3.82 \(\int \frac{(e x)^{-1+n}}{(a+b \text{sech}(c+d x^n))^2} \, dx\)
Optimal. Leaf size=157 \[ -\frac{2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )}{\sqrt{a+b}}\right )}{a^2 d e n (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a+b \text{sech}\left (c+d x^n\right )\right )}+\frac{(e x)^n}{a^2 e n} \]
[Out]
(e*x)^n/(a^2*e*n) - (2*b*(2*a^2 - b^2)*(e*x)^n*ArcTan[(Sqrt[a - b]*Tanh[(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*Tanh[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(a + b*Sech[c + d
*x^n]))
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Rubi [A] time = 0.298022, antiderivative size = 157, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used =
{5440, 5436, 3785, 3919, 3831, 2659, 208} \[ -\frac{2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )}{\sqrt{a+b}}\right )}{a^2 d e n (a-b)^{3/2} (a+b)^{3/2}}+\frac{b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{a d e n \left (a^2-b^2\right ) \left (a+b \text{sech}\left (c+d x^n\right )\right )}+\frac{(e x)^n}{a^2 e n} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^(-1 + n)/(a + b*Sech[c + d*x^n])^2,x]
[Out]
(e*x)^n/(a^2*e*n) - (2*b*(2*a^2 - b^2)*(e*x)^n*ArcTan[(Sqrt[a - b]*Tanh[(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*Tanh[c + d*x^n])/(a*(a^2 - b^2)*d*e*n*x^n*(a + b*Sech[c + d
*x^n]))
Rule 5440
Int[((e_)*(x_))^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[(e^IntPart[m]*(e*
x)^FracPart[m])/x^FracPart[m], Int[x^m*(a + b*Sech[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x]
Rule 5436
Int[(x_)^(m_.)*((a_.) + (b_.)*Sech[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Sech[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]
Rule 3785
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 3919
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 3831
Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[1/b, Int[1/(1 + (a*Sin[e
+ f*x])/b), x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]
Rule 2659
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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{(e x)^{-1+n}}{\left (a+b \text{sech}\left (c+d x^n\right )\right )^2} \, dx &=\frac{\left (x^{-n} (e x)^n\right ) \int \frac{x^{-1+n}}{\left (a+b \text{sech}\left (c+d x^n\right )\right )^2} \, dx}{e}\\ &=\frac{\left (x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int \frac{1}{(a+b \text{sech}(c+d x))^2} \, dx,x,x^n\right )}{e n}\\ &=\frac{b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text{sech}\left (c+d x^n\right )\right )}-\frac{\left (x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int \frac{-a^2+b^2+a b \text{sech}(c+d x)}{a+b \text{sech}(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 \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text{sech}\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 ) \operatorname{Subst}\left (\int \frac{\text{sech}(c+d x)}{a+b \text{sech}(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 \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text{sech}\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 ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a \cosh (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 \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text{sech}\left (c+d x^n\right )\right )}-\frac{\left (2 i \left (-a^2 b+b \left (-a^2+b^2\right )\right ) x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int \frac{1}{1+\frac{a}{b}+\left (1-\frac{a}{b}\right ) x^2} \, dx,x,i \tanh \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 \tan ^{-1}\left (\frac{\sqrt{a-b} \tanh \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 \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text{sech}\left (c+d x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.57379, size = 233, normalized size = 1.48 \[ \frac{x^{-n} (e x)^n \left (b \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+a b \sqrt{a^2-b^2} \sinh \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )}{\sqrt{a^2-b^2}}\right )\right )+a \cosh \left (c+d x^n\right ) \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \tan ^{-1}\left (\frac{(b-a) \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )}{\sqrt{a^2-b^2}}\right )\right )\right )}{a^2 d e n (a-b) (a+b) \sqrt{a^2-b^2} \left (a \cosh \left (c+d x^n\right )+b\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^(-1 + n)/(a + b*Sech[c + d*x^n])^2,x]
[Out]
((e*x)^n*(a*((a^2 - b^2)^(3/2)*(c + d*x^n) + (4*a^2*b - 2*b^3)*ArcTan[((-a + b)*Tanh[(c + d*x^n)/2])/Sqrt[a^2
- b^2]])*Cosh[c + d*x^n] + b*((a^2 - b^2)^(3/2)*(c + d*x^n) + (4*a^2*b - 2*b^3)*ArcTan[((-a + b)*Tanh[(c + d*x
^n)/2])/Sqrt[a^2 - b^2]] + a*b*Sqrt[a^2 - b^2]*Sinh[c + d*x^n])))/(a^2*(a - b)*(a + b)*Sqrt[a^2 - b^2]*d*e*n*x
^n*(b + a*Cosh[c + d*x^n]))
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Maple [C] time = 0.125, size = 491, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x)
[Out]
1/a^2/n*x*exp(-1/2*(-1+n)*(I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi-I*csgn(I*e)*csgn(I*e*x)^2*Pi-I*csgn(I*x)*csgn(
I*e*x)^2*Pi+I*csgn(I*e*x)^3*Pi-2*ln(e)-2*ln(x)))-2*b^2*exp(-1/2*(-1+n)*(I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi-I
*csgn(I*e)*csgn(I*e*x)^2*Pi-I*csgn(I*x)*csgn(I*e*x)^2*Pi+I*csgn(I*e*x)^3*Pi-2*ln(e)-2*ln(x)))*x*(b*exp(c+d*x^n
)+a)/a^2/(a^2-b^2)/d/n/(x^n)/(2*b*exp(c+d*x^n)+exp(2*c+2*d*x^n)*a+a)-2*b/a^2*(2*a^2-b^2)/(a^2-b^2)/n*exp(-1/2*
I*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*n*csgn(I*e)*csgn(I*e*x)^2)*exp(1/2*I*Pi*n*csgn(I*x)*csgn(
I*e*x)^2)*exp(-1/2*I*Pi*n*csgn(I*e*x)^3)*exp(1/2*I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(-1/2*I*Pi*csgn(I*e)
*csgn(I*e*x)^2)*exp(-1/2*I*Pi*csgn(I*x)*csgn(I*e*x)^2)*exp(1/2*I*Pi*csgn(I*e*x)^3)*e^n/e*exp(c)/d/(a^2*exp(2*c
)-exp(2*c)*b^2)^(1/2)*arctan(1/2*(2*a*exp(2*c+d*x^n)+2*exp(c)*b)/(a^2*exp(2*c)-exp(2*c)*b^2)^(1/2))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -2 \,{\left (2 \, a^{2} b e^{n} e^{c} - b^{3} e^{n} e^{c}\right )} \int \frac{e^{\left (d x^{n} + n \log \left (x\right )\right )}}{{\left (a^{5} e e^{\left (2 \, c\right )} - a^{3} b^{2} e e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d x^{n}\right )} + 2 \,{\left (a^{4} b e e^{c} - a^{2} b^{3} e e^{c}\right )} x e^{\left (d x^{n}\right )} +{\left (a^{5} e - a^{3} b^{2} e\right )} x}\,{d x} - \frac{2 \, a b^{2} e^{n} -{\left (a^{3} d e^{n} - a b^{2} d e^{n}\right )} x^{n} -{\left (a^{3} d e^{n} e^{\left (2 \, c\right )} - a b^{2} d e^{n} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n} + n \log \left (x\right )\right )} + 2 \,{\left (b^{3} e^{n} e^{c} -{\left (a^{2} b d e^{n} e^{c} - b^{3} d e^{n} e^{c}\right )} x^{n}\right )} e^{\left (d x^{n}\right )}}{a^{5} d e n - a^{3} b^{2} d e n +{\left (a^{5} d e n e^{\left (2 \, c\right )} - a^{3} b^{2} d e n e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n}\right )} + 2 \,{\left (a^{4} b d e n e^{c} - a^{2} b^{3} d e n e^{c}\right )} e^{\left (d x^{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x, algorithm="maxima")
[Out]
