Optimal. Leaf size=19 \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
[Out]
________________________________________________________________________________________
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]
[Out]
Rule 3770
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]