Integrand size = 10, antiderivative size = 56 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 i \sqrt {a \text {csch}^3(x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) (i \sinh (x))^{3/2}-2 \cosh (x) \sqrt {a \text {csch}^3(x)} \sinh (x) \] Output:
-2*I*(a*csch(x)^3)^(1/2)*EllipticE(cos(1/4*Pi+1/2*I*x),2^(1/2))*(I*sinh(x) )^(3/2)-2*cosh(x)*(a*csch(x)^3)^(1/2)*sinh(x)
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \sqrt {a \text {csch}^3(x)} \left (\cosh (x)-E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right ) \sqrt {i \sinh (x)}\right ) \sinh (x) \] Input:
Integrate[Sqrt[a*Csch[x]^3],x]
Output:
-2*Sqrt[a*Csch[x]^3]*(Cosh[x] - EllipticE[(Pi - (2*I)*x)/4, 2]*Sqrt[I*Sinh [x]])*Sinh[x]
Time = 0.39 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 4611, 3042, 4255, 3042, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sqrt {a \text {csch}^3(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sqrt {i a \sec \left (\frac {\pi }{2}+i x\right )^3}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle \frac {\sqrt {a \text {csch}^3(x)} \int (i \text {csch}(x))^{3/2}dx}{(i \text {csch}(x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \text {csch}^3(x)} \int (-\csc (i x))^{3/2}dx}{(i \text {csch}(x))^{3/2}}\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \frac {\sqrt {a \text {csch}^3(x)} \left (-\int \frac {1}{\sqrt {i \text {csch}(x)}}dx-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )}{(i \text {csch}(x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \text {csch}^3(x)} \left (-\int \frac {1}{\sqrt {-\csc (i x)}}dx-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )}{(i \text {csch}(x))^{3/2}}\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \frac {\sqrt {a \text {csch}^3(x)} \left (-\frac {\int \sqrt {i \sinh (x)}dx}{\sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )}{(i \text {csch}(x))^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a \text {csch}^3(x)} \left (-\frac {\int \sqrt {\sin (i x)}dx}{\sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}-2 i \cosh (x) \sqrt {i \text {csch}(x)}\right )}{(i \text {csch}(x))^{3/2}}\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \frac {\sqrt {a \text {csch}^3(x)} \left (-2 i \cosh (x) \sqrt {i \text {csch}(x)}-\frac {2 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)} \sqrt {i \text {csch}(x)}}\right )}{(i \text {csch}(x))^{3/2}}\) |
Input:
Int[Sqrt[a*Csch[x]^3],x]
Output:
(Sqrt[a*Csch[x]^3]*((-2*I)*Cosh[x]*Sqrt[I*Csch[x]] - ((2*I)*EllipticE[Pi/4 - (I/2)*x, 2])/(Sqrt[I*Csch[x]]*Sqrt[I*Sinh[x]])))/(I*Csch[x])^(3/2)
Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
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] + Simp[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart [p])) Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
\[\int \sqrt {a \operatorname {csch}\left (x \right )^{3}}d x\]
Input:
int((a*csch(x)^3)^(1/2),x)
Output:
int((a*csch(x)^3)^(1/2),x)
Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \sqrt {a \text {csch}^3(x)} \, dx=-2 \, \sqrt {2} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \sqrt {2} \sqrt {a} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) \] Input:
integrate((a*csch(x)^3)^(1/2),x, algorithm="fricas")
Output:
-2*sqrt(2)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + s inh(x)^2 - 1))*(cosh(x) + sinh(x)) - 2*sqrt(2)*sqrt(a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cosh(x) + sinh(x)))
\[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int \sqrt {a \operatorname {csch}^{3}{\left (x \right )}}\, dx \] Input:
integrate((a*csch(x)**3)**(1/2),x)
Output:
Integral(sqrt(a*csch(x)**3), x)
\[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int { \sqrt {a \operatorname {csch}\left (x\right )^{3}} \,d x } \] Input:
integrate((a*csch(x)^3)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a*csch(x)^3), x)
\[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int { \sqrt {a \operatorname {csch}\left (x\right )^{3}} \,d x } \] Input:
integrate((a*csch(x)^3)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a*csch(x)^3), x)
Timed out. \[ \int \sqrt {a \text {csch}^3(x)} \, dx=\int \sqrt {\frac {a}{{\mathrm {sinh}\left (x\right )}^3}} \,d x \] Input:
int((a/sinh(x)^3)^(1/2),x)
Output:
int((a/sinh(x)^3)^(1/2), x)
\[ \int \sqrt {a \text {csch}^3(x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\mathrm {csch}\left (x \right )}\, \mathrm {csch}\left (x \right )d x \right ) \] Input:
int((a*csch(x)^3)^(1/2),x)
Output:
sqrt(a)*int(sqrt(csch(x))*csch(x),x)