\(\int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx\) [174]

Optimal result

Integrand size = 13, antiderivative size = 14 \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=-2 i \log (i+\sinh (x))+\sinh (x) \]

[Out]

-2*I*ln(I+sinh(x))+sinh(x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 45} \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=\sinh (x)-2 i \log (\sinh (x)+i) \]

[In]

Int[Cosh[x]^3/(I + Sinh[x])^2,x]

[Out]

(-2*I)*Log[I + Sinh[x]] + Sinh[x]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2746

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x]
&& IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] ||  !IntegerQ[m + 1/2])

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=-2 i \log (i+\sinh (x))+\sinh (x) \]

[In]

Integrate[Cosh[x]^3/(I + Sinh[x])^2,x]

[Out]

(-2*I)*Log[I + Sinh[x]] + Sinh[x]

Maple [A] (verified)

Time = 25.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.79

method	result	size
risch	\(2 i x +\frac {{\mathrm e}^{x}}{2}-\frac {{\mathrm e}^{-x}}{2}-4 i \ln \left ({\mathrm e}^{x}+i\right )\)	\(25\)
default	\(-4 i \ln \left (\tanh \left (\frac {x}{2}\right )+i\right )+2 i \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )-1}+2 i \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {1}{\tanh \left (\frac {x}{2}\right )+1}\)	\(53\)

[In]

int(cosh(x)^3/(I+sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

2*I*x+1/2*exp(x)-1/2*exp(-x)-4*I*ln(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. 26 vs. \(2 (10) = 20\).

Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {1}{2} \, {\left (4 i \, x e^{x} - 8 i \, e^{x} \log \left (e^{x} + i\right ) + e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )} \]

[In]

integrate(cosh(x)^3/(I+sinh(x))^2,x, algorithm="fricas")

[Out]

1/2*(4*I*x*e^x - 8*I*e^x*log(e^x + I) + e^(2*x) - 1)*e^(-x)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (12) = 24\).

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.86 \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=2 i x + \frac {e^{x}}{2} - 4 i \log {\left (e^{x} + i \right )} - \frac {e^{- x}}{2} \]

[In]

integrate(cosh(x)**3/(I+sinh(x))**2,x)

[Out]

2*I*x + exp(x)/2 - 4*I*log(exp(x) + I) - exp(-x)/2

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. 23 vs. \(2 (10) = 20\).

Time = 0.19 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.64 \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=-2 i \, x - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - 4 i \, \log \left (e^{\left (-x\right )} - i\right ) \]

[In]

integrate(cosh(x)^3/(I+sinh(x))^2,x, algorithm="maxima")

[Out]

-2*I*x - 1/2*e^(-x) + 1/2*e^x - 4*I*log(e^(-x) - I)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 21 vs. \(2 (10) = 20\).

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=2 i \, x - \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - 4 i \, \log \left (e^{x} + i\right ) \]

[In]

integrate(cosh(x)^3/(I+sinh(x))^2,x, algorithm="giac")

[Out]

2*I*x - 1/2*e^(-x) + 1/2*e^x - 4*I*log(e^x + I)

Mupad [B] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {\cosh ^3(x)}{(i+\sinh (x))^2} \, dx=\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}+x\,2{}\mathrm {i}-\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,4{}\mathrm {i} \]

[In]

int(cosh(x)^3/(sinh(x) + 1i)^2,x)

[Out]

x*2i - exp(-x)/2 - log(exp(x) + 1i)*4i + exp(x)/2