-2*(2*a^2*b*e^n*e^c - b^3*e^n*e^c)*integrate(e^(d*x^n + n*log(x))/((a^5*e*e^(2*c) - a^3*b^2*e*e^(2*c))*x*e^(2*
d*x^n) + 2*(a^4*b*e*e^c - a^2*b^3*e*e^c)*x*e^(d*x^n) + (a^5*e - a^3*b^2*e)*x), x) - (2*a*b^2*e^n - (a^3*d*e^n
- a*b^2*d*e^n)*x^n - (a^3*d*e^n*e^(2*c) - a*b^2*d*e^n*e^(2*c))*e^(2*d*x^n + n*log(x)) + 2*(b^3*e^n*e^c - (a^2*
b*d*e^n*e^c - b^3*d*e^n*e^c)*x^n)*e^(d*x^n))/(a^5*d*e*n - a^3*b^2*d*e*n + (a^5*d*e*n*e^(2*c) - a^3*b^2*d*e*n*e
^(2*c))*e^(2*d*x^n) + 2*(a^4*b*d*e*n*e^c - a^2*b^3*d*e*n*e^c)*e^(d*x^n))
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Fricas [B] time = 2.91469, size = 9543, normalized size = 60.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x, algorithm="fricas")
[Out]
[((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*
log(e))*cosh(n*log(x)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 +
a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh
(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5
- 2*a^3*b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e)
) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x))
+ c)^2 + 2*((a^4*b - 2*a^2*b^3 + b^5)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) - (a^2*b^3 - b^5)*cosh((n - 1)*lo
g(e)) - (a^2*b^3 - b^5 - (a^4*b - 2*a^2*b^3 + b^5)*d*cosh(n*log(x)))*sinh((n - 1)*log(e)) + ((a^4*b - 2*a^2*b^
3 + b^5)*d*cosh((n - 1)*log(e)) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cos
h(n*log(x)) + d*sinh(n*log(x)) + c) - 2*(a^3*b^2 - a*b^4)*cosh((n - 1)*log(e)) - (((2*a^3*b - a*b^3)*sqrt(-a^2
+ b^2)*cosh((n - 1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x))
+ d*sinh(n*log(x)) + c)^2 + ((2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(
-a^2 + b^2)*sinh((n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (2*a^3*b - a*b^3)*sqrt(-a^
2 + b^2)*cosh((n - 1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*sinh((n - 1)*log(e)) + 2*((2*a^2*b^2 - b^4)
*sqrt(-a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^2*b^2 - b^4)*sqrt(-a^2 + b^2)*sinh((n - 1)*log(e)))*cosh(d*cosh(
n*log(x)) + d*sinh(n*log(x)) + c) + 2*((2*a^2*b^2 - b^4)*sqrt(-a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^2*b^2 -
b^4)*sqrt(-a^2 + b^2)*sinh((n - 1)*log(e)) + ((2*a^3*b - a*b^3)*sqrt(-a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^3
*b - a*b^3)*sqrt(-a^2 + b^2)*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*sinh(d*cosh(
n*log(x)) + d*sinh(n*log(x)) + c))*log((a*b + (b^2 + sqrt(-a^2 + b^2)*b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(
x)) + c) + (a^2 - b^2 - sqrt(-a^2 + b^2)*b)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(-a^2 + b^2)*a
)/(a*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + b)) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d*cosh((n - 1)*log(e))
*cosh(n*log(x)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 - 2*a^3*b^2 + a*b^4)
*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2
+ a*b^4)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^2*b^3 - b^
5)*cosh((n - 1)*log(e)) - (a^2*b^3 - b^5 - (a^4*b - 2*a^2*b^3 + b^5)*d*cosh(n*log(x)))*sinh((n - 1)*log(e)) +
((a^4*b - 2*a^2*b^3 + b^5)*d*cosh((n - 1)*log(e)) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh((n - 1)*log(e)))*sinh(n*l
og(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (2*a^3*b^2 - 2*a*b^4 - (a^5 - 2*a^3*b^2 + a*b^4)*d*cos
h(n*log(x)))*sinh((n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b
^4)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*
log(x)) + c)^2 + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(a^6*b -
2*a^4*b^3 + a^2*b^5)*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*n + 2*(
