\(\int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx\) [85]

Optimal result

Integrand size = 13, antiderivative size = 19 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]

[Out]

1/2*sinh(x)^2-1/3*I*sinh(x)^3

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3957, 2914, 2644, 30} \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]

[In]

Int[Cosh[x]^3/(I + Csch[x]),x]

[Out]

Sinh[x]^2/2 - (I/3)*Sinh[x]^3

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2644

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2914

Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]
), x_Symbol] :> Dist[1/a, Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Dist[1/(b*d), Int[Cos[e + f*x]
^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n,
-p]))

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]

[In]

Integrate[Cosh[x]^3/(I + Csch[x]),x]

[Out]

Sinh[x]^2/2 - (I/3)*Sinh[x]^3

Maple [A] (verified)

Time = 222.89 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79

method	result	size
derivativedivides	\(-\frac {i}{3 \operatorname {csch}\left (x \right )^{3}}+\frac {1}{2 \operatorname {csch}\left (x \right )^{2}}\)	\(15\)
default	\(-\frac {i}{3 \operatorname {csch}\left (x \right )^{3}}+\frac {1}{2 \operatorname {csch}\left (x \right )^{2}}\)	\(15\)
risch	\(-\frac {i {\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{8}+\frac {i {\mathrm e}^{x}}{8}-\frac {i {\mathrm e}^{-x}}{8}+\frac {{\mathrm e}^{-2 x}}{8}+\frac {i {\mathrm e}^{-3 x}}{24}\)	\(40\)

[In]

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

[Out]

-1/3*I/csch(x)^3+1/2/csch(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. 36 vs. \(2 (13) = 26\).

Time = 0.25 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{24} \, {\left (-i \, e^{\left (6 \, x\right )} + 3 \, e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} + i\right )} e^{\left (-3 \, x\right )} \]

[In]

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

[Out]

1/24*(-I*e^(6*x) + 3*e^(5*x) + 3*I*e^(4*x) - 3*I*e^(2*x) + 3*e^x + I)*e^(-3*x)

Sympy [F]

\[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\int \frac {\cosh ^{3}{\left (x \right )}}{\operatorname {csch}{\left (x \right )} + i}\, dx \]

[In]

integrate(cosh(x)**3/(I+csch(x)),x)

[Out]

Integral(cosh(x)**3/(csch(x) + I), x)

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. 39 vs. \(2 (13) = 26\).

Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {1}{24} \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - i\right )} e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (-x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} + \frac {1}{24} i \, e^{\left (-3 \, x\right )} \]

[In]

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

[Out]

1/24*(3*e^(-x) + 3*I*e^(-2*x) - I)*e^(3*x) - 1/8*I*e^(-x) + 1/8*e^(-2*x) + 1/24*I*e^(-3*x)

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. 35 vs. \(2 (13) = 26\).

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=-\frac {1}{24} \, {\left (3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} e^{\left (-3 \, x\right )} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {1}{8} i \, e^{x} \]

[In]

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

[Out]

-1/24*(3*I*e^(2*x) - 3*e^x - I)*e^(-3*x) - 1/24*I*e^(3*x) + 1/8*e^(2*x) + 1/8*I*e^x

Mupad [B] (verification not implemented)

Time = 2.23 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.05 \[ \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx=\frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{2\,x}}{8}+\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{8} \]

[In]

int(cosh(x)^3/(1/sinh(x) + 1i),x)

[Out]

exp(-2*x)/8 - (exp(-x)*1i)/8 + exp(2*x)/8 + (exp(-3*x)*1i)/24 - (exp(3*x)*1i)/24 + (exp(x)*1i)/8