\(\int \frac {(e x)^{-1+n}}{(a+b \sec (c+d x^n))^2} \, dx\) [81]

Optimal result

Integrand size = 22, antiderivative size = 157 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^n}{a^2 e n}-\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d e n}+\frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \sec \left (c+d x^n\right )\right )} \]

[Out]

(e*x)^n/a^2/e/n-2*b*(2*a^2-b^2)*(e*x)^n*arctanh((a-b)^(1/2)*tan(1/2*c+1/2*d*x^n)/(a+b)^(1/2))/a^2/(a-b)^(3/2)/
(a+b)^(3/2)/d/e/n/(x^n)+b^2*(e*x)^n*tan(c+d*x^n)/a/(a^2-b^2)/d/e/n/(x^n)/(a+b*sec(c+d*x^n))

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4293, 4289, 3870, 4004, 3916, 2738, 214} \[ \int \frac {(e x)^{-1+n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=-\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a^2 d e n (a-b)^{3/2} (a+b)^{3/2}}+\frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a+b \sec \left (c+d x^n\right )\right )}+\frac {(e x)^n}{a^2 e n} \]

[In]

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

[Out]

(e*x)^n/(a^2*e*n) - (2*b*(2*a^2 - b^2)*(e*x)^n*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x^n)/2])/Sqrt[a + b]])/(a^2*(a
- b)^(3/2)*(a + b)^(3/2)*d*e*n*x^n) + (b^2*(e*x)^n*Tan[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(a + b*Sec[c + d*x
^n]))

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 3870

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[c + d*x]*((a + b*Csc[c + d*x])^(n +
1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Dist[1/(a*(n + 1)*(a^2 - b^2)), Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^
2 - b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x]
&& NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 3916

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a/b)*Si
n[e + f*x]), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 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}& = \frac {\left (x^{-n} (e x)^n\right ) \int \frac {x^{-1+n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx}{e} \\ & = \frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{(a+b \sec (c+d x))^2} \, dx,x,x^n\right )}{e n} \\ & = \frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \sec \left (c+d x^n\right )\right )}-\frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {-a^2+b^2+a b \sec (c+d x)}{a+b \sec (c+d x)} \, dx,x,x^n\right )}{a \left (a^2-b^2\right ) e n} \\ & = \frac {(e x)^n}{a^2 e n}+\frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \sec \left (c+d x^n\right )\right )}+\frac {\left (\left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {\sec (c+d x)}{a+b \sec (c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2-b^2\right ) e n} \\ & = \frac {(e x)^n}{a^2 e n}+\frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \sec \left (c+d x^n\right )\right )}+\frac {\left (\left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a \cos (c+d x)}{b}} \, dx,x,x^n\right )}{a^2 b \left (a^2-b^2\right ) e n} \\ & = \frac {(e x)^n}{a^2 e n}+\frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \sec \left (c+d x^n\right )\right )}+\frac {\left (2 \left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}+\left (1-\frac {a}{b}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a^2 b \left (a^2-b^2\right ) d e n} \\ & = \frac {(e x)^n}{a^2 e n}-\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d e n}+\frac {b^2 x^{-n} (e x)^n \tan \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \sec \left (c+d x^n\right )\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.69 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.22 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\frac {x^{-n} (e x)^n \left (-2 b \left (-2 a^2+b^2\right ) \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right ) \left (b+a \cos \left (c+d x^n\right )\right )+\sqrt {a^2-b^2} \left (a \left (a^2-b^2\right ) \left (c+d x^n\right ) \cos \left (c+d x^n\right )+b \left (\left (a^2-b^2\right ) \left (c+d x^n\right )+a b \sin \left (c+d x^n\right )\right )\right )\right )}{a^2 (a-b) (a+b) \sqrt {a^2-b^2} d e n \left (b+a \cos \left (c+d x^n\right )\right )} \]

[In]

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

[Out]

((e*x)^n*(-2*b*(-2*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x^n)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x^n]) +
Sqrt[a^2 - b^2]*(a*(a^2 - b^2)*(c + d*x^n)*Cos[c + d*x^n] + b*((a^2 - b^2)*(c + d*x^n) + a*b*Sin[c + d*x^n])))
)/(a^2*(a - b)*(a + b)*Sqrt[a^2 - b^2]*d*e*n*x^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 3.

