3.65 \(\int \frac {\sinh (x)}{i+\text {csch}(x)} \, dx\)

Optimal. Leaf size=20 \[ x-2 i \cosh (x)-\frac {\cosh (x)}{\text {csch}(x)+i} \]

[Out]

x-2*I*cosh(x)-cosh(x)/(I+csch(x))

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Rubi [A]  time = 0.05, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3819, 3787, 2638, 8} \[ x-2 i \cosh (x)-\frac {\cosh (x)}{\text {csch}(x)+i} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]/(I + Csch[x]),x]

[Out]

x - (2*I)*Cosh[x] - Cosh[x]/(I + Csch[x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3819

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(f*(a + b*Csc[e + f*x])), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[
e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sinh (x)}{i+\text {csch}(x)} \, dx &=-\frac {\cosh (x)}{i+\text {csch}(x)}+\int (-2 i+\text {csch}(x)) \sinh (x) \, dx\\ &=-\frac {\cosh (x)}{i+\text {csch}(x)}-2 i \int \sinh (x) \, dx+\int 1 \, dx\\ &=x-2 i \cosh (x)-\frac {\cosh (x)}{i+\text {csch}(x)}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 35, normalized size = 1.75 \[ x-i \cosh (x)-\frac {2 \sinh \left (\frac {x}{2}\right )}{\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]/(I + Csch[x]),x]

[Out]

x - I*Cosh[x] - (2*Sinh[x/2])/(Cosh[x/2] + I*Sinh[x/2])

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fricas [B]  time = 0.41, size = 40, normalized size = 2.00 \[ \frac {{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} + {\left (-2 i \, x - 5 i\right )} e^{x} - i \, e^{\left (3 \, x\right )} - 1}{2 \, e^{\left (2 \, x\right )} - 2 i \, e^{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.13, size = 26, normalized size = 1.30 \[ x + \frac {{\left (5 \, e^{x} - i\right )} e^{\left (-x\right )}}{2 \, {\left (i \, e^{x} + 1\right )}} - \frac {1}{2} i \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.20, size = 51, normalized size = 2.55 \[ \frac {i}{\tanh \left (\frac {x}{2}\right )-1}-\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )-\frac {i}{\tanh \left (\frac {x}{2}\right )+1}+\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )-\frac {2}{\tanh \left (\frac {x}{2}\right )-i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(I+csch(x)),x)

[Out]

I/(tanh(1/2*x)-1)-ln(tanh(1/2*x)-1)-I/(tanh(1/2*x)+1)+ln(tanh(1/2*x)+1)-2/(tanh(1/2*x)-I)

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maxima [A]  time = 0.32, size = 31, normalized size = 1.55 \[ x - \frac {5 i \, e^{\left (-x\right )} - 1}{2 \, {\left (i \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )}\right )}} - \frac {1}{2} i \, e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.46, size = 24, normalized size = 1.20 \[ x-\frac {{\mathrm {e}}^{-x}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {2{}\mathrm {i}}{{\mathrm {e}}^x-\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)/(1/sinh(x) + 1i),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)/(I+csch(x)),x)

[Out]

Integral(sinh(x)/(csch(x) + I), x)

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