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

Optimal. Leaf size=19 \[ \frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3872, 2835, 2564, 30} \[ \frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]

Antiderivative was successfully verified.

[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 2564

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 2835

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 3872

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[((g*C
os[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*} \int \frac {\cosh ^3(x)}{i+\text {csch}(x)} \, dx &=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\\ &=-\operatorname {Subst}(\int x \, dx,x,i \sinh (x))+\operatorname {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*}

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Mathematica [A]  time = 0.01, size = 19, normalized size = 1.00 \[ \frac {\sinh ^2(x)}{2}-\frac {1}{3} i \sinh ^3(x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^3/(I + Csch[x]),x]

[Out]

Sinh[x]^2/2 - (I/3)*Sinh[x]^3

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fricas [B]  time = 0.40, size = 36, normalized size = 1.89 \[ \frac {1}{24} \, {\left (-i \, e^{\left (6 \, x\right )} + 3 \, e^{\left (5 \, x\right )} + 3 i \, e^{\left (4 \, x\right )} - 3 i \, e^{\left (2 \, x\right )} + 3 \, e^{x} + i\right )} e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.14, size = 35, normalized size = 1.84 \[ -\frac {1}{24} \, {\left (3 i \, e^{\left (2 \, x\right )} - 3 \, e^{x} - i\right )} e^{\left (-3 \, x\right )} - \frac {1}{24} i \, e^{\left (3 \, x\right )} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {1}{8} i \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.11, size = 15, normalized size = 0.79 \[ -\frac {i}{3 \mathrm {csch}\relax (x )^{3}}+\frac {1}{2 \mathrm {csch}\relax (x )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.30, size = 39, normalized size = 2.05 \[ \frac {1}{24} \, {\left (3 \, e^{\left (-x\right )} + 3 i \, e^{\left (-2 \, x\right )} - i\right )} e^{\left (3 \, x\right )} - \frac {1}{8} i \, e^{\left (-x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} + \frac {1}{24} i \, e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 0.14, size = 39, normalized size = 2.05 \[ \frac {{\mathrm {e}}^{-2\,x}}{8}-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{8}+\frac {{\mathrm {e}}^{2\,x}}{8}+\frac {{\mathrm {e}}^{-3\,x}\,1{}\mathrm {i}}{24}-\frac {{\mathrm {e}}^{3\,x}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{3}{\relax (x )}}{\operatorname {csch}{\relax (x )} + i}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cosh(x)**3/(csch(x) + I), x)

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