Time = 1.72 (sec) , antiderivative size = 638, normalized size of antiderivative = 4.06

method	result	size
risch	\(\frac {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}}}{a^{2} n}+\frac {2 i b^{2} e^{n} \left (-1\right )^{\frac {\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )}{2}} \left (b \,{\mathrm e}^{\frac {i \left (-\pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+\pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi n \operatorname {csgn}\left (i e x \right )^{3}-\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi \operatorname {csgn}\left (i e x \right )^{3}+2 d \,x^{n}+2 c \right )}{2}}+{\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e x \right ) \left (-\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) n +\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right ) n +\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) n -\operatorname {csgn}\left (i e x \right )^{2} n -\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )-\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+\operatorname {csgn}\left (i e x \right )^{2}\right )}{2}} a \right )}{a^{2} \left (a^{2}-b^{2}\right ) d n \left (a \,{\mathrm e}^{2 i \left (c +d \,x^{n}\right )}+2 b \,{\mathrm e}^{i \left (c +d \,x^{n}\right )}+a \right ) e}+\frac {2 i \arctan \left (\frac {2 a \,{\mathrm e}^{i \left (d \,x^{n}+2 c \right )}+2 \,{\mathrm e}^{i c} b}{2 \sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}}\right ) e^{n} \left (-2 a^{2}+b^{2}\right ) b \,{\mathrm e}^{\frac {i \left (-\pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+\pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi n \operatorname {csgn}\left (i e x \right )^{3}+\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )-\pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}-\pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}+\pi \operatorname {csgn}\left (i e x \right )^{3}+2 c \right )}{2}}}{\sqrt {a^{2} {\mathrm e}^{2 i c}-{\mathrm e}^{2 i c} b^{2}}\, d e n \left (-a^{2}+b^{2}\right ) a^{2}}\)	\(638\)

[In]

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

[Out]

1/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)*cs
gn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2*ln(x)+2*ln(e)))+2*I*b^2/a^2/(a^2-b^2)/d/n/(a*exp(2*I*(c+d*x^n))+2*b*exp(I*(c+
d*x^n))+a)*e^n*(-1)^(1/2*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*(b*exp(1/2*I*(-Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+
Pi*n*csgn(I*e)*csgn(I*e*x)^2+Pi*n*csgn(I*x)*csgn(I*e*x)^2-Pi*n*csgn(I*e*x)^3-Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csg
n(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e*x)^3+2*d*x^n+2*c))+exp(1/2*I*Pi*csgn(I*e*x)*(-csgn(I*e)*csgn(I*x)*n+csgn(I*e)
*csgn(I*e*x)*n+csgn(I*x)*csgn(I*e*x)*n-csgn(I*e*x)^2*n-csgn(I*e)*csgn(I*e*x)-csgn(I*x)*csgn(I*e*x)+csgn(I*e*x)
^2))*a)/e+2*I*arctan(1/2*(2*a*exp(I*(d*x^n+2*c))+2*exp(I*c)*b)/(a^2*exp(2*I*c)-exp(2*I*c)*b^2)^(1/2))/(a^2*exp
(2*I*c)-exp(2*I*c)*b^2)^(1/2)/d/e*e^n/n/(-a^2+b^2)*(-2*a^2+b^2)/a^2*b*exp(1/2*I*(-Pi*n*csgn(I*e)*csgn(I*x)*csg
n(I*e*x)+Pi*n*csgn(I*e)*csgn(I*e*x)^2+Pi*n*csgn(I*x)*csgn(I*e*x)^2-Pi*n*csgn(I*e*x)^3+Pi*csgn(I*e)*csgn(I*x)*c
sgn(I*e*x)-Pi*csgn(I*e)*csgn(I*e*x)^2-Pi*csgn(I*x)*csgn(I*e*x)^2+Pi*csgn(I*e*x)^3+2*c))

