3.313 \(\int \frac{\csc ^{\frac{5}{2}}(a+b \log (c x^n))}{x} \, dx\)

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

[Out]

(-2*Cos[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]]^(3/2))/(3*b*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)

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Rubi [A]  time = 0.0583563, antiderivative size = 98, normalized size of antiderivative = 1., 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 = {3768, 3771, 2641} \[ \frac{2 \sqrt{\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{3 b n}-\frac{2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(-2*Cos[a + b*Log[c*x^n]]*Csc[a + b*Log[c*x^n]]^(3/2))/(3*b*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 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]

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]

Rubi steps

\begin{align*} \int \frac{\csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \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 ) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\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 ) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\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 ) \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\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*}

Mathematica [A]  time = 0.186015, size = 73, normalized size = 0.74 \[ -\frac{2 \csc ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \left (\sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right ) \text{EllipticF}\left (\frac{1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right ),2\right )+\cos \left (a+b \log \left (c x^n\right )\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 1.118, size = 131, normalized size = 1.3 \begin{align*}{\frac{1}{3\,n\cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) b} \left ( \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 ) }{\it EllipticF} \left ( \sqrt{\sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +1},{\frac{\sqrt{2}}{2}} \right ) \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) -2\, \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \right ) \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3/n/sin(a+b*ln(c*x^n))^(3/2)*((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))*sin(a+b*ln(c*x^n))-2*cos(a+b*ln(c*x^n))^2)/cos(
a+b*ln(c*x^n))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError