3.77 \(\int (e x)^{-1+2 n} (a+b \text {sech}(c+d x^n))^2 \, dx\)
Optimal. Leaf size=208 \[ \frac {a^2 (e x)^{2 n}}{2 e n}-\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{d x^n+c}\right )}{d^2 e n}+\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{d x^n+c}\right )}{d^2 e n}+\frac {4 a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cosh \left (c+d x^n\right )\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tanh \left (c+d x^n\right )}{d e n} \]
[Out]
1/2*a^2*(e*x)^(2*n)/e/n+4*a*b*(e*x)^(2*n)*arctan(exp(c+d*x^n))/d/e/n/(x^n)-b^2*(e*x)^(2*n)*ln(cosh(c+d*x^n))/d
^2/e/n/(x^(2*n))-2*I*a*b*(e*x)^(2*n)*polylog(2,-I*exp(c+d*x^n))/d^2/e/n/(x^(2*n))+2*I*a*b*(e*x)^(2*n)*polylog(
2,I*exp(c+d*x^n))/d^2/e/n/(x^(2*n))+b^2*(e*x)^(2*n)*tanh(c+d*x^n)/d/e/n/(x^n)
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Rubi [A] time = 0.21, antiderivative size = 208, normalized size of antiderivative = 1.00,
number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used
= {5440, 5436, 4190, 4180, 2279, 2391, 4184, 3475} \[ -\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac {a^2 (e x)^{2 n}}{2 e n}+\frac {4 a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cosh \left (c+d x^n\right )\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tanh \left (c+d x^n\right )}{d e n} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^(-1 + 2*n)*(a + b*Sech[c + d*x^n])^2,x]
[Out]
(a^2*(e*x)^(2*n))/(2*e*n) + (4*a*b*(e*x)^(2*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x^n) - (b^2*(e*x)^(2*n)*Log[Cosh[
c + d*x^n]])/(d^2*e*n*x^(2*n)) - ((2*I)*a*b*(e*x)^(2*n)*PolyLog[2, (-I)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + ((
2*I)*a*b*(e*x)^(2*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (b^2*(e*x)^(2*n)*Tanh[c + d*x^n])/(d*e*n
*x^n)
Rule 2279
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 2391
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 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rule 4180
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Rule 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 4190
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
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 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]
Rubi steps
\begin {align*} \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x (a+b \text {sech}(c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac {\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \left (a^2 x+2 a b x \text {sech}(c+d x)+b^2 x \text {sech}^2(c+d x)\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}+\frac {\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x \text {sech}(c+d x) \, dx,x,x^n\right )}{e n}+\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int x \text {sech}^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}+\frac {4 a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tanh \left (c+d x^n\right )}{d e n}-\frac {\left (2 i a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (2 i a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \log \left (1+i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}-\frac {\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \tanh (c+d x) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}+\frac {4 a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cosh \left (c+d x^n\right )\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tanh \left (c+d x^n\right )}{d e n}-\frac {\left (2 i a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}+\frac {\left (2 i a b x^{-2 n} (e x)^{2 n}\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}\\ &=\frac {a^2 (e x)^{2 n}}{2 e n}+\frac {4 a b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (\cosh \left (c+d x^n\right )\right )}{d^2 e n}-\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i a b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{2 n} \tanh \left (c+d x^n\right )}{d e n}\\ \end {align*}
