\(\int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx\) [186]

Optimal result

Integrand size = 13, antiderivative size = 16 \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=-\frac {i}{2 (1-i \sinh (x))^2} \]

[Out]

-1/2*I/(1-I*sinh(x))^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2746, 32} \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=-\frac {i}{2 (1-i \sinh (x))^2} \]

[In]

Int[Cosh[x]/(1 - I*Sinh[x])^3,x]

[Out]

(-1/2*I)/(1 - I*Sinh[x])^2

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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}& = i \text {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,-i \sinh (x)\right ) \\ & = -\frac {i}{2 (1-i \sinh (x))^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=\frac {i}{2 (i+\sinh (x))^2} \]

[In]

Integrate[Cosh[x]/(1 - I*Sinh[x])^3,x]

[Out]

(I/2)/(I + Sinh[x])^2

Maple [A] (verified)

Time = 58.09 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(-\frac {i}{2 \left (1-i \sinh \left (x \right )\right )^{2}}\)	\(13\)
default	\(-\frac {i}{2 \left (1-i \sinh \left (x \right )\right )^{2}}\)	\(13\)
risch	\(\frac {2 i {\mathrm e}^{2 x}}{\left ({\mathrm e}^{x}+i\right )^{4}}\)	\(15\)

[In]

int(cosh(x)/(1-I*sinh(x))^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*I/(1-I*sinh(x))^2

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

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.88 \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=\frac {2 i \, e^{\left (2 \, x\right )}}{e^{\left (4 \, x\right )} + 4 i \, e^{\left (3 \, x\right )} - 6 \, e^{\left (2 \, x\right )} - 4 i \, e^{x} + 1} \]

[In]

integrate(cosh(x)/(1-I*sinh(x))^3,x, algorithm="fricas")

[Out]

2*I*e^(2*x)/(e^(4*x) + 4*I*e^(3*x) - 6*e^(2*x) - 4*I*e^x + 1)

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. 36 vs. \(2 (12) = 24\).

Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=\frac {2 i e^{2 x}}{e^{4 x} + 4 i e^{3 x} - 6 e^{2 x} - 4 i e^{x} + 1} \]

[In]

integrate(cosh(x)/(1-I*sinh(x))**3,x)

[Out]

2*I*exp(2*x)/(exp(4*x) + 4*I*exp(3*x) - 6*exp(2*x) - 4*I*exp(x) + 1)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=-\frac {i}{2 \, {\left (-i \, \sinh \left (x\right ) + 1\right )}^{2}} \]

[In]

integrate(cosh(x)/(1-I*sinh(x))^3,x, algorithm="maxima")

[Out]

-1/2*I/(-I*sinh(x) + 1)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=\frac {2 i \, e^{\left (2 \, x\right )}}{{\left (e^{x} + i\right )}^{4}} \]

[In]

integrate(cosh(x)/(1-I*sinh(x))^3,x, algorithm="giac")

[Out]

2*I*e^(2*x)/(e^x + I)^4

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh (x)}{(1-i \sinh (x))^3} \, dx=\frac {{\mathrm {e}}^{2\,x}\,2{}\mathrm {i}}{{\left (-1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )}^4} \]

[In]

int(-cosh(x)/(sinh(x)*1i - 1)^3,x)

[Out]

(exp(2*x)*2i)/(exp(x)*1i - 1)^4