3.15 \(\int (b \text {csch}(c+d x))^{3/2} \, dx\)

Optimal. Leaf size=84 \[ -\frac {2 b \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{d}-\frac {2 i b^2 E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)}} \]

[Out]

-2*b*cosh(d*x+c)*(b*csch(d*x+c))^(1/2)/d+2*I*b^2*(sin(1/2*I*c+1/4*Pi+1/2*I*d*x)^2)^(1/2)/sin(1/2*I*c+1/4*Pi+1/
2*I*d*x)*EllipticE(cos(1/2*I*c+1/4*Pi+1/2*I*d*x),2^(1/2))/d/(b*csch(d*x+c))^(1/2)/(I*sinh(d*x+c))^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3768, 3771, 2639} \[ -\frac {2 b \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{d}-\frac {2 i b^2 E\left (\left .\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {i \sinh (c+d x)} \sqrt {b \text {csch}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(b*Csch[c + d*x])^(3/2),x]

[Out]

(-2*b*Cosh[c + d*x]*Sqrt[b*Csch[c + d*x]])/d - ((2*I)*b^2*EllipticE[(I*c - Pi/2 + I*d*x)/2, 2])/(d*Sqrt[b*Csch
[c + d*x]]*Sqrt[I*Sinh[c + d*x]])

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 3768

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

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps

\begin {align*} \int (b \text {csch}(c+d x))^{3/2} \, dx &=-\frac {2 b \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{d}+b^2 \int \frac {1}{\sqrt {b \text {csch}(c+d x)}} \, dx\\ &=-\frac {2 b \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{d}+\frac {b^2 \int \sqrt {i \sinh (c+d x)} \, dx}{\sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}}\\ &=-\frac {2 b \cosh (c+d x) \sqrt {b \text {csch}(c+d x)}}{d}-\frac {2 i b^2 E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{d \sqrt {b \text {csch}(c+d x)} \sqrt {i \sinh (c+d x)}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 60, normalized size = 0.71 \[ -\frac {2 b \sqrt {b \text {csch}(c+d x)} \left (\cosh (c+d x)-\sqrt {i \sinh (c+d x)} E\left (\left .\frac {1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(b*Csch[c + d*x])^(3/2),x]

[Out]

(-2*b*Sqrt[b*Csch[c + d*x]]*(Cosh[c + d*x] - EllipticE[((-2*I)*c + Pi - (2*I)*d*x)/4, 2]*Sqrt[I*Sinh[c + d*x]]
))/d

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fricas [F]  time = 2.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \operatorname {csch}\left (d x + c\right )} b \operatorname {csch}\left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*csch(d*x + c))*b*csch(d*x + c), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*csch(d*x + c))^(3/2), x)

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maple [F]  time = 0.24, size = 0, normalized size = 0.00 \[ \int \left (b \,\mathrm {csch}\left (d x +c \right )\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \operatorname {csch}\left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*csch(d*x + c))^(3/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {b}{\mathrm {sinh}\left (c+d\,x\right )}\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/sinh(c + d*x))^(3/2),x)

[Out]

int((b/sinh(c + d*x))^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \operatorname {csch}{\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*csch(c + d*x))**(3/2), x)

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