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 628, normalized size of antiderivative = 4.00 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\left [\frac {2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d e^{n - 1} x^{n} \cos \left (d x^{n} + c\right ) + 2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d e^{n - 1} x^{n} + 2 \, {\left (a^{3} b^{2} - a b^{4}\right )} e^{n - 1} \sin \left (d x^{n} + c\right ) + {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {a^{2} - b^{2}} e^{n - 1} \cos \left (d x^{n} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {a^{2} - b^{2}} e^{n - 1}\right )} \log \left (\frac {2 \, a b \cos \left (d x^{n} + c\right ) - {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{n} + c\right )^{2} + 2 \, a^{2} - b^{2} - 2 \, {\left (\sqrt {a^{2} - b^{2}} b \cos \left (d x^{n} + c\right ) + \sqrt {a^{2} - b^{2}} a\right )} \sin \left (d x^{n} + c\right )}{a^{2} \cos \left (d x^{n} + c\right )^{2} + 2 \, a b \cos \left (d x^{n} + c\right ) + b^{2}}\right )}{2 \, {\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d n \cos \left (d x^{n} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d n\right )}}, \frac {{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d e^{n - 1} x^{n} \cos \left (d x^{n} + c\right ) + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d e^{n - 1} x^{n} + {\left (a^{3} b^{2} - a b^{4}\right )} e^{n - 1} \sin \left (d x^{n} + c\right ) - {\left ({\left (2 \, a^{3} b - a b^{3}\right )} \sqrt {-a^{2} + b^{2}} e^{n - 1} \cos \left (d x^{n} + c\right ) + {\left (2 \, a^{2} b^{2} - b^{4}\right )} \sqrt {-a^{2} + b^{2}} e^{n - 1}\right )} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} b \cos \left (d x^{n} + c\right ) + \sqrt {-a^{2} + b^{2}} a}{{\left (a^{2} - b^{2}\right )} \sin \left (d x^{n} + c\right )}\right )}{{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d n \cos \left (d x^{n} + c\right ) + {\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d n}\right ] \]

[In]

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

[Out]

[1/2*(2*(a^5 - 2*a^3*b^2 + a*b^4)*d*e^(n - 1)*x^n*cos(d*x^n + c) + 2*(a^4*b - 2*a^2*b^3 + b^5)*d*e^(n - 1)*x^n
+ 2*(a^3*b^2 - a*b^4)*e^(n - 1)*sin(d*x^n + c) + ((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*e^(n - 1)*cos(d*x^n + c)
+ (2*a^2*b^2 - b^4)*sqrt(a^2 - b^2)*e^(n - 1))*log((2*a*b*cos(d*x^n + c) - (a^2 - 2*b^2)*cos(d*x^n + c)^2 + 2*
a^2 - b^2 - 2*(sqrt(a^2 - b^2)*b*cos(d*x^n + c) + sqrt(a^2 - b^2)*a)*sin(d*x^n + c))/(a^2*cos(d*x^n + c)^2 + 2
*a*b*cos(d*x^n + c) + b^2)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*cos(d*x^n + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d
*n), ((a^5 - 2*a^3*b^2 + a*b^4)*d*e^(n - 1)*x^n*cos(d*x^n + c) + (a^4*b - 2*a^2*b^3 + b^5)*d*e^(n - 1)*x^n + (
a^3*b^2 - a*b^4)*e^(n - 1)*sin(d*x^n + c) - ((2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*e^(n - 1)*cos(d*x^n + c) + (2*
a^2*b^2 - b^4)*sqrt(-a^2 + b^2)*e^(n - 1))*arctan(-(sqrt(-a^2 + b^2)*b*cos(d*x^n + c) + sqrt(-a^2 + b^2)*a)/((
a^2 - b^2)*sin(d*x^n + c))))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*cos(d*x^n + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d
*n)]

