Integrand size = 22, antiderivative size = 235 \[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{3 n}}{3 e n}-\frac {2 i b x^{-n} (e x)^{3 n} \arctan \left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{i \left (c+d x^n\right )}\right )}{d^3 e n} \]
1/3*a*(e*x)^(3*n)/e/n-2*I*b*(e*x)^(3*n)*arctan(exp(I*(c+d*x^n)))/d/e/n/(x^ n)+2*I*b*(e*x)^(3*n)*polylog(2,-I*exp(I*(c+d*x^n)))/d^2/e/n/(x^(2*n))-2*I* b*(e*x)^(3*n)*polylog(2,I*exp(I*(c+d*x^n)))/d^2/e/n/(x^(2*n))-2*b*(e*x)^(3 *n)*polylog(3,-I*exp(I*(c+d*x^n)))/d^3/e/n/(x^(3*n))+2*b*(e*x)^(3*n)*polyl og(3,I*exp(I*(c+d*x^n)))/d^3/e/n/(x^(3*n))
\[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx \]
Time = 0.43 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2010, 2009}
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.
\(\displaystyle \int (e x)^{3 n-1} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 2010 |
\(\displaystyle \int \left (a (e x)^{3 n-1}+b (e x)^{3 n-1} \sec \left (c+d x^n\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a (e x)^{3 n}}{3 e n}-\frac {2 i b x^{-n} (e x)^{3 n} \arctan \left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{i \left (d x^n+c\right )}\right )}{d^3 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}\) |
(a*(e*x)^(3*n))/(3*e*n) - ((2*I)*b*(e*x)^(3*n)*ArcTan[E^(I*(c + d*x^n))])/ (d*e*n*x^n) + ((2*I)*b*(e*x)^(3*n)*PolyLog[2, (-I)*E^(I*(c + d*x^n))])/(d^ 2*e*n*x^(2*n)) - ((2*I)*b*(e*x)^(3*n)*PolyLog[2, I*E^(I*(c + d*x^n))])/(d^ 2*e*n*x^(2*n)) - (2*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^(I*(c + d*x^n))])/(d^3 *e*n*x^(3*n)) + (2*b*(e*x)^(3*n)*PolyLog[3, I*E^(I*(c + d*x^n))])/(d^3*e*n *x^(3*n))
3.1.74.3.1 Defintions of rubi rules used
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] , x] /; FreeQ[{c, m}, x] && SumQ[u] && !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
\[\int \left (e x \right )^{3 n -1} \left (a +b \sec \left (c +d \,x^{n}\right )\right )d x\]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 655 vs. \(2 (215) = 430\).
Time = 0.34 (sec) , antiderivative size = 655, normalized size of antiderivative = 2.79 \[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\frac {2 \, a d^{3} e^{3 \, n - 1} x^{3 \, n} - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) - 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + 6 i \, b d e^{3 \, n - 1} x^{n} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + 3 \, b c^{2} e^{3 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) - 3 \, b c^{2} e^{3 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) + 3 \, b c^{2} e^{3 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) - 3 \, b c^{2} e^{3 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) - 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + 6 \, b e^{3 \, n - 1} {\rm polylog}\left (3, -i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right ) + 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - 3 \, {\left (b d^{2} e^{3 \, n - 1} x^{2 \, n} - b c^{2} e^{3 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right )}{6 \, d^{3} n} \]
1/6*(2*a*d^3*e^(3*n - 1)*x^(3*n) - 6*I*b*d*e^(3*n - 1)*x^n*dilog(I*cos(d*x ^n + c) + sin(d*x^n + c)) - 6*I*b*d*e^(3*n - 1)*x^n*dilog(I*cos(d*x^n + c) - sin(d*x^n + c)) + 6*I*b*d*e^(3*n - 1)*x^n*dilog(-I*cos(d*x^n + c) + sin (d*x^n + c)) + 6*I*b*d*e^(3*n - 1)*x^n*dilog(-I*cos(d*x^n + c) - sin(d*x^n + c)) + 3*b*c^2*e^(3*n - 1)*log(cos(d*x^n + c) + I*sin(d*x^n + c) + I) - 3*b*c^2*e^(3*n - 1)*log(cos(d*x^n + c) - I*sin(d*x^n + c) + I) + 3*b*c^2*e ^(3*n - 1)*log(-cos(d*x^n + c) + I*sin(d*x^n + c) + I) - 3*b*c^2*e^(3*n - 1)*log(-cos(d*x^n + c) - I*sin(d*x^n + c) + I) - 6*b*e^(3*n - 1)*polylog(3 , I*cos(d*x^n + c) + sin(d*x^n + c)) + 6*b*e^(3*n - 1)*polylog(3, I*cos(d* x^n + c) - sin(d*x^n + c)) - 6*b*e^(3*n - 1)*polylog(3, -I*cos(d*x^n + c) + sin(d*x^n + c)) + 6*b*e^(3*n - 1)*polylog(3, -I*cos(d*x^n + c) - sin(d*x ^n + c)) + 3*(b*d^2*e^(3*n - 1)*x^(2*n) - b*c^2*e^(3*n - 1))*log(I*cos(d*x ^n + c) + sin(d*x^n + c) + 1) - 3*(b*d^2*e^(3*n - 1)*x^(2*n) - b*c^2*e^(3* n - 1))*log(I*cos(d*x^n + c) - sin(d*x^n + c) + 1) + 3*(b*d^2*e^(3*n - 1)* x^(2*n) - b*c^2*e^(3*n - 1))*log(-I*cos(d*x^n + c) + sin(d*x^n + c) + 1) - 3*(b*d^2*e^(3*n - 1)*x^(2*n) - b*c^2*e^(3*n - 1))*log(-I*cos(d*x^n + c) - sin(d*x^n + c) + 1))/(d^3*n)
\[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{3 n - 1} \left (a + b \sec {\left (c + d x^{n} \right )}\right )\, dx \]
\[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \]
2*b*e^(3*n)*integrate((x^(3*n)*cos(2*d*x^n + 2*c)*cos(d*x^n + c) + x^(3*n) *sin(2*d*x^n + 2*c)*sin(d*x^n + c) + x^(3*n)*cos(d*x^n + c))/(e*x*cos(2*d* x^n + 2*c)^2 + e*x*sin(2*d*x^n + 2*c)^2 + 2*e*x*cos(2*d*x^n + 2*c) + e*x), x) + 1/3*(e*x)^(3*n)*a/(e*n)
\[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \]
Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int \left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{3\,n-1} \,d x \]