\(\int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx\) [55]

Optimal result

Integrand size = 17, antiderivative size = 123 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}} \]

[Out]

2*arctanh(coth(d*x+c)*a^(1/2)/(a+I*a*csch(d*x+c))^(1/2))/a^(3/2)/d-1/2*coth(d*x+c)/d/(a+I*a*csch(d*x+c))^(3/2)
-5/4*arctanh(1/2*coth(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*csch(d*x+c))^(1/2))/a^(3/2)/d*2^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3862, 4005, 3859, 209, 3880} \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}} \]

[In]

Int[(a + I*a*Csch[c + d*x])^(-3/2),x]

[Out]

(2*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Coth[c + d*
x])/(Sqrt[2]*Sqrt[a + I*a*Csch[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Coth[c + d*x]/(2*d*(a + I*a*Csch[c + d*x])
^(3/2))

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 3862

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

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
]

Rule 4005

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}-\frac {\int \frac {-2 a+\frac {1}{2} i a \text {csch}(c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx}{2 a^2} \\ & = -\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}+\frac {\int \sqrt {a+i a \text {csch}(c+d x)} \, dx}{a^2}-\frac {(5 i) \int \frac {\text {csch}(c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}} \, dx}{4 a} \\ & = -\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}}-\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 )}{a d}+\frac {(5 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 )}{2 a d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{a^{3/2} d}-\frac {5 \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {2} \sqrt {a+i a \text {csch}(c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {\coth (c+d x)}{2 d (a+i a \text {csch}(c+d x))^{3/2}} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(327\) vs. \(2(123)=246\).

Time = 3.42 (sec) , antiderivative size = 327, normalized size of antiderivative = 2.66 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\frac {\left (-2 \sqrt {a}-8 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+5 \sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+i \text {csch}(c+d x) \left (2 \sqrt {a}-8 \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}+5 \sqrt {2} \arctan \left (\frac {\sqrt {i a (i+\text {csch}(c+d x))}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {i a (i+\text {csch}(c+d x))}\right )\right ) \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )}{4 a^{3/2} d (i+\text {csch}(c+d x)) \sqrt {a+i a \text {csch}(c+d x)} \left (\cosh \left (\frac {1}{2} (c+d x)\right )-i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \]

[In]

Integrate[(a + I*a*Csch[c + d*x])^(-3/2),x]

[Out]

((-2*Sqrt[a] - 8*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]]*Sqrt[I*a*(I + Csch[c + d*x])] + 5*Sqrt[2]*ArcTa
n[Sqrt[I*a*(I + Csch[c + d*x])]/(Sqrt[2]*Sqrt[a])]*Sqrt[I*a*(I + Csch[c + d*x])] + I*Csch[c + d*x]*(2*Sqrt[a]
- 8*ArcTan[Sqrt[I*a*(I + Csch[c + d*x])]/Sqrt[a]]*Sqrt[I*a*(I + Csch[c + d*x])] + 5*Sqrt[2]*ArcTan[Sqrt[I*a*(I
+ Csch[c + d*x])]/(Sqrt[2]*Sqrt[a])]*Sqrt[I*a*(I + Csch[c + d*x])]))*(Cosh[(c + d*x)/2] + I*Sinh[(c + d*x)/2]
))/(4*a^(3/2)*d*(I + Csch[c + d*x])*Sqrt[a + I*a*Csch[c + d*x]]*(Cosh[(c + d*x)/2] - I*Sinh[(c + d*x)/2]))

Maple [F]

\[\int \frac {1}{\left (a +i a \,\operatorname {csch}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

[In]

int(1/(a+I*a*csch(d*x+c))^(3/2),x)

[Out]

int(1/(a+I*a*csch(d*x+c))^(3/2),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. 873 vs. \(2 (96) = 192\).

Time = 0.30 (sec) , antiderivative size = 873, normalized size of antiderivative = 7.10 \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+I*a*csch(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

-1/4*(5*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(2*(2*sqrt(1/2)
*(a^2*d*e^(2*d*x + 2*c) - a^2*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) + a*e^(d*x + c) - I*a)*e^(-d*
x - c)) - 5*sqrt(1/2)*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(-2*(2*sqrt
(1/2)*(a^2*d*e^(2*d*x + 2*c) - a^2*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) - a*e^(d*x + c) + I*a)*e
^(-d*x - c)) - 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(2*((a*d*e^(2*d*
x + 2*c) - a*d)*sqrt(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) + e^(d*x + c) + I)*e^(-d*x - c)/(a*d)) + 2*(a^
2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(-2*((a*d*e^(2*d*x + 2*c) - a*d)*sqr
t(a/(e^(2*d*x + 2*c) - 1))*sqrt(1/(a^3*d^2)) - e^(d*x + c) - I)*e^(-d*x - c)/(a*d)) - 2*(a^2*d*e^(2*d*x + 2*c)
+ 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*log(2*((a^2*d*e^(2*d*x + 2*c) - I*a^2*d*e^(d*x + c) - 2*a^
2*d)*sqrt(1/(a^3*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)/(a*d)) + 2*(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x + c) - a^2*d)*sqrt(1/(a^3*d^2))*lo
g(-2*((a^2*d*e^(2*d*x + 2*c) - I*a^2*d*e^(d*x + c) - 2*a^2*d)*sqrt(1/(a^3*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)/(a*d)) + 2*sqrt(a/(e^(2*d*x +
2*c) - 1))*(e^(3*d*x + 3*c) - I*e^(2*d*x + 2*c) - e^(d*x + c) + I))/(a^2*d*e^(2*d*x + 2*c) + 2*I*a^2*d*e^(d*x
+ c) - a^2*d)

Sympy [F]

\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{\left (i a \operatorname {csch}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(a+I*a*csch(d*x+c))**(3/2),x)

[Out]

Integral((I*a*csch(c + d*x) + a)**(-3/2), x)

Maxima [F]

\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(a+I*a*csch(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((I*a*csch(d*x + c) + a)^(-3/2), x)

Giac [F]

\[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (i \, a \operatorname {csch}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/(a+I*a*csch(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((I*a*csch(d*x + c) + a)^(-3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+i a \text {csch}(c+d x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a\,1{}\mathrm {i}}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

[In]

int(1/(a + (a*1i)/sinh(c + d*x))^(3/2),x)

[Out]

int(1/(a + (a*1i)/sinh(c + d*x))^(3/2), x)