2 + (2*a*(a^2 + 2*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*(a^2 + b^2)^(3/2)) - (a^2*Cosh[x])
/(b*(a^2 + b^2)*(a + b*Sinh[x]))

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[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 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(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 2869

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] -
Dist[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {i \int \frac {-i a b+i \left (a^2+b^2\right ) \sinh (x)}{a+b \sinh (x)} \, dx}{b \left (a^2+b^2\right )} \\ & = \frac {x}{b^2}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (a \left (a^2+2 b^2\right )\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{b^2 \left (a^2+b^2\right )} \\ & = \frac {x}{b^2}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (2 a \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^2 \left (a^2+b^2\right )} \\ & = \frac {x}{b^2}-\frac {a^2 \cosh (x)}{b \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {\left (4 a \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b^2 \left (a^2+b^2\right )} \\ & = \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))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.04 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x+\frac {2 a \left (a^2+2 b^2\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}-\frac {a^2 b \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{b^2} \]

[In]

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

[Out]

(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

Maple [A] (verified)

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	\(\frac {x}{b^{2}}+\frac {2 a^{2} \left ({\mathrm e}^{x} a -b \right )}{b^{2} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}+\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{2}}+\frac {2 a \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}-\frac {a^{3} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} b^{2}}-\frac {2 a \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\)	\(286\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

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

Time = 0.29 (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 )} \]

[In]

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

[Out]

(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*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*co
sh(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. \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.80 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (a^{2} + 2 \, b^{2}\right )} a \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (a^{3} e^{\left (-x\right )} + a^{2} b\right )}}{a^{2} b^{3} + b^{5} + 2 \, {\left (a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} - {\left (a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )}} + \frac {x}{b^{2}} \]

[In]

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

[Out]

-(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)

none

Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.58 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=-\frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{{\left (a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a^{3} e^{x} - a^{2} b\right )}}{{\left (a^{2} b^{2} + b^{4}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac {x}{b^{2}} \]

[In]

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

[Out]

-(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 = 1.55 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.75 \[ \int \frac {\sinh ^2(x)}{(a+b \sinh (x))^2} \, dx=\frac {x}{b^2}-\frac {\frac {2\,a^2}{a^2\,b+b^3}-\frac {2\,a^3\,{\mathrm {e}}^x}{b\,\left (a^2\,b+b^3\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^3+2\,a\,b^2\right )}{b^3\,\left (a^2+b^2\right )}-\frac {2\,a\,\left (a^2+2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (a^2+2\,b^2\right )}{b^2\,{\left (a^2+b^2\right )}^{3/2}}+\frac {a\,\ln \left (\frac {2\,{\mathrm {e}}^x\,\left (a^3+2\,a\,b^2\right )}{b^3\,\left (a^2+b^2\right )}+\frac {2\,a\,\left (a^2+2\,b^2\right )\,\left (b-a\,{\mathrm {e}}^x\right )}{b^3\,{\left (a^2+b^2\right )}^{3/2}}\right )\,\left (a^2+2\,b^2\right )}{b^2\,{\left (a^2+b^2\right )}^{3/2}} \]

[In]

int(sinh(x)^2/(a + b*sinh(x))^2,x)

[Out]

x/b^2 - ((2*a^2)/(a^2*b + b^3) - (2*a^3*exp(x))/(b*(a^2*b + b^3)))/(2*a*exp(x) - b + b*exp(2*x)) - (a*log((2*e
xp(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))