3.1.82 \(\int \frac {(e x)^{-1+n}}{(a+b \text {sech}(c+d x^n))^2} \, dx\) [82]

Optimal. Leaf size=157 \[ \frac {(e x)^n}{a^2 e n}-\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \text {ArcTan}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a+b}}\right )}{a^2 (a-b)^{3/2} (a+b)^{3/2} d e n}+\frac {b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{a \left (a^2-b^2\right ) d e n \left (a+b \text {sech}\left (c+d x^n\right )\right )} \]

[Out]

(e*x)^n/a^2/e/n-2*b*(2*a^2-b^2)*(e*x)^n*arctan((a-b)^(1/2)*tanh(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*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 = 0.22, antiderivative size = 157, normalized size of antiderivative = 1.00, 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 = {5548, 5544, 3870, 4004, 3916, 2738, 214} \begin {gather*} -\frac {2 b \left (2 a^2-b^2\right ) x^{-n} (e x)^n \text {ArcTan}\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} \end {gather*}

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 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 5544

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 5548

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]

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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.44, size = 233, normalized size = 1.48 \begin {gather*} \frac {x^{-n} (e x)^n \left (a \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )\right ) \cosh \left (c+d x^n\right )+b \left (\left (a^2-b^2\right )^{3/2} \left (c+d x^n\right )+\left (4 a^2 b-2 b^3\right ) \text {ArcTan}\left (\frac {(-a+b) \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2-b^2}}\right )+a b \sqrt {a^2-b^2} \sinh \left (c+d x^n\right )\right )\right )}{a^2 (a-b) (a+b) \sqrt {a^2-b^2} d e n \left (b+a \cosh \left (c+d x^n\right )\right )} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 3.38, size = 491, normalized size = 3.13

method	result	size
risch	\(\frac {x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{a^{2} n}-\frac {2 b^{2} {\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}} x \left (b \,{\mathrm e}^{c +d \,x^{n}}+a \right ) x^{-n}}{a^{2} \left (a^{2}-b^{2}\right ) d n \left (a \,{\mathrm e}^{2 c +2 d \,x^{n}}+2 b \,{\mathrm e}^{c +d \,x^{n}}+a \right )}-\frac {2 b \left (2 a^{2}-b^{2}\right ) {\mathrm e}^{-\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi n \mathrm {csgn}\left (i e x \right )^{3}}{2}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i e x \right )^{3}}{2}} e^{n} {\mathrm e}^{c} \arctan \left (\frac {2 a \,{\mathrm e}^{2 c +d \,x^{n}}+2 \,{\mathrm e}^{c} b}{2 \sqrt {a^{2} {\mathrm e}^{2 c}-b^{2} {\mathrm e}^{2 c}}}\right )}{a^{2} \left (a^{2}-b^{2}\right ) n e d \sqrt {a^{2} {\mathrm e}^{2 c}-b^{2} {\mathrm e}^{2 c}}}\)	\(491\)

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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^(c + n) - b^3*e^(c + n))*integrate(e^(d*x^n + n*log(x))/((a^5 - a^3*b^2)*x*e + (a^5*e^(2*c) - a^
3*b^2*e^(2*c))*x*e^(2*d*x^n + 1) + 2*(a^4*b*e^c - a^2*b^3*e^c)*x*e^(d*x^n + 1)), x) - (2*a*b^2*e^n - (a^3*d*e^
n - a*b^2*d*e^n)*x^n - (a^3*d*e^(2*c + n) - a*b^2*d*e^(2*c + n))*e^(2*d*x^n + n*log(x)) + 2*(b^3*e^(c + n) - (
a^2*b*d*e^(c + n) - b^3*d*e^(c + n))*x^n)*e^(d*x^n))/((a^5*d*n - a^3*b^2*d*n)*e + (a^5*d*n*e^(2*c) - a^3*b^2*d
*n*e^(2*c))*e^(2*d*x^n + 1) + 2*(a^4*b*d*n*e^c - a^2*b^3*d*n*e^c)*e^(d*x^n + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1569 vs. \(2 (148) = 296\).
time = 0.43, size = 3169, normalized size = 20.18 \begin {gather*} \text {Too large to display} \end {gather*}