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Mathematica [B] time = 2.88, size = 501, normalized size = 2.41 \[ \frac {\text {csch}^5(c) x^{-2 n} (e x)^{2 n} \text {sech}\left (c+d x^n\right ) \left (-a^2 d^2 \sqrt {-\text {csch}^2(c)} x^{2 n} \sinh \left (d x^n\right )+a^2 d^2 \sqrt {-\text {csch}^2(c)} x^{2 n} \sinh \left (2 c+d x^n\right )+8 a b \cosh \left (c+d x^n\right ) \text {Li}_2\left (-e^{-d x^n-\tanh ^{-1}(\coth (c))}\right )-8 a b \cosh \left (c+d x^n\right ) \text {Li}_2\left (e^{-d x^n-\tanh ^{-1}(\coth (c))}\right )+8 a b d x^n \cosh \left (c+d x^n\right ) \log \left (1-e^{-\tanh ^{-1}(\coth (c))-d x^n}\right )-8 a b d x^n \cosh \left (c+d x^n\right ) \log \left (e^{-\tanh ^{-1}(\coth (c))-d x^n}+1\right )+8 a b \tanh ^{-1}(\coth (c)) \cosh \left (c+d x^n\right ) \log \left (1-e^{-\tanh ^{-1}(\coth (c))-d x^n}\right )-8 a b \tanh ^{-1}(\coth (c)) \cosh \left (c+d x^n\right ) \log \left (e^{-\tanh ^{-1}(\coth (c))-d x^n}+1\right )+8 a b \sqrt {-\text {csch}^2(c)} \tanh ^{-1}(\coth (c)) \sinh \left (d x^n\right ) \tan ^{-1}\left (\cosh (c) \tanh \left (\frac {d x^n}{2}\right )+\sinh (c)\right )-8 a b \sqrt {-\text {csch}^2(c)} \tanh ^{-1}(\coth (c)) \sinh \left (2 c+d x^n\right ) \tan ^{-1}\left (\cosh (c) \tanh \left (\frac {d x^n}{2}\right )+\sinh (c)\right )-2 b^2 d \sqrt {-\text {csch}^2(c)} x^n \cosh \left (d x^n\right )+2 b^2 d \sqrt {-\text {csch}^2(c)} x^n \cosh \left (2 c+d x^n\right )+2 b^2 \sqrt {-\text {csch}^2(c)} \sinh \left (d x^n\right ) \log \left (\cosh \left (c+d x^n\right )\right )-2 b^2 \sqrt {-\text {csch}^2(c)} \sinh \left (2 c+d x^n\right ) \log \left (\cosh \left (c+d x^n\right )\right )\right )}{4 d^2 e n \left (-\text {csch}^2(c)\right )^{5/2}} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(e*x)^(-1 + 2*n)*(a + b*Sech[c + d*x^n])^2,x]
[Out]
((e*x)^(2*n)*Csch[c]^5*Sech[c + d*x^n]*(-2*b^2*d*x^n*Cosh[d*x^n]*Sqrt[-Csch[c]^2] + 2*b^2*d*x^n*Cosh[2*c + d*x
^n]*Sqrt[-Csch[c]^2] + 8*a*b*d*x^n*Cosh[c + d*x^n]*Log[1 - E^(-(d*x^n) - ArcTanh[Coth[c]])] + 8*a*b*ArcTanh[Co
th[c]]*Cosh[c + d*x^n]*Log[1 - E^(-(d*x^n) - ArcTanh[Coth[c]])] - 8*a*b*d*x^n*Cosh[c + d*x^n]*Log[1 + E^(-(d*x
^n) - ArcTanh[Coth[c]])] - 8*a*b*ArcTanh[Coth[c]]*Cosh[c + d*x^n]*Log[1 + E^(-(d*x^n) - ArcTanh[Coth[c]])] + 8
*a*b*Cosh[c + d*x^n]*PolyLog[2, -E^(-(d*x^n) - ArcTanh[Coth[c]])] - 8*a*b*Cosh[c + d*x^n]*PolyLog[2, E^(-(d*x^
n) - ArcTanh[Coth[c]])] - a^2*d^2*x^(2*n)*Sqrt[-Csch[c]^2]*Sinh[d*x^n] + 8*a*b*ArcTan[Sinh[c] + Cosh[c]*Tanh[(
d*x^n)/2]]*ArcTanh[Coth[c]]*Sqrt[-Csch[c]^2]*Sinh[d*x^n] + 2*b^2*Sqrt[-Csch[c]^2]*Log[Cosh[c + d*x^n]]*Sinh[d*
x^n] + a^2*d^2*x^(2*n)*Sqrt[-Csch[c]^2]*Sinh[2*c + d*x^n] - 8*a*b*ArcTan[Sinh[c] + Cosh[c]*Tanh[(d*x^n)/2]]*Ar
cTanh[Coth[c]]*Sqrt[-Csch[c]^2]*Sinh[2*c + d*x^n] - 2*b^2*Sqrt[-Csch[c]^2]*Log[Cosh[c + d*x^n]]*Sinh[2*c + d*x
^n]))/(4*d^2*e*n*x^(2*n)*(-Csch[c]^2)^(5/2))
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fricas [B] time = 0.54, size = 2964, normalized size = 14.25 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n))^2,x, algorithm="fricas")
[Out]
1/2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 + 4*b^2*c*cosh((2*n - 1)*log(e)) + (a^2*d^2*cosh((2*n - 1
)*log(e))*cosh(n*log(x))^2 + 4*b^2*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*b^2*c*cosh((2*n - 1)*log(e)) +
(a^2*d^2*cosh((2*n - 1)*log(e)) + a^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^2*cosh(n*log(x))^2
+ 4*b^2*d*cosh(n*log(x)) + 4*b^2*c)*sinh((2*n - 1)*log(e)) + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))
+ 2*b^2*d*cosh((2*n - 1)*log(e)) + (a^2*d^2*cosh(n*log(x)) + 2*b^2*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))
*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 + 4*b^2*
d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*b^2*c*cosh((2*n - 1)*log(e)) + (a^2*d^2*cosh((2*n - 1)*log(e)) + a
^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^2*cosh(n*log(x))^2 + 4*b^2*d*cosh(n*log(x)) + 4*b^2*c
