Integrand size = 17, antiderivative size = 67 \[ \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.82 \[ \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {4 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )+2 \tan ^2\left (a+b \log \left (c x^n\right )\right )-\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]
-1/4*(4*Log[Cos[a + b*Log[c*x^n]]] + 2*Tan[a + b*Log[c*x^n]]^2 - Tan[a + b *Log[c*x^n]]^4)/(b*n)
Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3039, 3042, 3954, 3042, 3954, 3042, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \tan ^5\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \tan \left (a+b \log \left (c x^n\right )\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\int \tan ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\int \tan \left (a+b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\int \tan \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \tan \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {\tan ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}-\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b}}{n}\) |
(-(Log[Cos[a + b*Log[c*x^n]]]/b) - Tan[a + b*Log[c*x^n]]^2/(2*b) + Tan[a + b*Log[c*x^n]]^4/(4*b))/n
3.2.74.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.64 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85
| method | result | size |
| derivativedivides | \(\frac {\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}-\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}+\frac {\ln \left (1+{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )}{2}}{n b}\) | \(57\) |
| default | \(\frac {\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}-\frac {{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}+\frac {\ln \left (1+{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )}{2}}{n b}\) | \(57\) |
| parallelrisch | \(-\frac {-{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}+2 {\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}-2 \ln \left (1+{\tan \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )}{4 b n}\) | \(58\) |
| risch | \(-i \ln \left (x \right )+\frac {2 i a}{n b}+\frac {2 i \ln \left (c \right )}{n}+\frac {2 i \ln \left (x^{n}\right )}{n}-\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}+\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}+\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}-\frac {4 \left (x^{n}\right )^{2 i b} c^{2 i b} \left (c^{4 i b} \left (x^{n}\right )^{4 i b} {\mathrm e}^{-3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{6 i a}+c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{-2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{4 i a}+{\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}\right )}{b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}^{4}}-\frac {\ln \left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}+1\right )}{b n}\) | \(667\) |
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (63) = 126\).
Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.93 \[ \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {{\left (\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \frac {1}{2}\right ) + 4 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 2}{2 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} + 2 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n\right )}} \]
-1/2*((cos(2*b*n*log(x) + 2*b*log(c) + 2*a)^2 + 2*cos(2*b*n*log(x) + 2*b*l og(c) + 2*a) + 1)*log(1/2*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 1/2) + 4* cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + 2)/(b*n*cos(2*b*n*log(x) + 2*b*log( c) + 2*a)^2 + 2*b*n*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n)
Time = 3.92 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.22 \[ \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \log {\left (x \right )} \tan ^{5}{\left (a \right )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\left (x \right )} \tan ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\frac {\log {\left (\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{2 b n} + \frac {\tan ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b n} - \frac {\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{2 b n} & \text {otherwise} \end {cases} \]
Piecewise((log(x)*tan(a)**5, Eq(b, 0) & (Eq(b, 0) | Eq(n, 0))), (log(x)*ta n(a + b*log(c))**5, Eq(n, 0)), (log(tan(a + b*log(c*x**n))**2 + 1)/(2*b*n) + tan(a + b*log(c*x**n))**4/(4*b*n) - tan(a + b*log(c*x**n))**2/(2*b*n), True))
Leaf count of result is larger than twice the leaf count of optimal. 4466 vs. \(2 (63) = 126\).
Time = 0.31 (sec) , antiderivative size = 4466, normalized size of antiderivative = 66.66 \[ \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]
-1/2*(32*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*cos(6*b*log(x^n) + 6*a)^2 + 48*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 32*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + 32* (cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*sin(6*b*log(x^n) + 6*a)^2 + 48*(co s(4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^2 + 32*(cos(2 *b*log(c))^2 + sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 + 8*((cos(8*b* log(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n ) + 6*a) + (cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log( c)))*cos(4*b*log(x^n) + 4*a) + (cos(8*b*log(c))*cos(2*b*log(c)) + sin(8*b* log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + (cos(6*b*log(c))*sin(8* b*log(c)) - cos(8*b*log(c))*sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + (co s(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*sin(4*b*l og(x^n) + 4*a) + (cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2* b*log(c)))*sin(2*b*log(x^n) + 2*a))*cos(8*b*log(x^n) + 8*a) + 8*(10*(cos(6 *b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log( x^n) + 4*a) + 8*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b *log(c)))*cos(2*b*log(x^n) + 2*a) + 10*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 8*(cos(2*b*log( c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + cos(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) + 8*(10*(cos(4*b*log(c)...
Timed out. \[ \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]
Time = 32.62 (sec) , antiderivative size = 247, normalized size of antiderivative = 3.69 \[ \int \frac {\tan ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\ln \left (x\right )\,1{}\mathrm {i}+\frac {8}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}-\frac {4}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}+\frac {4}{b\,n\,\left (4\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+6\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+4\,{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+{\mathrm {e}}^{a\,8{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,8{}\mathrm {i}}+1\right )}-\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}{b\,n}-\frac {8}{b\,n\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+3\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+1\right )} \]
log(x)*1i + 8/(b*n*(2*exp(a*2i)*(c*x^n)^(b*2i) + exp(a*4i)*(c*x^n)^(b*4i) + 1)) - 4/(b*n*(exp(a*2i)*(c*x^n)^(b*2i) + 1)) + 4/(b*n*(4*exp(a*2i)*(c*x^ n)^(b*2i) + 6*exp(a*4i)*(c*x^n)^(b*4i) + 4*exp(a*6i)*(c*x^n)^(b*6i) + exp( a*8i)*(c*x^n)^(b*8i) + 1)) - log(exp(a*2i)*(c*x^n)^(b*2i) + 1)/(b*n) - 8/( b*n*(3*exp(a*2i)*(c*x^n)^(b*2i) + 3*exp(a*4i)*(c*x^n)^(b*4i) + exp(a*6i)*( c*x^n)^(b*6i) + 1))