3.53 \(\int \sqrt{a+i a \text{csch}(c+d x)} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.0197913, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3774, 203} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[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 3774

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 203

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

Rubi steps

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

Mathematica [A]  time = 0.890078, size = 80, normalized size = 2. \[ \frac{2 (-1)^{3/4} \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\text{csch}(c+d x)+i}\right )}{d (\text{csch}(c+d x)-i) \sqrt{\text{csch}(c+d x)+i}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.52, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+ia{\rm csch} \left (dx+c\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i \, a \operatorname{csch}\left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.39565, size = 814, normalized size = 20.35 \begin{align*} -\frac{1}{2} \, \sqrt{\frac{a}{d^{2}}} \log \left (\frac{2 \,{\left (-\left (i - 4\right ) \, d e^{\left (3 \, d x + 3 \, c\right )} - \left (4 i + 1\right ) \, d\right )} \sqrt{\frac{a}{d^{2}}} + \sqrt{\frac{a e^{\left (2 \, d x + 2 \, c\right )} + 2 i \, a e^{\left (d x + c\right )} - a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (\left (2 i - 8\right ) \, e^{\left (3 \, d x + 3 \, c\right )} + \left (8 i + 2\right ) \, e^{\left (2 \, d x + 2 \, c\right )} - \left (2 i - 8\right ) \, e^{\left (d x + c\right )} - 8 i - 2\right )}}{\left (20 i + 48\right ) \, e^{\left (2 \, d x + 2 \, c\right )} + \left (48 i - 20\right ) \, e^{\left (d x + c\right )}}\right ) + \frac{1}{2} \, \sqrt{\frac{a}{d^{2}}} \log \left (\frac{2 \,{\left (\left (i - 4\right ) \, d e^{\left (3 \, d x + 3 \, c\right )} + \left (4 i + 1\right ) \, d\right )} \sqrt{\frac{a}{d^{2}}} + \sqrt{\frac{a e^{\left (2 \, d x + 2 \, c\right )} + 2 i \, a e^{\left (d x + c\right )} - a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (\left (2 i - 8\right ) \, e^{\left (3 \, d x + 3 \, c\right )} + \left (8 i + 2\right ) \, e^{\left (2 \, d x + 2 \, c\right )} - \left (2 i - 8\right ) \, e^{\left (d x + c\right )} - 8 i - 2\right )}}{\left (20 i + 48\right ) \, e^{\left (2 \, d x + 2 \, c\right )} + \left (48 i - 20\right ) \, e^{\left (d x + c\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(a/d^2)*log((2*(-(I - 4)*d*e^(3*d*x + 3*c) - (4*I + 1)*d)*sqrt(a/d^2) + sqrt((a*e^(2*d*x + 2*c) + 2*I
*a*e^(d*x + c) - a)/(e^(2*d*x + 2*c) - 1))*((2*I - 8)*e^(3*d*x + 3*c) + (8*I + 2)*e^(2*d*x + 2*c) - (2*I - 8)*
e^(d*x + c) - 8*I - 2))/((20*I + 48)*e^(2*d*x + 2*c) + (48*I - 20)*e^(d*x + c))) + 1/2*sqrt(a/d^2)*log((2*((I
- 4)*d*e^(3*d*x + 3*c) + (4*I + 1)*d)*sqrt(a/d^2) + sqrt((a*e^(2*d*x + 2*c) + 2*I*a*e^(d*x + c) - a)/(e^(2*d*x
+ 2*c) - 1))*((2*I - 8)*e^(3*d*x + 3*c) + (8*I + 2)*e^(2*d*x + 2*c) - (2*I - 8)*e^(d*x + c) - 8*I - 2))/((20*
I + 48)*e^(2*d*x + 2*c) + (48*I - 20)*e^(d*x + c)))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{i \, a \operatorname{csch}\left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)