\(\int \sqrt {a-i a \text {csch}(c+d x)} \, dx\) [56]

Optimal result

Integrand size = 17, antiderivative size = 40 \[ \int \sqrt {a-i a \text {csch}(c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3859, 209} \[ \int \sqrt {a-i a \text {csch}(c+d x)} \, dx=\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a-i a \text {csch}(c+d x)}}\right )}{d} \]

[In]

Int[Sqrt[a - I*a*Csch[c + d*x]],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.00 \[ \int \sqrt {a-i a \text {csch}(c+d x)} \, dx=-\frac {2 (-1)^{3/4} \arctan \left ((-1)^{3/4} \sqrt {-i+\text {csch}(c+d x)}\right ) \coth (c+d x) \sqrt {a-i a \text {csch}(c+d x)}}{d \sqrt {-i+\text {csch}(c+d x)} (i+\text {csch}(c+d x))} \]

[In]

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

[Out]

(-2*(-1)^(3/4)*ArcTan[(-1)^(3/4)*Sqrt[-I + Csch[c + d*x]]]*Coth[c + d*x]*Sqrt[a - I*a*Csch[c + d*x]])/(d*Sqrt[
-I + Csch[c + d*x]]*(I + Csch[c + d*x]))

Maple [F]

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

[In]

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

[Out]

int((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. 383 vs. \(2 (32) = 64\).

Time = 0.29 (sec) , antiderivative size = 383, normalized size of antiderivative = 9.58 \[ \int \sqrt {a-i a \text {csch}(c+d x)} \, dx=\frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {a}{d^{2}}} + a e^{\left (d x + c\right )} - i \, a\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (-\frac {2 \, {\left ({\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {a}{d^{2}}} - a e^{\left (d x + c\right )} + i \, a\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {2 \, {\left ({\left (a e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, a e^{\left (2 \, d x + 2 \, c\right )} - a e^{\left (d x + c\right )} - 2 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} + {\left (a d e^{\left (2 \, d x + 2 \, c\right )} + i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a}{d^{2}}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {2 \, {\left ({\left (a e^{\left (3 \, d x + 3 \, c\right )} + 2 i \, a e^{\left (2 \, d x + 2 \, c\right )} - a e^{\left (d x + c\right )} - 2 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} - {\left (a d e^{\left (2 \, d x + 2 \, c\right )} + i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a}{d^{2}}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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