Integrand size = 22, antiderivative size = 79 \[ \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^n}{e n}+\frac {2 a b x^{-n} (e x)^n \text {arctanh}\left (\sin \left (c+d x^n\right )\right )}{d e n}+\frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{d e n} \]
a^2*(e*x)^n/e/n+2*a*b*(e*x)^n*arctanh(sin(c+d*x^n))/d/e/n/(x^n)+b^2*(e*x)^ n*tan(c+d*x^n)/d/e/n/(x^n)
Time = 0.49 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.68 \[ \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx=\frac {x^{-n} (e x)^n \left (a^2 d x^n+2 a b \text {arctanh}\left (\sin \left (c+d x^n\right )\right )+b^2 \tan \left (c+d x^n\right )\right )}{d e n} \]
Time = 0.45 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.71, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4696, 4692, 3042, 4260, 3042, 4254, 24, 4257}
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.
\(\displaystyle \int (e x)^{n-1} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx\) |
\(\Big \downarrow \) 4696 |
\(\displaystyle \frac {x^{-n} (e x)^n \int x^{n-1} \left (a+b \sec \left (d x^n+c\right )\right )^2dx}{e}\) |
\(\Big \downarrow \) 4692 |
\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \sec \left (d x^n+c\right )\right )^2dx^n}{e n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x^{-n} (e x)^n \int \left (a+b \csc \left (d x^n+c+\frac {\pi }{2}\right )\right )^2dx^n}{e n}\) |
\(\Big \downarrow \) 4260 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (2 a b \int \sec \left (d x^n+c\right )dx^n+b^2 \int \sec ^2\left (d x^n+c\right )dx^n+a^2 x^n\right )}{e n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (2 a b \int \csc \left (d x^n+c+\frac {\pi }{2}\right )dx^n+b^2 \int \csc \left (d x^n+c+\frac {\pi }{2}\right )^2dx^n+a^2 x^n\right )}{e n}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (2 a b \int \csc \left (d x^n+c+\frac {\pi }{2}\right )dx^n-\frac {b^2 \int 1d\left (-\tan \left (d x^n+c\right )\right )}{d}+a^2 x^n\right )}{e n}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (2 a b \int \csc \left (d x^n+c+\frac {\pi }{2}\right )dx^n+a^2 x^n+\frac {b^2 \tan \left (c+d x^n\right )}{d}\right )}{e n}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle \frac {x^{-n} (e x)^n \left (a^2 x^n+\frac {2 a b \text {arctanh}\left (\sin \left (c+d x^n\right )\right )}{d}+\frac {b^2 \tan \left (c+d x^n\right )}{d}\right )}{e n}\) |
3.1.75.3.1 Defintions of rubi rules used
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Simp[2*a*b Int[Csc[c + d*x], x], x] + Simp[b^2 Int[Csc[c + d*x]^2, x] , x]) /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sec[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m + 1)/n], 0] && IntegerQ[p]
Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x _Symbol] :> Simp[e^IntPart[m]*((e*x)^FracPart[m]/x^FracPart[m]) Int[x^m*( a + b*Sec[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.02 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.49
method | result | size |
risch | \(\frac {a^{2} x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-i \pi \operatorname {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{n}+\frac {2 i x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-i \pi \operatorname {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}} b^{2} x^{-n}}{d n \left (1+{\mathrm e}^{2 i \left (c +d \,x^{n}\right )}\right )}-\frac {4 i \arctan \left ({\mathrm e}^{i \left (c +d \,x^{n}\right )}\right ) e^{n} a b \,{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right )\right ) \left (-\operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e \right )\right )}{2}}}{d e n}\) | \(276\) |
