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

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Rubi [A]  time = 0.02, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 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])

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]

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*}

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Mathematica [A]  time = 0.93, size = 80, normalized size = 2.00 \[ \frac {2 (-1)^{3/4} \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\text {csch}(c+d x)+i}\right )}{d (\text {csch}(c+d x)-i) \sqrt {\text {csch}(c+d x)+i}} \]

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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fricas [B]  time = 0.95, size = 386, normalized size = 9.65 \[ \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {{\left (2 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {a}{d^{2}}} + 2 \, a e^{\left (d x + c\right )} + 2 i \, a\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (-\frac {{\left (2 \, {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt {\frac {a}{d^{2}}} - 2 \, a e^{\left (d x + c\right )} - 2 i \, a\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {{\left ({\left (2 \, a e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a e^{\left (d x + c\right )} + 4 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} + 2 \, {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a}{d^{2}}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {a}{d^{2}}} \log \left (\frac {{\left ({\left (2 \, a e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, a e^{\left (d x + c\right )} + 4 i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} - 2 \, {\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a}{d^{2}}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]

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*(d*e^(2*d*x + 2*c) - d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(a/d^2) + 2*a*e^(d*x + c) + 2
*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) - 2*a*e^(d*x + c) - 2*I*a)*e^(-d*x - c)/d) + 1/2*sqrt(a/d^2)*log(((2*a*e^(3*d*x + 3*c) - 4*I*a*e^(2*d*x +
2*c) - 2*a*e^(d*x + c) + 4*I*a)*sqrt(a/(e^(2*d*x + 2*c) - 1)) + 2*(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) - 4*I*a*e^(2*d*x + 2*c) - 2
*a*e^(d*x + c) + 4*I*a)*sqrt(a/(e^(2*d*x + 2*c) - 1)) - 2*(a*d*e^(2*d*x + 2*c) - I*a*d*e^(d*x + c) - 2*a*d)*sq
rt(a/d^2))*e^(-2*d*x - 2*c)/d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}\,{d x} \]

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)

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maple [F]  time = 2.39, size = 0, normalized size = 0.00 \[ \int \sqrt {a +i a \,\mathrm {csch}\left (d x +c \right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {i a \operatorname {csch}{\left (c + d x \right )} + a}\, dx \]

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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