3.51 \(\int (a+i a \text{csch}(c+d x))^{5/2} \, dx\)

Optimal. Leaf size=107 \[ \frac{14 a^3 \coth (c+d x)}{3 d \sqrt{a+i a \text{csch}(c+d x)}}+\frac{2 a^2 \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)}}{3 d}+\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d} \]

[Out]

(2*a^(5/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/d + (14*a^3*Coth[c + d*x])/(3*d*Sqrt[
a + I*a*Csch[c + d*x]]) + (2*a^2*Coth[c + d*x]*Sqrt[a + I*a*Csch[c + d*x]])/(3*d)

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Rubi [A]  time = 0.12708, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {3775, 3915, 3774, 203, 3792} \[ \frac{14 a^3 \coth (c+d x)}{3 d \sqrt{a+i a \text{csch}(c+d x)}}+\frac{2 a^2 \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)}}{3 d}+\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Csch[c + d*x])^(5/2),x]

[Out]

(2*a^(5/2)*ArcTanh[(Sqrt[a]*Coth[c + d*x])/Sqrt[a + I*a*Csch[c + d*x]]])/d + (14*a^3*Coth[c + d*x])/(3*d*Sqrt[
a + I*a*Csch[c + d*x]]) + (2*a^2*Coth[c + d*x]*Sqrt[a + I*a*Csch[c + d*x]])/(3*d)

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 3915

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Dist[c, In
t[Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[d, Int[Sqrt[a + b*Csc[e + f*x]]*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]

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])

Rule 3792

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*b*Cot[e + f*x])/
(f*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 (a+i a \text{csch}(c+d x))^{5/2} \, dx &=\frac{2 a^2 \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)}}{3 d}+\frac{1}{3} (2 a) \int \sqrt{a+i a \text{csch}(c+d x)} \left (\frac{3 a}{2}+\frac{7}{2} i a \text{csch}(c+d x)\right ) \, dx\\ &=\frac{2 a^2 \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)}}{3 d}+\frac{1}{3} \left (7 i a^2\right ) \int \text{csch}(c+d x) \sqrt{a+i a \text{csch}(c+d x)} \, dx+a^2 \int \sqrt{a+i a \text{csch}(c+d x)} \, dx\\ &=\frac{14 a^3 \coth (c+d x)}{3 d \sqrt{a+i a \text{csch}(c+d x)}}+\frac{2 a^2 \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)}}{3 d}-\frac{\left (2 i a^3\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^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{d}+\frac{14 a^3 \coth (c+d x)}{3 d \sqrt{a+i a \text{csch}(c+d x)}}+\frac{2 a^2 \coth (c+d x) \sqrt{a+i a \text{csch}(c+d x)}}{3 d}\\ \end{align*}

Mathematica [A]  time = 1.40158, size = 136, normalized size = 1.27 \[ \frac{2 a^2 \sqrt{a+i a \text{csch}(c+d x)} \left (\coth (c+d x)+\frac{14 \sinh \left (\frac{1}{2} (c+d x)\right )}{\cosh \left (\frac{1}{2} (c+d x)\right )-i \sinh \left (\frac{1}{2} (c+d x)\right )}+\frac{3 (-1)^{3/4} \coth (c+d x) \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\text{csch}(c+d x)+i}\right )}{(\text{csch}(c+d x)-i) \sqrt{\text{csch}(c+d x)+i}}-7 i\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Csch[c + d*x])^(5/2),x]

[Out]

(2*a^2*Sqrt[a + I*a*Csch[c + d*x]]*(-7*I + Coth[c + d*x] + (3*(-1)^(3/4)*ArcTanh[(-1)^(3/4)*Sqrt[I + Csch[c +
d*x]]]*Coth[c + d*x])/((-I + Csch[c + d*x])*Sqrt[I + Csch[c + d*x]]) + (14*Sinh[(c + d*x)/2])/(Cosh[(c + d*x)/
2] - I*Sinh[(c + d*x)/2])))/(3*d)

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Maple [F]  time = 0.258, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia{\rm csch} \left (dx+c\right ) \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*csch(d*x+c))^(5/2),x)

[Out]

int((a+I*a*csch(d*x+c))^(5/2),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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*csch(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((I*a*csch(d*x + c) + a)^(5/2), x)

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Fricas [B]  time = 2.41947, size = 1403, normalized size = 13.11 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*csch(d*x+c))^(5/2),x, algorithm="fricas")

[Out]

-(2*sqrt(a^5/d^2)*(3*d*e^(3*d*x + 3*c) + 3*I*d*e^(2*d*x + 2*c) - 3*d*e^(d*x + c) - 3*I*d)*log((2*sqrt(a^5/d^2)
*(-(I - 4)*d*e^(3*d*x + 3*c) - (4*I + 1)*d) + ((2*I - 8)*a^2*e^(3*d*x + 3*c) + (8*I + 2)*a^2*e^(2*d*x + 2*c) -
(2*I - 8)*a^2*e^(d*x + c) - (8*I + 2)*a^2)*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^2*e^(2*d*x + 2*c) + (48*I - 20)*a^2*e^(d*x + c))) - 2*sqrt(a^5/d^2)*(3*d*e^(3*d*x + 3*c)
+ 3*I*d*e^(2*d*x + 2*c) - 3*d*e^(d*x + c) - 3*I*d)*log((2*sqrt(a^5/d^2)*((I - 4)*d*e^(3*d*x + 3*c) + (4*I + 1
)*d) + ((2*I - 8)*a^2*e^(3*d*x + 3*c) + (8*I + 2)*a^2*e^(2*d*x + 2*c) - (2*I - 8)*a^2*e^(d*x + c) - (8*I + 2)*
a^2)*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^2*e^(2*d*x + 2*c)
+ (48*I - 20)*a^2*e^(d*x + c))) - (64*a^2*e^(3*d*x + 3*c) - 48*I*a^2*e^(2*d*x + 2*c) - 48*a^2*e^(d*x + c) + 6
4*I*a^2)*sqrt((a*e^(2*d*x + 2*c) + 2*I*a*e^(d*x + c) - a)/(e^(2*d*x + 2*c) - 1)))/(12*d*e^(3*d*x + 3*c) + 12*I
*d*e^(2*d*x + 2*c) - 12*d*e^(d*x + c) - 12*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{5}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*csch(d*x+c))**(5/2),x)

[Out]

Integral((I*a*csch(c + d*x) + a)**(5/2), 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{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*csch(d*x+c))^(5/2),x, algorithm="giac")

[Out]

integrate((I*a*csch(d*x + c) + a)^(5/2), x)