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

Optimal. Leaf size=55 \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

ArcTanh[Sin[a + b*Log[c*x^n]]]/(2*b*n) + (Sec[a + b*Log[c*x^n]]*Tan[a + b*Log[c*x^n]])/(2*b*n)

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Rubi [A]  time = 0.0384104, antiderivative size = 55, 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 = {3768, 3770} \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Sec[a + b*Log[c*x^n]]^3/x,x]

[Out]

ArcTanh[Sin[a + b*Log[c*x^n]]]/(2*b*n) + (Sec[a + b*Log[c*x^n]]*Tan[a + b*Log[c*x^n]])/(2*b*n)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
 1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*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 ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \sec ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac{\operatorname{Subst}\left (\int \sec (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{2 n}\\ &=\frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\sec \left (a+b \log \left (c x^n\right )\right ) \tan \left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end{align*}

Mathematica [A]  time = 0.0726142, size = 55, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac{\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTanh[Sin[a + b*Log[c*x^n]]]/(2*b*n) + (Sec[a + b*Log[c*x^n]]*Tan[a + b*Log[c*x^n]])/(2*b*n)

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Maple [A]  time = 0.039, size = 64, normalized size = 1.2 \begin{align*}{\frac{\sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \tan \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{2\,bn}}+{\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 ) }{2\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(a+b*ln(c*x^n))^3/x,x)

[Out]

1/2*sec(a+b*ln(c*x^n))*tan(a+b*ln(c*x^n))/b/n+1/2/b/n*ln(sec(a+b*ln(c*x^n))+tan(a+b*ln(c*x^n)))

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Maxima [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(sec(a+b*log(c*x^n))^3/x,x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.507579, size = 311, normalized size = 5.65 \begin{align*} \frac{\cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) - \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \log \left (-\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 1\right ) + 2 \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(cos(b*n*log(x) + b*log(c) + a)^2*log(sin(b*n*log(x) + b*log(c) + a) + 1) - cos(b*n*log(x) + b*log(c) + a)
^2*log(-sin(b*n*log(x) + b*log(c) + a) + 1) + 2*sin(b*n*log(x) + b*log(c) + a))/(b*n*cos(b*n*log(x) + b*log(c)
 + a)^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{3}{\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))**3/x,x)

[Out]

Integral(sec(a + b*log(c*x**n))**3/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 )^{3}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sec(b*log(c*x^n) + a)^3/x, x)