)*sinh((2*n - 1)*log(e)) + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 2*b^2*d*cosh((2*n - 1)*log(e)) +
(a^2*d^2*cosh(n*log(x)) + 2*b^2*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*l
og(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 +
4*b^2*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*b^2*c*cosh((2*n - 1)*log(e)) + (a^2*d^2*cosh((2*n - 1)*log(
e)) + a^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^2*cosh(n*log(x))^2 + 4*b^2*d*cosh(n*log(x)) +
4*b^2*c)*sinh((2*n - 1)*log(e)) + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 2*b^2*d*cosh((2*n - 1)*lo
g(e)) + (a^2*d^2*cosh(n*log(x)) + 2*b^2*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*s
inh(n*log(x)) + c)^2 + (a^2*d^2*cosh((2*n - 1)*log(e)) + a^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + ((
4*I*a*b*cosh((2*n - 1)*log(e)) + 4*I*a*b*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)
^2 + 4*I*a*b*cosh((2*n - 1)*log(e)) + (8*I*a*b*cosh((2*n - 1)*log(e)) + 8*I*a*b*sinh((2*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) + (4*I*a*b*cosh((2*n - 1
)*log(e)) + 4*I*a*b*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 4*I*a*b*sinh((2*
n - 1)*log(e)))*dilog(I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + I*sinh(d*cosh(n*log(x)) + d*sinh(n*log
(x)) + c)) + ((-4*I*a*b*cosh((2*n - 1)*log(e)) - 4*I*a*b*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sin
h(n*log(x)) + c)^2 - 4*I*a*b*cosh((2*n - 1)*log(e)) + (-8*I*a*b*cosh((2*n - 1)*log(e)) - 8*I*a*b*sinh((2*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) + (-4*I
*a*b*cosh((2*n - 1)*log(e)) - 4*I*a*b*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2
- 4*I*a*b*sinh((2*n - 1)*log(e)))*dilog(-I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - I*sinh(d*cosh(n*log
(x)) + d*sinh(n*log(x)) + c)) - 2*(((2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (2*I*a*b*c + b^2)*sinh((2*n - 1
)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (2*
I*a*b*c + b^2)*sinh((2*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) + ((2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (2*I*a*b*c + b^2)*sinh((2*n - 1)*log(e)))*s
inh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (2*I*a*b*c + b^2)*
sinh((2*n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*l
og(x)) + c) + I) - 2*(((-2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (-2*I*a*b*c + b^2)*sinh((2*n - 1)*log(e)))*
cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((-2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (-2*I*a*b*c +
b^2)*sinh((2*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*l
og(x)) + c) + ((-2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (-2*I*a*b*c + b^2)*sinh((2*n - 1)*log(e)))*sinh(d*c
osh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (-2*I*a*b*c + b^2)*cosh((2*n - 1)*log(e)) + (-2*I*a*b*c + b^2)*sinh(
(2*n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)
) + c) - I) + (-4*I*a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*I*a*b*c*cosh((2*n - 1)*log(e)) + (-4*I*a*b
*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*I*a*b*c*cosh((2*n - 1)*log(e)) + (-4*I*a*b*d*cosh(n*log(x)) - 4*I
*a*b*c)*sinh((2*n - 1)*log(e)) + (-4*I*a*b*d*cosh((2*n - 1)*log(e)) - 4*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(n
*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (-8*I*a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x))
- 8*I*a*b*c*cosh((2*n - 1)*log(e)) + (-8*I*a*b*d*cosh(n*log(x)) - 8*I*a*b*c)*sinh((2*n - 1)*log(e)) + (-8*I*a*
b*d*cosh((2*n - 1)*log(e)) - 8*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(
n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (-4*I*a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x
)) - 4*I*a*b*c*cosh((2*n - 1)*log(e)) + (-4*I*a*b*d*cosh(n*log(x)) - 4*I*a*b*c)*sinh((2*n - 1)*log(e)) + (-4*I
