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

Optimal. Leaf size=72 \[ \frac{2 a^2 \coth (c+d x)}{d \sqrt{a+i a \text{csch}(c+d x)}}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d} \]

[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]])

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Rubi [A]  time = 0.0415648, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {3775, 21, 3774, 203} \[ \frac{2 a^2 \coth (c+d x)}{d \sqrt{a+i a \text{csch}(c+d x)}}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(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 3775

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]

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 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 (a+i a \text{csch}(c+d x))^{3/2} \, dx &=\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 ) \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 a^{3/2} \tanh ^{-1}\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]  time = 1.17214, size = 100, normalized size = 1.39 \[ -\frac{2 i a \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)} \left (\sqrt{\text{csch}(c+d x)+i}-\sqrt [4]{-1} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\text{csch}(c+d x)+i}\right )\right )}{d (\text{csch}(c+d x)-i) \sqrt{\text{csch}(c+d x)+i}} \]

Antiderivative was successfully verified.

[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]])

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Maple [F]  time = 0.206, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia{\rm csch} \left (dx+c\right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \operatorname{csch}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 2.40457, size = 1089, normalized size = 15.12 \begin{align*} -\frac{2 \,{\left (d e^{\left (d x + c\right )} + i \, d\right )} \sqrt{\frac{a^{3}}{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^{3}}{d^{2}}} +{\left (\left (2 i - 8\right ) \, a e^{\left (3 \, d x + 3 \, c\right )} + \left (8 i + 2\right ) \, a e^{\left (2 \, d x + 2 \, c\right )} - \left (2 i - 8\right ) \, a e^{\left (d x + c\right )} - \left (8 i + 2\right ) \, a\right )} \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 (20 i + 48\right ) \, a e^{\left (2 \, d x + 2 \, c\right )} + \left (48 i - 20\right ) \, a e^{\left (d x + c\right )}}\right ) - 2 \,{\left (d e^{\left (d x + c\right )} + i \, d\right )} \sqrt{\frac{a^{3}}{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^{3}}{d^{2}}} +{\left (\left (2 i - 8\right ) \, a e^{\left (3 \, d x + 3 \, c\right )} + \left (8 i + 2\right ) \, a e^{\left (2 \, d x + 2 \, c\right )} - \left (2 i - 8\right ) \, a e^{\left (d x + c\right )} - \left (8 i + 2\right ) \, a\right )} \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 (20 i + 48\right ) \, a e^{\left (2 \, d x + 2 \, c\right )} + \left (48 i - 20\right ) \, a e^{\left (d x + c\right )}}\right ) -{\left (8 \, a e^{\left (d x + c\right )} - 8 i \, a\right )} \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}}}{4 \, d e^{\left (d x + c\right )} + 4 i \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (i a \operatorname{csch}{\left (c + d x \right )} + a\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \operatorname{csch}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)