3.54 \(\int \frac {1}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx\)

Optimal. Leaf size=91 \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]

[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)

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Rubi [A]  time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3776, 3774, 203, 3795} \[ \frac {2 \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{\sqrt {a} d} \]

Antiderivative was successfully verified.

[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 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]

Rule 3776

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 3795

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

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Mathematica [A]  time = 1.12, size = 118, normalized size = 1.30 \[ \frac {\sqrt {a} \coth (c+d x) \left (2 \tan ^{-1}\left (\frac {\sqrt {i a (\text {csch}(c+d x)+i)}}{\sqrt {a}}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {i a (\text {csch}(c+d x)+i)}}{\sqrt {2} \sqrt {a}}\right )\right )}{d \sqrt {i a (\text {csch}(c+d x)+i)} \sqrt {a+i a \text {csch}(c+d x)}} \]

Antiderivative was successfully verified.

[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[I*a*(I + Csch[c + d*x])]*Sqrt[a + I*a*Csch[c + d*x]])

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fricas [B]  time = 1.50, size = 561, normalized size = 6.16 \[ -\frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left ({\left (2 \, \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}}} + 2 \, a e^{\left (d x + c\right )} - 2 i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {1}{a d^{2}}} \log \left (-{\left (2 \, \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}}} - 2 \, a e^{\left (d x + c\right )} + 2 i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{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 {1}{a d^{2}}} + 2 \, e^{\left (d x + c\right )} + 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac {1}{2} \, \sqrt {\frac {1}{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 {1}{a d^{2}}} - 2 \, e^{\left (d x + c\right )} - 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac {1}{2} \, \sqrt {\frac {1}{a d^{2}}} \log \left (\frac {{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} + \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + 4 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 {{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt {\frac {1}{a d^{2}}} - \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} {\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + 4 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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maple [F]  time = 0.82, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a +i a \,\mathrm {csch}\left (d x +c \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {i \, a \operatorname {csch}\left (d x + c\right ) + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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