\(\int (e x)^{-1+2 n} (a+b \sec (c+d x^n)) \, dx\) [73]

Optimal result

Integrand size = 22, antiderivative size = 149 \[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \arctan \left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n} \]

[Out]

1/2*a*(e*x)^(2*n)/e/n-2*I*b*(e*x)^(2*n)*arctan(exp(I*(c+d*x^n)))/d/e/n/(x^n)+I*b*(e*x)^(2*n)*polylog(2,-I*exp(
I*(c+d*x^n)))/d^2/e/n/(x^(2*n))-I*b*(e*x)^(2*n)*polylog(2,I*exp(I*(c+d*x^n)))/d^2/e/n/(x^(2*n))

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {14, 4293, 4289, 4266, 2317, 2438} \[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \arctan \left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-i e^{i \left (d x^n+c\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,i e^{i \left (d x^n+c\right )}\right )}{d^2 e n} \]

[In]

Int[(e*x)^(-1 + 2*n)*(a + b*Sec[c + d*x^n]),x]

[Out]

(a*(e*x)^(2*n))/(2*e*n) - ((2*I)*b*(e*x)^(2*n)*ArcTan[E^(I*(c + d*x^n))])/(d*e*n*x^n) + (I*b*(e*x)^(2*n)*PolyL
og[2, (-I)*E^(I*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - (I*b*(e*x)^(2*n)*PolyLog[2, I*E^(I*(c + d*x^n))])/(d^2*e*n*
x^(2*n))

Rule 14

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]]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 4266

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E
^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))],
x], x] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e,
f}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 4289

Int[(x_)^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(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]

Rule 4293

Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sec[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \int \left (a (e x)^{-1+2 n}+b (e x)^{-1+2 n} \sec \left (c+d x^n\right )\right ) \, dx \\ & = \frac {a (e x)^{2 n}}{2 e n}+b \int (e x)^{-1+2 n} \sec \left (c+d x^n\right ) \, dx \\ & = \frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \sec \left (c+d x^n\right ) \, dx}{e} \\ & = \frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int x \sec (c+d x) \, dx,x,x^n\right )}{e n} \\ & = \frac {a (e x)^{2 n}}{2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \arctan \left (e^{i \left (c+d x^n\right )}\right )}{d e n}-\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,x^n\right )}{d e n} \\ & = \frac {a (e x)^{2 n}}{2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \arctan \left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \left (c+d x^n\right )}\right )}{d^2 e n} \\ & = \frac {a (e x)^{2 n}}{2 e n}-\frac {2 i b x^{-n} (e x)^{2 n} \arctan \left (e^{i \left (c+d x^n\right )}\right )}{d e n}+\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-i e^{i \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,i e^{i \left (c+d x^n\right )}\right )}{d^2 e n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.87 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.26 \[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\frac {(e x)^{2 n} \cos \left (c+d x^n\right ) \left (a+\frac {b x^{-2 n} \left (\left (-2 c+\pi -2 d x^n\right ) \left (\log \left (1-i e^{-i \left (c+d x^n\right )}\right )-\log \left (1+i e^{-i \left (c+d x^n\right )}\right )\right )-(-2 c+\pi ) \log \left (\cot \left (\frac {1}{4} \left (2 c+\pi +2 d x^n\right )\right )\right )+2 i \left (\operatorname {PolyLog}\left (2,-i e^{-i \left (c+d x^n\right )}\right )-\operatorname {PolyLog}\left (2,i e^{-i \left (c+d x^n\right )}\right )\right )\right )}{d^2}\right ) \left (a+b \sec \left (c+d x^n\right )\right )}{2 e n \left (b+a \cos \left (c+d x^n\right )\right )} \]

[In]

Integrate[(e*x)^(-1 + 2*n)*(a + b*Sec[c + d*x^n]),x]

[Out]

((e*x)^(2*n)*Cos[c + d*x^n]*(a + (b*((-2*c + Pi - 2*d*x^n)*(Log[1 - I/E^(I*(c + d*x^n))] - Log[1 + I/E^(I*(c +
d*x^n))]) - (-2*c + Pi)*Log[Cot[(2*c + Pi + 2*d*x^n)/4]] + (2*I)*(PolyLog[2, (-I)/E^(I*(c + d*x^n))] - PolyLo
g[2, I/E^(I*(c + d*x^n))])))/(d^2*x^(2*n)))*(a + b*Sec[c + d*x^n]))/(2*e*n*(b + a*Cos[c + d*x^n]))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.93 (sec) , antiderivative size = 829, normalized size of antiderivative = 5.56

method	result	size
risch	\(\text {Expression too large to display}\)	\(829\)

[In]

int((e*x)^(2*n-1)*(a+b*sec(c+d*x^n)),x,method=_RETURNVERBOSE)

[Out]

