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

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

[Out]

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

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Rubi [A]  time = 0.0603745, 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, 2641} \[ \frac{2 \sin \left (a+b \log \left (c x^n\right )\right )}{3 b n \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )}}+\frac{2 \sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.133149, size = 72, normalized size = 0.77 \[ \frac{\sqrt{\sec \left (a+b \log \left (c x^n\right )\right )} \left (2 \sqrt{\cos \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{2} \left (a+b \log \left (c x^n\right )\right ),2\right )+\sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 2.337, size = 247, normalized size = 2.7 \begin{align*} -{\frac{2}{3\,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 ( 4\,\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}+\sqrt{ \left ( \sin \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( a/2+1/2\,b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \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))^(3/2),x)

[Out]

-2/3/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)*(4*cos(1/2*a+1/2*b*ln(c*x^n))*s
in(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)*(2*sin(1/2*a+1/2*b*ln(c*x^n))^2-1)^(1/2)*Elli
pticF(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{3}{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))^(3/2),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Timed out