\(\int \frac {1}{a+i a \text {csch}(a+b x)} \, dx\) [49]

Optimal result

Integrand size = 15, antiderivative size = 32 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3862, 8} \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {x}{a}-\frac {\coth (a+b x)}{b (a+i a \text {csch}(a+b x))} \]

[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 3862

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*} \text {integral}& = -\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] (verified)

Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {1}{b}+\frac {x}{a}-\frac {2 \sinh \left (\frac {1}{2} (a+b x)\right )}{a b \left (\cosh \left (\frac {1}{2} (a+b x)\right )-i \sinh \left (\frac {1}{2} (a+b x)\right )\right )} \]

[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]))

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {x}{a}+\frac {2 i}{b a \left ({\mathrm e}^{b x +a}+i\right )}\)	\(27\)
parallelrisch	\(\frac {i x b +\tanh \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (b x -2 i\right )}{b a \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i\right )}\)	\(44\)
derivativedivides	\(\frac {\ln \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {2}{\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i}-\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{a b}\)	\(54\)
default	\(\frac {\ln \left (1+\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )-\frac {2}{\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )+i}-\ln \left (\tanh \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}{a b}\)	\(54\)

[In]

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

[Out]

x/a+2*I/b/a/(exp(b*x+a)+I)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {b x e^{\left (b x + a\right )} + i \, b x + 2 i}{a b e^{\left (b x + a\right )} + i \, a b} \]

[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)

Sympy [F]

\[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=- \frac {i \int \frac {1}{\operatorname {csch}{\left (a + b x \right )} - i}\, dx}{a} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {b x + a}{a b} + \frac {2 i}{{\left (a e^{\left (-b x - a\right )} - i \, a\right )} b} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {\frac {b x + a}{a} + \frac {2 i}{a {\left (e^{\left (b x + a\right )} + i\right )}}}{b} \]

[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

Mupad [B] (verification not implemented)

Time = 2.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.81 \[ \int \frac {1}{a+i a \text {csch}(a+b x)} \, dx=\frac {x}{a}+\frac {2{}\mathrm {i}}{a\,b\,\left ({\mathrm {e}}^{a+b\,x}+1{}\mathrm {i}\right )} \]

[In]

int(1/(a + (a*1i)/sinh(a + b*x)),x)

[Out]

x/a + 2i/(a*b*(exp(a + b*x) + 1i))