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

((a^4 - a^2*b^2)*d*e^n*x^n*cos(2*d*x^n + 2*c)^2 + 4*(a^2*b^2 - b^4)*d*e^n*x^n*cos(d*x^n + c)^2 + (a^4 - a^2*b^
2)*d*e^n*x^n*sin(2*d*x^n + 2*c)^2 + 2*a*b^3*e^n*sin(d*x^n + c) + 4*(a^2*b^2 - b^4)*d*e^n*x^n*sin(d*x^n + c)^2
+ 4*(a^3*b - a*b^3)*d*e^n*x^n*cos(d*x^n + c) + (a^4 - a^2*b^2)*d*e^n*x^n - 2*(a*b^3*e^n*sin(d*x^n + c) - 2*(a^
3*b - a*b^3)*d*e^n*x^n*cos(d*x^n + c) - (a^4 - a^2*b^2)*d*e^n*x^n)*cos(2*d*x^n + 2*c) - 2*((2*a^8*b - 3*a^6*b^
3 + a^4*b^5)*d*e^(n + 1)*n*cos(2*d*x^n + 2*c)^2*cos(c) + 4*(2*a^6*b^3 - 3*a^4*b^5 + a^2*b^7)*d*e^(n + 1)*n*cos
(d*x^n + c)^2*cos(c) + (2*a^8*b - 3*a^6*b^3 + a^4*b^5)*d*e^(n + 1)*n*cos(c)*sin(2*d*x^n + 2*c)^2 + 4*(2*a^7*b^
2 - 3*a^5*b^4 + a^3*b^6)*d*e^(n + 1)*n*cos(c)*sin(2*d*x^n + 2*c)*sin(d*x^n + c) + 4*(2*a^6*b^3 - 3*a^4*b^5 + a
^2*b^7)*d*e^(n + 1)*n*cos(c)*sin(d*x^n + c)^2 + 4*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6)*d*e^(n + 1)*n*cos(d*x^n +
c)*cos(c) + (2*a^8*b - 3*a^6*b^3 + a^4*b^5)*d*e^(n + 1)*n*cos(c) + 2*(2*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6)*d*e^
(n + 1)*n*cos(d*x^n + c)*cos(c) + (2*a^8*b - 3*a^6*b^3 + a^4*b^5)*d*e^(n + 1)*n*cos(c))*cos(2*d*x^n + 2*c))*in
tegrate((a^3*x^n*cos(2*d*x^n + 2*c)*cos(d*x^n) + a^3*x^n*sin(2*d*x^n + 2*c)*sin(d*x^n) + 2*(a^2*b - b^3)*x^n*c
os(d*x^n)^2*cos(c) + 2*(a^2*b - b^3)*x^n*cos(c)*sin(d*x^n)^2 + (a^3 - a*b^2)*x^n*cos(d*x^n) - (a*b^2*x^n*cos(d
*x^n)*cos(2*c) + a*b^2*x^n*sin(d*x^n)*sin(2*c))*cos(2*d*x^n) - (a*b^2*x^n*cos(2*c)*sin(d*x^n) - a*b^2*x^n*cos(
d*x^n)*sin(2*c))*sin(2*d*x^n))/(a^8*e*x*cos(2*d*x^n + 2*c)^2 + a^8*e*x*sin(2*d*x^n + 2*c)^2 + (a^4*b^4*cos(2*c
)^2 + a^4*b^4*sin(2*c)^2)*e*x*cos(2*d*x^n)^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*
b^4 + a^2*b^6)*sin(c)^2)*e*x*cos(d*x^n)^2 + 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*e*x*cos(d*x^n)*cos(c) + (a^4*b^4*c
os(2*c)^2 + a^4*b^4*sin(2*c)^2)*e*x*sin(2*d*x^n)^2 + 4*((a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 -
2*a^4*b^4 + a^2*b^6)*sin(c)^2)*e*x*sin(d*x^n)^2 - 4*(a^7*b - 2*a^5*b^3 + a^3*b^5)*e*x*sin(d*x^n)*sin(c) + (a^8
- 2*a^6*b^2 + a^4*b^4)*e*x - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))
*e*x*cos(d*x^n) + (a^6*b^2 - a^4*b^4)*e*x*cos(2*c) + 2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b
^5)*cos(2*c)*sin(c))*e*x*sin(d*x^n))*cos(2*d*x^n) - 2*(a^6*b^2*e*x*cos(2*d*x^n)*cos(2*c) - a^6*b^2*e*x*sin(2*d
*x^n)*sin(2*c) - 2*(a^7*b - a^5*b^3)*e*x*cos(d*x^n)*cos(c) + 2*(a^7*b - a^5*b^3)*e*x*sin(d*x^n)*sin(c) - (a^8
- a^6*b^2)*e*x)*cos(2*d*x^n + 2*c) + 2*(2*((a^5*b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*
sin(c))*e*x*cos(d*x^n) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*e*x*sin
(d*x^n) + (a^6*b^2 - a^4*b^4)*e*x*sin(2*c))*sin(2*d*x^n) - 2*(a^6*b^2*e*x*cos(2*c)*sin(2*d*x^n) + a^6*b^2*e*x*
cos(2*d*x^n)*sin(2*c) - 2*(a^7*b - a^5*b^3)*e*x*cos(c)*sin(d*x^n) - 2*(a^7*b - a^5*b^3)*e*x*cos(d*x^n)*sin(c))
*sin(2*d*x^n + 2*c)), x) - 2*((2*a^8*b - 3*a^6*b^3 + a^4*b^5)*d*e^(n + 1)*n*cos(2*d*x^n + 2*c)^2*sin(c) + 4*(2
