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

Optimal. Leaf size=18 \[ \frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

Tan[a + b*Log[c*x^n]]/(b*n)

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Rubi [A]  time = 0.027744, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3767, 8} \[ \frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*Log[c*x^n]]^2/x,x]

[Out]

Tan[a + b*Log[c*x^n]]/(b*n)

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\sec ^2\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sec ^2(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\operatorname{Subst}\left (\int 1 \, dx,x,-\tan \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ &=\frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.0913382, size = 18, normalized size = 1. \[ \frac{\tan \left (a+b \log \left (c x^n\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tan[a + b*Log[c*x^n]]/(b*n)

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Maple [A]  time = 0.033, size = 19, normalized size = 1.1 \begin{align*}{\frac{\tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tan(a+b*ln(c*x^n))/b/n

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Maxima [B]  time = 1.16738, size = 223, normalized size = 12.39 \begin{align*} \frac{2 \,{\left (\cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )\right )}}{2 \, b n \cos \left (2 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) +{\left (b \cos \left (2 \, b \log \left (c\right )\right )^{2} + b \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )^{2} - 2 \, b n \sin \left (2 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) +{\left (b \cos \left (2 \, b \log \left (c\right )\right )^{2} + b \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )^{2} + b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(cos(2*b*log(x^n) + 2*a)*sin(2*b*log(c)) + cos(2*b*log(c))*sin(2*b*log(x^n) + 2*a))/(2*b*n*cos(2*b*log(c))*c
os(2*b*log(x^n) + 2*a) + (b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 - 2*b*n*sin(2
*b*log(c))*sin(2*b*log(x^n) + 2*a) + (b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*sin(2*b*log(x^n) + 2*a)^2 +
b*n)

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Fricas [A]  time = 0.467164, size = 93, normalized size = 5.17 \begin{align*} \frac{\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(b*n*log(x) + b*log(c) + a)/(b*n*cos(b*n*log(x) + b*log(c) + a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sec(a + b*log(c*x**n))**2/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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