\(\int \frac {1}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx\) [57]

Optimal result

Integrand size = 17, antiderivative size = 91 \[ \int \frac {1}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]

[Out]

2*arctanh(coth(d*x+c)*a^(1/2)/(a-I*a*csch(d*x+c))^(1/2))/d/a^(1/2)-arctanh(1/2*coth(d*x+c)*a^(1/2)*2^(1/2)/(a-
I*a*csch(d*x+c))^(1/2))*2^(1/2)/d/a^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3861, 3859, 209, 3880} \[ \int \frac {1}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]

[In]

Int[1/Sqrt[a - I*a*Csch[c + d*x]],x]

[Out]

(2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a - I*a*Csch[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*ArcTanh[(Sqrt[a]*Coth[
c + d*x])/(Sqrt[2]*Sqrt[a - I*a*Csch[c + d*x]])])/(Sqrt[a]*d)

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 3859

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(a + x^2), x], x, b*(C
ot[c + d*x]/Sqrt[a + b*Csc[c + d*x]])], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 3861

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

Rule 3880

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

Rubi steps \begin{align*} \text {integral}& = i \int \frac {\text {csch}(c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx+\frac {\int \sqrt {a-i a \text {csch}(c+d x)} \, dx}{a} \\ & = \frac {(2 i) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {i a \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d}-\frac {(2 i) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {i a \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a-i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx=\frac {\sqrt {a} \left (2 \arctan \left (\frac {\sqrt {-i a (-i+\text {csch}(c+d x))}}{\sqrt {a}}\right )-\sqrt {2} \arctan \left (\frac {\sqrt {-i a (-i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right )\right ) \coth (c+d x)}{d \sqrt {a (-1-i \text {csch}(c+d x))} \sqrt {a-i a \text {csch}(c+d x)}} \]

[In]

Integrate[1/Sqrt[a - I*a*Csch[c + d*x]],x]

[Out]

(Sqrt[a]*(2*ArcTan[Sqrt[(-I)*a*(-I + Csch[c + d*x])]/Sqrt[a]] - Sqrt[2]*ArcTan[Sqrt[(-I)*a*(-I + Csch[c + d*x]
)]/(Sqrt[2]*Sqrt[a])])*Coth[c + d*x])/(d*Sqrt[a*(-1 - I*Csch[c + d*x])]*Sqrt[a - I*a*Csch[c + d*x]])

Maple [F]

\[\int \frac {1}{\sqrt {a -i a \,\operatorname {csch}\left (d x +c \right )}}d x\]

[In]

int(1/(a-I*a*csch(d*x+c))^(1/2),x)

[Out]

int(1/(a-I*a*csch(d*x+c))^(1/2),x)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (72) = 144\).

Time = 0.28 (sec) , antiderivative size = 551, normalized size of antiderivative = 6.05 \[ \int \frac {1}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx=-\frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (2 \, {\left (\sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} + a e^{\left (d x + c\right )} + i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-2 \, {\left (\sqrt {2} {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} - a e^{\left (d x + c\right )} - i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} + e^{\left (d x + c\right )} - i\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {1}{a d^{2}}} - e^{\left (d x + c\right )} + i\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} + i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} - 2 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} + i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} - \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} - 2 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]

[In]

integrate(1/(a-I*a*csch(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(2)*sqrt(1/(a*d^2))*log(2*(sqrt(2)*(a*d*e^(2*d*x + 2*c) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(
a*d^2)) + a*e^(d*x + c) + I*a)*e^(-d*x - c)) + 1/2*sqrt(2)*sqrt(1/(a*d^2))*log(-2*(sqrt(2)*(a*d*e^(2*d*x + 2*c
) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) - a*e^(d*x + c) - I*a)*e^(-d*x - c)) + 1/2*sqrt(1/(a*d^
2))*log(2*((d*e^(2*d*x + 2*c) - d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) + e^(d*x + c) - I)*e^(-d*x -
c)/d) - 1/2*sqrt(1/(a*d^2))*log(-2*((d*e^(2*d*x + 2*c) - d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a*d^2)) - e^
(d*x + c) + I)*e^(-d*x - c)/d) + 1/2*sqrt(1/(a*d^2))*log(2*((a*d*e^(2*d*x + 2*c) + I*a*d*e^(d*x + c) - 2*a*d)*
sqrt(1/(a*d^2)) + sqrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) + 2*I*e^(2*d*x + 2*c) - e^(d*x + c) - 2*I))*e
^(-2*d*x - 2*c)/d) - 1/2*sqrt(1/(a*d^2))*log(-2*((a*d*e^(2*d*x + 2*c) + I*a*d*e^(d*x + c) - 2*a*d)*sqrt(1/(a*d
^2)) - sqrt(a/(e^(2*d*x + 2*c) - 1))*(e^(3*d*x + 3*c) + 2*I*e^(2*d*x + 2*c) - e^(d*x + c) - 2*I))*e^(-2*d*x -
2*c)/d)

Sympy [F]

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

[In]

integrate(1/(a-I*a*csch(d*x+c))**(1/2),x)

[Out]

Integral(1/sqrt(-I*a*csch(c + d*x) + a), x)

Maxima [F]

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

[In]

integrate(1/(a-I*a*csch(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/sqrt(-I*a*csch(d*x + c) + a), x)

Giac [F]

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

[In]

integrate(1/(a-I*a*csch(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(1/sqrt(-I*a*csch(d*x + c) + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a-i a \text {csch}(c+d x)}} \, dx=\int \frac {1}{\sqrt {a-\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}}} \,d x \]

[In]

int(1/(a - (a*1i)/sinh(c + d*x))^(1/2),x)

[Out]

int(1/(a - (a*1i)/sinh(c + d*x))^(1/2), x)