*a^6*b^3 - 3*a^4*b^5 + a^2*b^7)*d*e^(n + 1)*n*cos(d*x^n + c)^2*sin(c) + (2*a^8*b - 3*a^6*b^3 + a^4*b^5)*d*e^(n
+ 1)*n*sin(2*d*x^n + 2*c)^2*sin(c) + 4*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6)*d*e^(n + 1)*n*sin(2*d*x^n + 2*c)*sin
(d*x^n + c)*sin(c) + 4*(2*a^6*b^3 - 3*a^4*b^5 + a^2*b^7)*d*e^(n + 1)*n*sin(d*x^n + c)^2*sin(c) + 4*(2*a^7*b^2
- 3*a^5*b^4 + a^3*b^6)*d*e^(n + 1)*n*cos(d*x^n + c)*sin(c) + (2*a^8*b - 3*a^6*b^3 + a^4*b^5)*d*e^(n + 1)*n*sin
(c) + 2*(2*(2*a^7*b^2 - 3*a^5*b^4 + a^3*b^6)*d*e^(n + 1)*n*cos(d*x^n + c)*sin(c) + (2*a^8*b - 3*a^6*b^3 + a^4*
b^5)*d*e^(n + 1)*n*sin(c))*cos(2*d*x^n + 2*c))*integrate((a^3*x^n*cos(d*x^n)*sin(2*d*x^n + 2*c) - a^3*x^n*cos(
2*d*x^n + 2*c)*sin(d*x^n) + 2*(a^2*b - b^3)*x^n*cos(d*x^n)^2*sin(c) + 2*(a^2*b - b^3)*x^n*sin(d*x^n)^2*sin(c)
- (a^3 - a*b^2)*x^n*sin(d*x^n) + (a*b^2*x^n*cos(2*c)*sin(d*x^n) - a*b^2*x^n*cos(d*x^n)*sin(2*c))*cos(2*d*x^n)
- (a*b^2*x^n*cos(d*x^n)*cos(2*c) + a*b^2*x^n*sin(d*x^n)*sin(2*c))*sin(2*d*x^n))/(a^8*e*x*cos(2*d*x^n + 2*c)^2
+ a^8*e*x*sin(2*d*x^n + 2*c)^2 + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*e*x*cos(2*d*x^n)^2 + 4*((a^6*b^2 -
2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*e*x*cos(d*x^n)^2 + 4*(a^7*b - 2*a^5*
b^3 + a^3*b^5)*e*x*cos(d*x^n)*cos(c) + (a^4*b^4*cos(2*c)^2 + a^4*b^4*sin(2*c)^2)*e*x*sin(2*d*x^n)^2 + 4*((a^6*
b^2 - 2*a^4*b^4 + a^2*b^6)*cos(c)^2 + (a^6*b^2 - 2*a^4*b^4 + a^2*b^6)*sin(c)^2)*e*x*sin(d*x^n)^2 - 4*(a^7*b -
2*a^5*b^3 + a^3*b^5)*e*x*sin(d*x^n)*sin(c) + (a^8 - 2*a^6*b^2 + a^4*b^4)*e*x - 2*(2*((a^5*b^3 - a^3*b^5)*cos(2
*c)*cos(c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*e*x*cos(d*x^n) + (a^6*b^2 - a^4*b^4)*e*x*cos(2*c) + 2*((a^5*
b^3 - a^3*b^5)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*e*x*sin(d*x^n))*cos(2*d*x^n) - 2*(a^6*b^
2*e*x*cos(2*d*x^n)*cos(2*c) - a^6*b^2*e*x*sin(2*d*x^n)*sin(2*c) - 2*(a^7*b - a^5*b^3)*e*x*cos(d*x^n)*cos(c) +
2*(a^7*b - a^5*b^3)*e*x*sin(d*x^n)*sin(c) - (a^8 - a^6*b^2)*e*x)*cos(2*d*x^n + 2*c) + 2*(2*((a^5*b^3 - a^3*b^5
)*cos(c)*sin(2*c) - (a^5*b^3 - a^3*b^5)*cos(2*c)*sin(c))*e*x*cos(d*x^n) - 2*((a^5*b^3 - a^3*b^5)*cos(2*c)*cos(
c) + (a^5*b^3 - a^3*b^5)*sin(2*c)*sin(c))*e*x*sin(d*x^n) + (a^6*b^2 - a^4*b^4)*e*x*sin(2*c))*sin(2*d*x^n) - 2*
(a^6*b^2*e*x*cos(2*c)*sin(2*d*x^n) + a^6*b^2*e*x*cos(2*d*x^n)*sin(2*c) - 2*(a^7*b - a^5*b^3)*e*x*cos(c)*sin(d*
x^n) - 2*(a^7*b - a^5*b^3)*e*x*cos(d*x^n)*sin(c))*sin(2*d*x^n + 2*c)), x) + 2*(a*b^3*e^n*cos(d*x^n + c) + a^2*
b^2*e^n + 2*(a^3*b - a*b^3)*d*e^n*x^n*sin(d*x^n + c))*sin(2*d*x^n + 2*c))/((a^6 - a^4*b^2)*d*e*n*cos(2*d*x^n +
2*c)^2 + 4*(a^4*b^2 - a^2*b^4)*d*e*n*cos(d*x^n + c)^2 + (a^6 - a^4*b^2)*d*e*n*sin(2*d*x^n + 2*c)^2 + 4*(a^5*b
- a^3*b^3)*d*e*n*sin(2*d*x^n + 2*c)*sin(d*x^n + c) + 4*(a^4*b^2 - a^2*b^4)*d*e*n*sin(d*x^n + c)^2 + 4*(a^5*b
- a^3*b^3)*d*e*n*cos(d*x^n + c) + (a^6 - a^4*b^2)*d*e*n + 2*(2*(a^5*b - a^3*b^3)*d*e*n*cos(d*x^n + c) + (a^6 -
a^4*b^2)*d*e*n)*cos(2*d*x^n + 2*c))