*a*b*d*cosh((2*n - 1)*log(e)) - 4*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c)^2 + (-4*I*a*b*d*cosh(n*log(x)) - 4*I*a*b*c)*sinh((2*n - 1)*log(e)) + (-4*I*a*b*d*cosh((2*n -
1)*log(e)) - 4*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*log(I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x))
+ c) + I*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) + (4*I*a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)
) + 4*I*a*b*c*cosh((2*n - 1)*log(e)) + (4*I*a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*I*a*b*c*cosh((2*n
- 1)*log(e)) + (4*I*a*b*d*cosh(n*log(x)) + 4*I*a*b*c)*sinh((2*n - 1)*log(e)) + (4*I*a*b*d*cosh((2*n - 1)*log(e
)) + 4*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (8*I*
a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 8*I*a*b*c*cosh((2*n - 1)*log(e)) + (8*I*a*b*d*cosh(n*log(x)) + 8
*I*a*b*c)*sinh((2*n - 1)*log(e)) + (8*I*a*b*d*cosh((2*n - 1)*log(e)) + 8*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(
n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (4*I*
a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + 4*I*a*b*c*cosh((2*n - 1)*log(e)) + (4*I*a*b*d*cosh(n*log(x)) + 4
*I*a*b*c)*sinh((2*n - 1)*log(e)) + (4*I*a*b*d*cosh((2*n - 1)*log(e)) + 4*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(
n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + (4*I*a*b*d*cosh(n*log(x)) + 4*I*a*b*c)*sinh((2*n
- 1)*log(e)) + (4*I*a*b*d*cosh((2*n - 1)*log(e)) + 4*I*a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*log(-I*co
sh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - I*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) + (a^2*d^2*
cosh(n*log(x))^2 + 4*b^2*c)*sinh((2*n - 1)*log(e)) + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + a^2*d^
2*cosh(n*log(x))*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))/(d^2*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^
2 + 2*d^2*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + d^2*
n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + d^2*n)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{2 \, n - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n))^2,x, algorithm="giac")
[Out]
integrate((b*sech(d*x^n + c) + a)^2*(e*x)^(2*n - 1), x)
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maple [F] time = 0.58, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{-1+2 n} \left (a +b \,\mathrm {sech}\left (c +d \,x^{n}\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n))^2,x)
[Out]
int((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n))^2,x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ 4 \, a b e^{2 \, n} \int \frac {e^{\left (d x^{n} + 2 \, n \log \relax (x) + c\right )}}{e x e^{\left (2 \, d x^{n} + 2 \, c\right )} + e x}\,{d x} + b^{2} {\left (\frac {2 \, e^{2 \, n} e^{\left (2 \, d x^{n} + n \log \relax (x) + 2 \, c\right )}}{d e n e^{\left (2 \, d x^{n} + 2 \, c\right )} + d e n} - \frac {e^{2 \, n - 1} \log \left ({\left (e^{\left (2 \, d x^{n} + 2 \, c\right )} + 1\right )} e^{\left (-2 \, c\right )}\right )}{d^{2} n}\right )} + \frac {\left (e x\right )^{2 \, n} a^{2}}{2 \, e n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n))^2,x, algorithm="maxima")
[Out]
4*a*b*e^(2*n)*integrate(e^(d*x^n + 2*n*log(x) + c)/(e*x*e^(2*d*x^n + 2*c) + e*x), x) + b^2*(2*e^(2*n)*e^(2*d*x
^n + n*log(x) + 2*c)/(d*e*n*e^(2*d*x^n + 2*c) + d*e*n) - e^(2*n - 1)*log((e^(2*d*x^n + 2*c) + 1)*e^(-2*c))/(d^
2*n)) + 1/2*(e*x)^(2*n)*a^2/(e*n)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2\,{\left (e\,x\right )}^{2\,n-1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b/cosh(c + d*x^n))^2*(e*x)^(2*n - 1),x)
[Out]
int((a + b/cosh(c + d*x^n))^2*(e*x)^(2*n - 1), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{2 n - 1} \left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)**(-1+2*n)*(a+b*sech(c+d*x**n))**2,x)
[Out]
Integral((e*x)**(2*n - 1)*(a + b*sech(c + d*x**n))**2, x)
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