\(\int (a+i a \text {csch}(c+d x))^{3/2} \, dx\) [52]

Optimal result

Integrand size = 17, antiderivative size = 72 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3860, 21, 3859, 209} \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \coth (c+d x)}{\sqrt {a+i a \text {csch}(c+d x)}}\right )}{d}+\frac {2 a^2 \coth (c+d x)}{d \sqrt {a+i a \text {csch}(c+d x)}} \]

[In]

Int[(a + I*a*Csch[c + d*x])^(3/2),x]

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 3860

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*Cot[c + d*x]*((a + b*Csc[c + d*x])^(
n - 2)/(d*(n - 1))), x] + Dist[a/(n - 1), Int[(a + b*Csc[c + d*x])^(n - 2)*(a*(n - 1) + b*(3*n - 4)*Csc[c + d*
x]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]

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

Mathematica [A] (verified)

Time = 1.19 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.39 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=-\frac {2 i a \coth (c+d x) \sqrt {a+i a \text {csch}(c+d x)} \left (-\sqrt [4]{-1} \text {arctanh}\left ((-1)^{3/4} \sqrt {i+\text {csch}(c+d x)}\right )+\sqrt {i+\text {csch}(c+d x)}\right )}{d (-i+\text {csch}(c+d x)) \sqrt {i+\text {csch}(c+d x)}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

int((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. 461 vs. \(2 (60) = 120\).

Time = 0.27 (sec) , antiderivative size = 461, normalized size of antiderivative = 6.40 \[ \int (a+i a \text {csch}(c+d x))^{3/2} \, dx=\frac {\sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} + i \, a^{2} + {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) - \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left (a^{2} e^{\left (d x + c\right )} + i \, a^{2} - {\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt {\frac {a^{3}}{d^{2}}} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-d x - c\right )}}{d}\right ) + \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} + {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \sqrt {\frac {a^{3}}{d^{2}}} d \log \left (-\frac {2 \, {\left ({\left (a d e^{\left (2 \, d x + 2 \, c\right )} - i \, a d e^{\left (d x + c\right )} - 2 \, a d\right )} \sqrt {\frac {a^{3}}{d^{2}}} - {\left (a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 2 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} e^{\left (d x + c\right )} + 2 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) + 4 \, {\left (a e^{\left (d x + c\right )} - i \, a\right )} \sqrt {\frac {a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}}{2 \, d} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

integrate((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]

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

[In]

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

[In]

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

[Out]

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