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

Optimal. Leaf size=19 \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

ArcTanh[Sin[a + b*Log[c*x^n]]]/(b*n)

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Rubi [A]  time = 0.0163493, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3770} \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*Log[c*x^n]]/x,x]

[Out]

ArcTanh[Sin[a + b*Log[c*x^n]]]/(b*n)

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.040198, size = 19, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[a + b*Log[c*x^n]]/x,x]

[Out]

ArcTanh[Sin[a + b*Log[c*x^n]]]/(b*n)

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Maple [A]  time = 0.022, size = 32, normalized size = 1.7 \begin{align*}{\frac{\ln \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +\tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07383, size = 42, normalized size = 2.21 \begin{align*} \frac{\log \left (\sec \left (b \log \left (c x^{n}\right ) + a\right ) + \tan \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 0.499456, size = 130, normalized size = 6.84 \begin{align*} \frac{\log \left (\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) - \log \left (-\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\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))/x,x, algorithm="fricas")

[Out]

1/2*(log(sin(b*n*log(x) + b*log(c) + a) + 1) - log(-sin(b*n*log(x) + b*log(c) + a) + 1))/(b*n)

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Sympy [A]  time = 4.00749, size = 51, normalized size = 2.68 \begin{align*} - \begin{cases} - \log{\left (x \right )} \sec{\left (a \right )} & \text{for}\: b = 0 \\- \log{\left (x \right )} \sec{\left (a + b \log{\left (c \right )} \right )} & \text{for}\: n = 0 \\- \frac{\log{\left (\tan{\left (a + b \log{\left (c x^{n} \right )} \right )} + \sec{\left (a + b \log{\left (c x^{n} \right )} \right )} \right )}}{b n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-log(x)*sec(a), Eq(b, 0)), (-log(x)*sec(a + b*log(c)), Eq(n, 0)), (-log(tan(a + b*log(c*x**n)) + s
ec(a + b*log(c*x**n)))/(b*n), True))

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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 )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*log(c*x^n) + a)/x, x)