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

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[a + b*x]/(b*(a - I*a*Csch[a + b*x]))

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Rubi [A]  time = 0.0153436, antiderivative size = 32, normalized size of antiderivative = 1., 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 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]

Rule 8

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

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*}

Mathematica [A]  time = 0.103366, 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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Maple [A]  time = 0.043, size = 63, normalized size = 2. \begin{align*}{\frac{1}{ab}\ln \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) +1 \right ) }-2\,{\frac{1}{ab \left ( \tanh \left ( 1/2\,bx+a/2 \right ) -i \right ) }}-{\frac{1}{ab}\ln \left ( \tanh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13904, size = 47, normalized size = 1.47 \begin{align*} \frac{b x + a}{a b} - \frac{2 i}{{\left (a e^{\left (-b x - a\right )} + i \, a\right )} b} \end{align*}

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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Fricas [A]  time = 1.59477, size = 80, normalized size = 2.5 \begin{align*} \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} \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{1}{i \operatorname{csch}{\left (a + b x \right )} - 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.17111, size = 42, normalized size = 1.31 \begin{align*} \frac{b x + a}{a b} - \frac{2 i}{a b{\left (e^{\left (b x + a\right )} - i\right )}} \end{align*}

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*b) - 2*I/(a*b*(e^(b*x + a) - I))