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) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*cosh(n*log(x)) + ((a^5
- 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*sinh(n*log(x)))*cosh(d*cosh(n*lo
g(x)) + d*sinh(n*log(x)) + c)^2 + (((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh
(n - 1))*cosh(n*log(x)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*
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)
+ (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(n - 1))*cosh(n*log(x)) - (a^2*b^3 - b^5)*cosh(n - 1) + ((a^4*b - 2*a^2*b^3
+ b^5)*d*cosh(n - 1) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(n - 1))*sinh(n*log(x)) - (a^2*b^3 - b^5)*sinh(n - 1))*
cosh(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) + (a^5 - 2*a^3*b^2 +
a*b^4)*d*sinh(n - 1))*cosh(n*log(x)) - 2*(a^3*b^2 - a*b^4)*cosh(n - 1) - (sqrt(-a^2 + b^2)*((2*a^3*b - a*b^3)*
cosh(n - 1) + (2*a^3*b - a*b^3)*sinh(n - 1))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + sqrt(-a^2 + b^2
)*((2*a^3*b - a*b^3)*cosh(n - 1) + (2*a^3*b - a*b^3)*sinh(n - 1))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c
)^2 + 2*sqrt(-a^2 + b^2)*((2*a^2*b^2 - b^4)*cosh(n - 1) + (2*a^2*b^2 - b^4)*sinh(n - 1))*cosh(d*cosh(n*log(x))
+ d*sinh(n*log(x)) + c) + 2*(sqrt(-a^2 + b^2)*((2*a^3*b - a*b^3)*cosh(n - 1) + (2*a^3*b - a*b^3)*sinh(n - 1))
*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(-a^2 + b^2)*((2*a^2*b^2 - b^4)*cosh(n - 1) + (2*a^2*b^2
- b^4)*sinh(n - 1)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(-a^2 + b^2)*((2*a^3*b - a*b^3)*cosh(
n - 1) + (2*a^3*b - a*b^3)*sinh(n - 1)))*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^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n - 1)
+ (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*cosh(n*log(x)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 -
2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((a^4*b - 2*
a^2*b^3 + b^5)*d*cosh(n - 1) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(n - 1))*cosh(n*log(x)) - (a^2*b^3 - b^5)*cosh(
n - 1) + ((a^4*b - 2*a^2*b^3 + b^5)*d*cosh(n - 1) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(n - 1))*sinh(n*log(x)) -
(a^2*b^3 - b^5)*sinh(n - 1))*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) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*sinh(n*log(x)) - 2*(a^3*b^2 - a*b^4)*sinh(n - 1))/((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*si
nh(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) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*cosh(n*log(x)) + ((a^5 -
2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*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) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(
n - 1))*cosh(n*log(x)) + ((a^5 - 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*s
inh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(sqrt(a^2 - b^2)*((2*a^3*b - a*b^3)*cosh(n
- 1) + (2*a^3*b - a*b^3)*sinh(n - 1))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + sqrt(a^2 - b^2)*((2*a^
3*b - a*b^3)*cosh(n - 1) + (2*a^3*b - a*b^3)*sinh(n - 1))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*
sqrt(a^2 - b^2)*((2*a^2*b^2 - b^4)*cosh(n - 1) + (2*a^2*b^2 - b^4)*sinh(n - 1))*cosh(d*cosh(n*log(x)) + d*sinh
(n*log(x)) + c) + 2*(sqrt(a^2 - b^2)*((2*a^3*b - a*b^3)*cosh(n - 1) + (2*a^3*b - a*b^3)*sinh(n - 1))*cosh(d*co
sh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(a^2 - b^2)*((2*a^2*b^2 - b^4)*cosh(n - 1) + (2*a^2*b^2 - b^4)*sinh
(n - 1)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(a^2 - b^2)*((2*a^3*b - a*b^3)*cosh(n - 1) + (2*
a^3*b - a*b^3)*sinh(n - 1)))*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) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(n - 1))*cosh(n*log(x)) - (a^2*b^3 - b^5)*cosh(n
- 1) + ((a^4*b - 2*a^2*b^3 + b^5)*d*cosh(n - 1) + (a^4*b - 2*a^2*b^3 + b^5)*d*sinh(n - 1))*sinh(n*log(x)) - (
a^2*b^3 - b^5)*sinh(n - 1))*cosh(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) + (a^5 - 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*cosh(n*log(x)) - 2*(a^3*b^2 - a*b^4)*cosh(n - 1) + 2*((((a^5
- 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 - 2*...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{n - 1}}{\left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \end {gather*}

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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (e\,x\right )}^{n-1}}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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