a^2/n*x*exp(1/2*(-1+n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*Pi*csgn(I* e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)*csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2*ln(x)+2 *ln(e)))+2*I*x*exp(1/2*(-1+n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*Pi* csgn(I*e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)*csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2* ln(x)+2*ln(e)))*b^2/d/n/(x^n)/(1+exp(2*I*(c+d*x^n)))-4*I*arctan(exp(I*(c+d *x^n)))/d/e*e^n/n*a*b*exp(1/2*I*Pi*csgn(I*e*x)*(-1+n)*(csgn(I*e*x)-csgn(I* x))*(-csgn(I*e*x)+csgn(I*e)))
Time = 0.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.43 \[ \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx=\frac {a^{2} d e^{n - 1} x^{n} \cos \left (d x^{n} + c\right ) + a b e^{n - 1} \cos \left (d x^{n} + c\right ) \log \left (\sin \left (d x^{n} + c\right ) + 1\right ) - a b e^{n - 1} \cos \left (d x^{n} + c\right ) \log \left (-\sin \left (d x^{n} + c\right ) + 1\right ) + b^{2} e^{n - 1} \sin \left (d x^{n} + c\right )}{d n \cos \left (d x^{n} + c\right )} \]
(a^2*d*e^(n - 1)*x^n*cos(d*x^n + c) + a*b*e^(n - 1)*cos(d*x^n + c)*log(sin (d*x^n + c) + 1) - a*b*e^(n - 1)*cos(d*x^n + c)*log(-sin(d*x^n + c) + 1) + b^2*e^(n - 1)*sin(d*x^n + c))/(d*n*cos(d*x^n + c))
\[ \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx=\int \left (e x\right )^{n - 1} \left (a + b \sec {\left (c + d x^{n} \right )}\right )^{2}\, dx \]
\[ \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{n - 1} \,d x } \]
(e*x)^n*a^2/(e*n) + 2*(b^2*e^n*sin(2*d*x^n + 2*c) + 2*(a*b*d*e^(n + 1)*n*c os(2*d*x^n + 2*c)^2 + a*b*d*e^(n + 1)*n*sin(2*d*x^n + 2*c)^2 + 2*a*b*d*e^( n + 1)*n*cos(2*d*x^n + 2*c) + a*b*d*e^(n + 1)*n)*integrate((x^n*cos(2*d*x^ n + 2*c)*cos(d*x^n + c) + x^n*sin(2*d*x^n + 2*c)*sin(d*x^n + c) + x^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*c os(2*d*x^n + 2*c) + e*x), x))/(d*e*n*cos(2*d*x^n + 2*c)^2 + d*e*n*sin(2*d* x^n + 2*c)^2 + 2*d*e*n*cos(2*d*x^n + 2*c) + d*e*n)
\[ \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \sec \left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{n - 1} \,d x } \]
Time = 15.51 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.28 \[ \int (e x)^{-1+n} \left (a+b \sec \left (c+d x^n\right )\right )^2 \, dx=\frac {a^2\,x\,{\left (e\,x\right )}^{n-1}}{n}+\frac {b^2\,x\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}}{d\,n\,x^n\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x^n\,2{}\mathrm {i}}+1\right )}+\frac {2\,a\,b\,x\,\ln \left (-a\,b\,{\left (e\,x\right )}^{n-1}\,4{}\mathrm {i}-4\,a\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\right )\,{\left (e\,x\right )}^{n-1}}{d\,n\,x^n}-\frac {2\,a\,b\,x\,\ln \left (a\,b\,{\left (e\,x\right )}^{n-1}\,4{}\mathrm {i}-4\,a\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\right )\,{\left (e\,x\right )}^{n-1}}{d\,n\,x^n} \]
(a^2*x*(e*x)^(n - 1))/n + (b^2*x*(e*x)^(n - 1)*2i)/(d*n*x^n*(exp(c*2i + d* x^n*2i) + 1)) + (2*a*b*x*log(- a*b*(e*x)^(n - 1)*4i - 4*a*b*exp(c*1i)*exp( d*x^n*1i)*(e*x)^(n - 1))*(e*x)^(n - 1))/(d*n*x^n) - (2*a*b*x*log(a*b*(e*x) ^(n - 1)*4i - 4*a*b*exp(c*1i)*exp(d*x^n*1i)*(e*x)^(n - 1))*(e*x)^(n - 1))/ (d*n*x^n)