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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 = 0.07, size = 55, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n}+\frac {\tan \left (a+b \log \left (c x^n\right )\right ) \sec \left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

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)

________________________________________________________________________________________

fricas [A]  time = 3.08, size = 100, normalized size = 1.82 \[ \frac {\cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} \log \left (\sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1\right ) - \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2} \log \left (-\sin \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1\right ) + 2 \, \sin \left (b n \log \relax (x) + b \log \relax (c) + a\right )}{4 \, b n \cos \left (b n \log \relax (x) + b \log \relax (c) + a\right )^{2}} \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec \left (b \log \left (c x^{n}\right ) + a\right )^{3}}{x}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.11, size = 64, normalized size = 1.16 \[ \frac {\sec \left (a +b \ln \left (c \,x^{n}\right )\right ) \tan \left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}+\frac {\ln \left (\sec \left (a +b \ln \left (c \,x^{n}\right )\right )+\tan \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 n b} \]

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

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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

________________________________________________________________________________________

mupad [B]  time = 6.30, size = 178, normalized size = 3.24 \[ \frac {\ln \left (-\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{x}\right )}{2\,b\,n}-\frac {\ln \left (\frac {1{}\mathrm {i}}{x}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}}{x}\right )}{2\,b\,n}+\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,2{}\mathrm {i}}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}-\frac {{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,1{}\mathrm {i}}\,1{}\mathrm {i}}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*cos(a + b*log(c*x^n))^3),x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

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)

________________________________________________________________________________________