3.277 \(\int \frac{1}{x \sec ^{\frac{5}{2}}(a+b \log (c x^n))} \, dx\)

Optimal. Leaf size=93 \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{6 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]

[Out]

(6*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(5*b*n) + (2*Si
n[a + b*Log[c*x^n]])/(5*b*n*Sec[a + b*Log[c*x^n]]^(3/2))

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Rubi [A]  time = 0.061194, antiderivative size = 93, 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 = {3769, 3771, 2639} \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{6 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{5 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2]*Sqrt[Sec[a + b*Log[c*x^n]]])/(5*b*n) + (2*Si
n[a + b*Log[c*x^n]])/(5*b*n*Sec[a + b*Log[c*x^n]]^(3/2))

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]

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]

Rubi steps

\begin{align*} \int \frac{1}{x \sec ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sec ^{\frac{5}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{\sec (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac{\left (3 \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int \sqrt{\cos (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{5 n}\\ &=\frac{6 \sqrt{\cos \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 )\right )\right |2\right ) \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}{5 b n}+\frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{5 b n \sec ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}

Mathematica [A]  time = 0.175315, size = 83, normalized size = 0.89 \[ \frac{\sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \left (\sin \left (a+b \log \left (c x^n\right )\right )+\sin \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+12 \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )\right )}{10 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[Sec[a + b*Log[c*x^n]]]*(12*Sqrt[Cos[a + b*Log[c*x^n]]]*EllipticE[(a + b*Log[c*x^n])/2, 2] + Sin[a + b*Lo
g[c*x^n]] + Sin[3*(a + b*Log[c*x^n])]))/(10*b*n)

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Maple [B]  time = 2.312, size = 280, normalized size = 3. \begin{align*} -{\frac{2}{5\,bn}\sqrt{ \left ( 2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}} \left ( -8\,\cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{6}+8\,\cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}-3\,\sqrt{2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{\it EllipticE} \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) ,\sqrt{2} \right ) -2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/5/n*((2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1)*sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-8*cos(1/2*a+1/2*b*ln(c*x^n))*
sin(1/2*a+1/2*b*ln(c*x^n))^6+8*cos(1/2*a+1/2*b*ln(c*x^n))*sin(1/2*a+1/2*b*ln(c*x^n))^4-3*(2*sin(1/2*a+1/2*b*ln
(c*x^n))^2-1)^(1/2)*(sin(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*EllipticE(cos(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))-2*sin(1
/2*a+1/2*b*ln(c*x^n))^2*cos(1/2*a+1/2*b*ln(c*x^n)))/(-2*sin(1/2*a+1/2*b*ln(c*x^n))^4+sin(1/2*a+1/2*b*ln(c*x^n)
)^2)^(1/2)/sin(1/2*a+1/2*b*ln(c*x^n))/(2*cos(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{x \sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(x*sec(b*log(c*x^n) + a)^(5/2)), 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(1/x/sec(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

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

[Out]

Timed out