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 18.23 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.94 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \sec \left (c+d x^n\right )\right )^2} \, dx=\frac {\frac {b^2\,x\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}}{a\,d\,n\,x^n\,\left (a^2-b^2\right )}+\frac {b^3\,x\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x^n\,1{}\mathrm {i}}\,{\left (e\,x\right )}^{n-1}\,2{}\mathrm {i}}{a^2\,d\,n\,x^n\,\left (a^2-b^2\right )}}{a+a\,{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x^n\,2{}\mathrm {i}}+2\,b\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x^n\,1{}\mathrm {i}}}+\frac {x\,{\left (e\,x\right )}^{n-1}}{a^2\,n}+\frac {b\,x\,\ln \left (-2\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,\left (b^3\,x\,{\left (e\,x\right )}^{n-1}-2\,a^2\,b\,x\,{\left (e\,x\right )}^{n-1}\right )-\frac {b\,x\,\left (a^4-a^2\,b^2\right )\,\left (a+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\right )\,{\left (e\,x\right )}^{n-1}\,\left (2\,a^2-b^2\right )\,2{}\mathrm {i}}{a^2\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )\,{\left (e\,x\right )}^{n-1}\,\left (2\,a^2-b^2\right )}{a^2\,d\,n\,x^n\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {b\,x\,\ln \left (-2\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\,\left (b^3\,x\,{\left (e\,x\right )}^{n-1}-2\,a^2\,b\,x\,{\left (e\,x\right )}^{n-1}\right )+\frac {b\,x\,\left (a^4-a^2\,b^2\right )\,\left (a+b\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,{\mathrm {e}}^{d\,x^n\,1{}\mathrm {i}}\right )\,{\left (e\,x\right )}^{n-1}\,\left (2\,a^2-b^2\right )\,2{}\mathrm {i}}{a^2\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )\,{\left (e\,x\right )}^{n-1}\,\left (2\,a^2-b^2\right )}{a^2\,d\,n\,x^n\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}} \]