(a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*
d*n)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)), ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*
log(x)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(
n*log(x))*sinh((n - 1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)
*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((a^5 - 2*a^3*b^2 +
a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n - 1)*log(e)
) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*log(e)))*sinh
(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*cosh((n -
1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x))
+ c)^2 + ((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*cosh((n - 1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*sinh((n -
1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*cosh((n - 1)*
log(e)) + (2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*sinh((n - 1)*log(e)) + 2*((2*a^2*b^2 - b^4)*sqrt(a^2 - b^2)*cosh((
n - 1)*log(e)) + (2*a^2*b^2 - b^4)*sqrt(a^2 - b^2)*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(
x)) + c) + 2*((2*a^2*b^2 - b^4)*sqrt(a^2 - b^2)*cosh((n - 1)*log(e)) + (2*a^2*b^2 - b^4)*sqrt(a^2 - b^2)*sinh(
(n - 1)*log(e)) + ((2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*cosh((n - 1)*log(e)) + (2*a^3*b - a*b^3)*sqrt(a^2 - b^2)*
sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x))
+ c))*arctan(-(sqrt(a^2 - b^2)*a*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(a^2 - b^2)*a*sinh(d*cosh
(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(a^2 - b^2)*b)/(a^2 - b^2)) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d*cosh((n
- 1)*log(e))*cosh(n*log(x)) - (a^2*b^3 - b^5)*cosh((n - 1)*log(e)) - (a^2*b^3 - b^5 - (a^4*b - 2*a^2*b^3 + b^5
)*d*cosh(n*log(x)))*sinh((n - 1)*log(e)) + ((a^4*b - 2*a^2*b^3 + b^5)*d*cosh((n - 1)*log(e)) + (a^4*b - 2*a^2*
b^3 + b^5)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 2*(a^3*b^2
- a*b^4)*cosh((n - 1)*log(e)) + 2*((a^4*b - 2*a^2*b^3 + b^5)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + ((a^5 - 2
*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n -
1)*log(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*log
(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^2*b^3 - b^5)*cosh((n - 1)*log(e)) - (
a^2*b^3 - b^5 - (a^4*b - 2*a^2*b^3 + b^5)*d*cosh(n*log(x)))*sinh((n - 1)*log(e)) + ((a^4*b - 2*a^2*b^3 + b^5)*
d*cosh((n - 1)*log(e)) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x
)) + d*sinh(n*log(x)) + c) - (2*a^3*b^2 - 2*a*b^4 - (a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n*log(x)))*sinh((n - 1)*l
og(e)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*log(e)))
*sinh(n*log(x)))/((a^7 - 2*a^5*b^2 + a^3*b^4)*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (a^7 - 2*a
^5*b^2 + a^3*b^4)*d*n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(a^6*b - 2*a^4*b^3 + a^2*b^5)*d*n*co
sh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^7 - 2*a^5*b^2 + a^3*b^4)*d*n + 2*((a^7 - 2*a^5*b^2 + a^3*b^4)
*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^6*b - 2*a^4*b^3 + a^2*b^5)*d*n)*sinh(d*cosh(n*log(x))
+ d*sinh(n*log(x)) + c))]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**(-1+n)/(a+b*sech(c+d*x**n))**2,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{n - 1}}{{\left (b \operatorname{sech}\left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+n)/(a+b*sech(c+d*x^n))^2,x, algorithm="giac")
[Out]
integrate((e*x)^(n - 1)/(b*sech(d*x^n + c) + a)^2, x)