3.49 \(\int \frac {1}{a+i a \text {csch}(a+b x)} \, dx\)

Optimal. Leaf size=32 \[ \frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))} \]

[Out]

x/a-coth(b*x+a)/b/(a+I*a*csch(b*x+a))

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3777, 8} \[ \frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Csch[a + b*x])^(-1),x]

[Out]

x/a - Coth[a + b*x]/(b*(a + I*a*Csch[a + b*x]))

Rule 8

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

Rule 3777

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

Rubi steps

\begin {align*} \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx &=-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))}+\frac {\int a \, dx}{a^2}\\ &=\frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 54, normalized size = 1.69 \[ -\frac {2 \sinh \left (\frac {1}{2} (a+b x)\right )}{a b \left (\cosh \left (\frac {1}{2} (a+b x)\right )-i \sinh \left (\frac {1}{2} (a+b x)\right )\right )}+\frac {x}{a}+\frac {1}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Csch[a + b*x])^(-1),x]

[Out]

b^(-1) + x/a - (2*Sinh[(a + b*x)/2])/(a*b*(Cosh[(a + b*x)/2] - I*Sinh[(a + b*x)/2]))

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fricas [A]  time = 1.43, size = 32, normalized size = 1.00 \[ \frac {b x e^{\left (b x + a\right )} + i \, b x + 2 i}{a b e^{\left (b x + a\right )} + i \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*csch(b*x+a)),x, algorithm="fricas")

[Out]

(b*x*e^(b*x + a) + I*b*x + 2*I)/(a*b*e^(b*x + a) + I*a*b)

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giac [A]  time = 0.13, size = 29, normalized size = 0.91 \[ \frac {\frac {b x + a}{a} + \frac {2 i}{a {\left (e^{\left (b x + a\right )} + i\right )}}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*csch(b*x+a)),x, algorithm="giac")

[Out]

((b*x + a)/a + 2*I/(a*(e^(b*x + a) + I)))/b

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maple [A]  time = 0.25, size = 63, normalized size = 1.97 \[ -\frac {2}{b a \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i\right )}-\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{b a}+\frac {\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )}{b a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+I*a*csch(b*x+a)),x)

[Out]

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

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maxima [A]  time = 0.36, size = 35, normalized size = 1.09 \[ \frac {b x + a}{a b} + \frac {2 i}{{\left (a e^{\left (-b x - a\right )} - i \, a\right )} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*csch(b*x+a)),x, algorithm="maxima")

[Out]

(b*x + a)/(a*b) + 2*I/((a*e^(-b*x - a) - I*a)*b)

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mupad [B]  time = 1.55, size = 26, normalized size = 0.81 \[ \frac {x}{a}+\frac {2{}\mathrm {i}}{a\,b\,\left ({\mathrm {e}}^{a+b\,x}+1{}\mathrm {i}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + (a*1i)/sinh(a + b*x)),x)

[Out]

x/a + 2i/(a*b*(exp(a + b*x) + 1i))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {i \int \frac {1}{\operatorname {csch}{\left (a + b x \right )} - i}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+I*a*csch(b*x+a)),x)

[Out]

-I*Integral(1/(csch(a + b*x) - I), x)/a

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