[In]

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

[Out]

((b^2*x*(e*x)^(n - 1)*2i)/(a*d*n*x^n*(a^2 - b^2)) + (b^3*x*exp(c*1i + d*x^n*1i)*(e*x)^(n - 1)*2i)/(a^2*d*n*x^n
*(a^2 - b^2)))/(a + a*exp(c*2i + d*x^n*2i) + 2*b*exp(c*1i + d*x^n*1i)) + (x*(e*x)^(n - 1))/(a^2*n) + (b*x*log(
- 2*exp(c*1i)*exp(d*x^n*1i)*(b^3*x*(e*x)^(n - 1) - 2*a^2*b*x*(e*x)^(n - 1)) - (b*x*(a^4 - a^2*b^2)*(a + b*exp(
c*1i)*exp(d*x^n*1i))*(e*x)^(n - 1)*(2*a^2 - b^2)*2i)/(a^2*(a + b)^(3/2)*(a - b)^(3/2)))*(e*x)^(n - 1)*(2*a^2 -
b^2))/(a^2*d*n*x^n*(a + b)^(3/2)*(a - b)^(3/2)) - (b*x*log((b*x*(a^4 - a^2*b^2)*(a + b*exp(c*1i)*exp(d*x^n*1i
))*(e*x)^(n - 1)*(2*a^2 - b^2)*2i)/(a^2*(a + b)^(3/2)*(a - b)^(3/2)) - 2*exp(c*1i)*exp(d*x^n*1i)*(b^3*x*(e*x)^
(n - 1) - 2*a^2*b*x*(e*x)^(n - 1)))*(e*x)^(n - 1)*(2*a^2 - b^2))/(a^2*d*n*x^n*(a + b)^(3/2)*(a - b)^(3/2))