\(\int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx\) [42]

Optimal result

Integrand size = 13, antiderivative size = 22 \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=-i x+\cosh (x)+\frac {i \cosh (x)}{i+\sinh (x)} \]

[Out]

-I*x+cosh(x)+I*cosh(x)/(I+sinh(x))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2825, 2814, 2727} \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=-i x+\cosh (x)+\frac {i \cosh (x)}{\sinh (x)+i} \]

[In]

Int[Sinh[x]^2/(I + Sinh[x]),x]

[Out]

(-I)*x + Cosh[x] + (I*Cosh[x])/(I + Sinh[x])

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[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 2825

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b^2
)*(Cos[e + f*x]/(d*f)), x] + Dist[1/d, Int[Simp[a^2*d - b*(b*c - 2*a*d)*Sin[e + f*x], x]/(c + d*Sin[e + f*x]),
x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps \begin{align*} \text {integral}& = \cosh (x)-i \int \frac {\sinh (x)}{i+\sinh (x)} \, dx \\ & = -i x+\cosh (x)-\int \frac {1}{i+\sinh (x)} \, dx \\ & = -i x+\cosh (x)+\frac {i \cosh (x)}{i+\sinh (x)} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(79\) vs. \(2(22)=44\).

Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.59 \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=\frac {\cosh (x) \left (2 i+\frac {2 i \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right )}{\sqrt {\cosh ^2(x)}}+\sinh (x)+\frac {2 \arcsin \left (\frac {\sqrt {1-i \sinh (x)}}{\sqrt {2}}\right ) \sinh (x)}{\sqrt {\cosh ^2(x)}}\right )}{i+\sinh (x)} \]

[In]

Integrate[Sinh[x]^2/(I + Sinh[x]),x]

[Out]

(Cosh[x]*(2*I + ((2*I)*ArcSin[Sqrt[1 - I*Sinh[x]]/Sqrt[2]])/Sqrt[Cosh[x]^2] + Sinh[x] + (2*ArcSin[Sqrt[1 - I*S
inh[x]]/Sqrt[2]]*Sinh[x])/Sqrt[Cosh[x]^2]))/(I + Sinh[x])

Maple [A] (verified)

Time = 1.78 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14

method	result	size
risch	\(-i x +\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}+\frac {2}{{\mathrm e}^{x}+i}\)	\(25\)
parallelrisch	\(\frac {\cosh \left (2 x \right )+i \sinh \left (2 x \right )+\left (6 i+2 x \right ) \sinh \left (x \right )-2 i \cosh \left (x \right ) x +2 i x -1}{2 i \sinh \left (x \right )+2 \cosh \left (x \right )-2}\)	\(47\)
default	\(\frac {2 i}{\tanh \left (\frac {x}{2}\right )+i}+i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}-i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )+\frac {1}{\tanh \left (\frac {x}{2}\right )+1}\)	\(52\)

[In]

int(sinh(x)^2/(I+sinh(x)),x,method=_RETURNVERBOSE)

[Out]

-I*x+1/2*exp(x)+1/2*exp(-x)+2/(exp(x)+I)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (16) = 32\).

Time = 0.25 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=\frac {{\left (-2 i \, x + i\right )} e^{\left (2 \, x\right )} + {\left (2 \, x + 5\right )} e^{x} + e^{\left (3 \, x\right )} + i}{2 \, {\left (e^{\left (2 \, x\right )} + i \, e^{x}\right )}} \]

[In]

integrate(sinh(x)^2/(I+sinh(x)),x, algorithm="fricas")

[Out]

1/2*((-2*I*x + I)*e^(2*x) + (2*x + 5)*e^x + e^(3*x) + I)/(e^(2*x) + I*e^x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=- i x + \frac {e^{x}}{2} + \frac {e^{- x}}{2} + \frac {2}{e^{x} + i} \]

[In]

integrate(sinh(x)**2/(I+sinh(x)),x)

[Out]

-I*x + exp(x)/2 + exp(-x)/2 + 2/(exp(x) + I)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (16) = 32\).

Time = 0.19 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=-i \, x + \frac {5 \, e^{\left (-x\right )} - i}{2 \, {\left (-i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )}\right )}} + \frac {1}{2} \, e^{\left (-x\right )} \]

[In]

integrate(sinh(x)^2/(I+sinh(x)),x, algorithm="maxima")

[Out]

-I*x + 1/2*(5*e^(-x) - I)/(-I*e^(-x) + e^(-2*x)) + 1/2*e^(-x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.18 \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=-i \, x + \frac {{\left (5 \, e^{x} + i\right )} e^{\left (-x\right )}}{2 \, {\left (e^{x} + i\right )}} + \frac {1}{2} \, e^{x} \]

[In]

integrate(sinh(x)^2/(I+sinh(x)),x, algorithm="giac")

[Out]

-I*x + 1/2*(5*e^x + I)*e^(-x)/(e^x + I) + 1/2*e^x

Mupad [B] (verification not implemented)

Time = 1.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sinh ^2(x)}{i+\sinh (x)} \, dx=\frac {{\mathrm {e}}^{-x}}{2}-x\,1{}\mathrm {i}+\frac {{\mathrm {e}}^x}{2}+\frac {2}{{\mathrm {e}}^x+1{}\mathrm {i}} \]

[In]

int(sinh(x)^2/(sinh(x) + 1i),x)

[Out]

exp(-x)/2 - x*1i + exp(x)/2 + 2/(exp(x) + 1i)