3.4.19 \(\int \frac {1}{x \csc ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [319]

Optimal. Leaf size=98 \[ -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{5 b n} \]

[Out]

-2/5*cos(a+b*ln(c*x^n))/b/n/csc(a+b*ln(c*x^n))^(3/2)-6/5*(sin(1/2*a+1/4*Pi+1/2*b*ln(c*x^n))^2)^(1/2)/sin(1/2*a
+1/4*Pi+1/2*b*ln(c*x^n))*EllipticE(cos(1/2*a+1/4*Pi+1/2*b*ln(c*x^n)),2^(1/2))*csc(a+b*ln(c*x^n))^(1/2)*sin(a+b
*ln(c*x^n))^(1/2)/b/n

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Rubi [A]
time = 0.04, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3854, 3856, 2719} \begin {gather*} \frac {6 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{5 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Csc[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(-2*Cos[a + b*Log[c*x^n]])/(5*b*n*Csc[a + b*Log[c*x^n]]^(3/2)) + (6*Sqrt[Csc[a + b*Log[c*x^n]]]*EllipticE[(a -
Pi/2 + b*Log[c*x^n])/2, 2]*Sqrt[Sin[a + b*Log[c*x^n]]])/(5*b*n)

Rule 2719

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]

Rule 3854

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

Rule 3856

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 \frac {1}{x \csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\csc ^{\frac {5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {\csc (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\left (3 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int \sqrt {\sin (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{5 b n \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {6 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{5 b n}\\ \end {align*}

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Mathematica [A]
time = 0.23, size = 88, normalized size = 0.90 \begin {gather*} -\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (3 E\left (\left .\frac {1}{4} \left (-2 a+\pi -2 b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}+\cos \left (a+b \log \left (c x^n\right )\right ) \sin ^2\left (a+b \log \left (c x^n\right )\right )\right )}{5 b n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Csc[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(-2*Sqrt[Csc[a + b*Log[c*x^n]]]*(3*EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]] +
Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n]]^2))/(5*b*n)

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Maple [A]
time = 0.32, size = 205, normalized size = 2.09

method	result	size
derivativedivides	\(\frac {\frac {2 \left (\sin ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{5}-\frac {2 \left (\sin ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{5}-\frac {6 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticE \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{5}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(205\)
default	\(\frac {\frac {2 \left (\sin ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{5}-\frac {2 \left (\sin ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{5}-\frac {6 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticE \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{5}+\frac {3 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{5}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(205\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/csc(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/n*(2/5*sin(a+b*ln(c*x^n))^4-2/5*sin(a+b*ln(c*x^n))^2-6/5*(sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))
+2)^(1/2)*(-sin(a+b*ln(c*x^n)))^(1/2)*EllipticE((sin(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))+3/5*(sin(a+b*ln(c*x^
n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))+2)^(1/2)*(-sin(a+b*ln(c*x^n)))^(1/2)*EllipticF((sin(a+b*ln(c*x^n))+1)^(1/2
),1/2*2^(1/2)))/cos(a+b*ln(c*x^n))/sin(a+b*ln(c*x^n))^(1/2)/b

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(x*csc(b*log(c*x^n) + a)^(5/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.26, size = 130, normalized size = 1.33 \begin {gather*} \frac {3 \, \sqrt {2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + 3 \, \sqrt {-2 i} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + \frac {2 \, {\left (\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}}{\sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}}{5 \, b n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

1/5*(3*sqrt(2*I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*n*log(x) + b*log(c) + a) + I*sin(b*n*lo
g(x) + b*log(c) + a))) + 3*sqrt(-2*I)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*n*log(x) + b*log(c
) + a) - I*sin(b*n*log(x) + b*log(c) + a))) + 2*(cos(b*n*log(x) + b*log(c) + a)^3 - cos(b*n*log(x) + b*log(c)
+ a))/sqrt(sin(b*n*log(x) + b*log(c) + a)))/(b*n)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*ln(c*x**n))**(5/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5007 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

integrate(1/(x*csc(b*log(c*x^n) + a)^(5/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1/sin(a + b*log(c*x^n)))^(5/2)),x)

[Out]

int(1/(x*(1/sin(a + b*log(c*x^n)))^(5/2)), x)

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