3.317 \(\int \frac {1}{x \csc ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\)

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

[Out]

-2/3*cos(a+b*ln(c*x^n))/b/n/csc(a+b*ln(c*x^n))^(1/2)-2/3*(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))*EllipticF(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 = 0.06, 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 = {3769, 3771, 2641} \[ \frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x*Csc[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

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

Rule 2641

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

Rule 3769

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 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 \frac {1}{x \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\csc ^{\frac {3}{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 )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac {\operatorname {Subst}\left (\int \sqrt {\csc (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac {\left (\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 ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )}}+\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} F\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 )}}{3 b n}\\ \end {align*}

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Mathematica [A]  time = 0.16, size = 76, normalized size = 0.78 \[ -\frac {\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \left (\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right )\right |2\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Csc[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

-1/3*(Sqrt[Csc[a + b*Log[c*x^n]]]*(2*EllipticF[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2]*Sqrt[Sin[a + b*Log[c*x^n]]]
+ Sin[2*(a + b*Log[c*x^n])]))/(b*n)

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")

[Out]

integral(1/(x*csc(b*log(c*x^n) + a)^(3/2)), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

integrate(1/(x*csc(b*log(c*x^n) + a)^(3/2)), x)

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maple [A]  time = 0.17, size = 131, normalized size = 1.34 \[ \frac {\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )}{3}-\frac {2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right ) \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/csc(a+b*ln(c*x^n))^(3/2),x)

[Out]

1/n*(1/3*(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((s
in(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))-2/3*sin(a+b*ln(c*x^n))*cos(a+b*ln(c*x^n))^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 \[ \int \frac {1}{x \csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

integrate(1/(x*csc(b*log(c*x^n) + a)^(3/2)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(1/sin(a + b*log(c*x^n)))^(3/2)),x)

[Out]

int(1/(x*(1/sin(a + b*log(c*x^n)))^(3/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \csc ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/csc(a+b*ln(c*x**n))**(3/2),x)

[Out]

Integral(1/(x*csc(a + b*log(c*x**n))**(3/2)), x)

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