I/e/n/d*b*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(e^n)^2*ln(1+exp(I*x^n*d)*(-exp(2*I*c))^(1/2))*(-exp(2*I*
c))^(1/2)*x^n*exp(-1/2*I*(2*Pi*n*csgn(I*e*x)^3-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*csgn(I*e*x)^2+2
*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-Pi*csgn(I*e*x)^3+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*csgn(I*e*x)^2+2
*c))-I/e/n/d*b*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(e^n)^2*ln(1-exp(I*x^n*d)*(-exp(2*I*c))^(1/2))*(-exp
(2*I*c))^(1/2)*x^n*exp(-1/2*I*(2*Pi*n*csgn(I*e*x)^3-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*csgn(I*e*x
)^2+2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-Pi*csgn(I*e*x)^3+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*csgn(I*e*x
)^2+2*c))+1/e/n/d^2*b*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(e^n)^2*(-exp(2*I*c))^(1/2)*dilog(1+exp(I*x^n
*d)*(-exp(2*I*c))^(1/2))*exp(-1/2*I*(2*Pi*n*csgn(I*e*x)^3-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x)*csgn
(I*e*x)^2+2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-Pi*csgn(I*e*x)^3+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x)*csgn
(I*e*x)^2+2*c))-1/e/n/d^2*b*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(e^n)^2*(-exp(2*I*c))^(1/2)*dilog(1-exp
(I*x^n*d)*(-exp(2*I*c))^(1/2))*exp(-1/2*I*(2*Pi*n*csgn(I*e*x)^3-2*Pi*n*csgn(I*e)*csgn(I*e*x)^2-2*Pi*n*csgn(I*x
)*csgn(I*e*x)^2+2*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)-Pi*csgn(I*e*x)^3+Pi*csgn(I*e)*csgn(I*e*x)^2+Pi*csgn(I*x
)*csgn(I*e*x)^2+2*c))+1/2*a/n*x*exp(1/2*(2*n-1)*(-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)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 470 vs. \(2 (133) = 266\).

Time = 0.30 (sec) , antiderivative size = 470, normalized size of antiderivative = 3.15 \[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\frac {a d^{2} e^{2 \, n - 1} x^{2 \, n} - b c e^{2 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) + b c e^{2 \, n - 1} \log \left (\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) - b c e^{2 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) + i \, \sin \left (d x^{n} + c\right ) + i\right ) + b c e^{2 \, n - 1} \log \left (-\cos \left (d x^{n} + c\right ) - i \, \sin \left (d x^{n} + c\right ) + i\right ) - i \, b e^{2 \, n - 1} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) - i \, b e^{2 \, n - 1} {\rm Li}_2\left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + i \, b e^{2 \, n - 1} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right )\right ) + i \, b e^{2 \, n - 1} {\rm Li}_2\left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right )\right ) + {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right ) + {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) + \sin \left (d x^{n} + c\right ) + 1\right ) - {\left (b d e^{2 \, n - 1} x^{n} + b c e^{2 \, n - 1}\right )} \log \left (-i \, \cos \left (d x^{n} + c\right ) - \sin \left (d x^{n} + c\right ) + 1\right )}{2 \, d^{2} n} \]

[In]

integrate((e*x)^(-1+2*n)*(a+b*sec(c+d*x^n)),x, algorithm="fricas")

[Out]

1/2*(a*d^2*e^(2*n - 1)*x^(2*n) - b*c*e^(2*n - 1)*log(cos(d*x^n + c) + I*sin(d*x^n + c) + I) + b*c*e^(2*n - 1)*
log(cos(d*x^n + c) - I*sin(d*x^n + c) + I) - b*c*e^(2*n - 1)*log(-cos(d*x^n + c) + I*sin(d*x^n + c) + I) + b*c
*e^(2*n - 1)*log(-cos(d*x^n + c) - I*sin(d*x^n + c) + I) - I*b*e^(2*n - 1)*dilog(I*cos(d*x^n + c) + sin(d*x^n
+ c)) - I*b*e^(2*n - 1)*dilog(I*cos(d*x^n + c) - sin(d*x^n + c)) + I*b*e^(2*n - 1)*dilog(-I*cos(d*x^n + c) + s
in(d*x^n + c)) + I*b*e^(2*n - 1)*dilog(-I*cos(d*x^n + c) - sin(d*x^n + c)) + (b*d*e^(2*n - 1)*x^n + b*c*e^(2*n
- 1))*log(I*cos(d*x^n + c) + sin(d*x^n + c) + 1) - (b*d*e^(2*n - 1)*x^n + b*c*e^(2*n - 1))*log(I*cos(d*x^n +
c) - sin(d*x^n + c) + 1) + (b*d*e^(2*n - 1)*x^n + b*c*e^(2*n - 1))*log(-I*cos(d*x^n + c) + sin(d*x^n + c) + 1)
- (b*d*e^(2*n - 1)*x^n + b*c*e^(2*n - 1))*log(-I*cos(d*x^n + c) - sin(d*x^n + c) + 1))/(d^2*n)

Sympy [F]

\[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{2 n - 1} \left (a + b \sec {\left (c + d x^{n} \right )}\right )\, dx \]

[In]

integrate((e*x)**(-1+2*n)*(a+b*sec(c+d*x**n)),x)

[Out]

Integral((e*x)**(2*n - 1)*(a + b*sec(c + d*x**n)), x)

Maxima [F]

\[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{2 \, n - 1} \,d x } \]

[In]

integrate((e*x)^(-1+2*n)*(a+b*sec(c+d*x^n)),x, algorithm="maxima")

[Out]

2*b*e^(2*n)*integrate((x^(2*n)*cos(2*d*x^n + 2*c)*cos(d*x^n + c) + x^(2*n)*sin(2*d*x^n + 2*c)*sin(d*x^n + c) +
x^(2*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) + 1/2*(e*x)^(2*n)*a/(e*n)

Giac [F]

\[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int { {\left (b \sec \left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{2 \, n - 1} \,d x } \]

[In]

integrate((e*x)^(-1+2*n)*(a+b*sec(c+d*x^n)),x, algorithm="giac")

[Out]

integrate((b*sec(d*x^n + c) + a)*(e*x)^(2*n - 1), x)

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+2 n} \left (a+b \sec \left (c+d x^n\right )\right ) \, dx=\int \left (a+\frac {b}{\cos \left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{2\,n-1} \,d x \]

[In]

int((a + b/cos(c + d*x^n))*(e*x)^(2*n - 1),x)

[Out]

int((a + b/cos(c + d*x^n))*(e*x)^(2*n - 1), x)