3.76 \(\int (e x)^{-1+n} (a+b \text{sech}(c+d x^n))^2 \, dx\)

Optimal. Leaf size=79 \[ \frac{a^2 (e x)^n}{e n}+\frac{2 a b x^{-n} (e x)^n \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )}{d e n}+\frac{b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{d e n} \]

[Out]

(a^2*(e*x)^n)/(e*n) + (2*a*b*(e*x)^n*ArcTan[Sinh[c + d*x^n]])/(d*e*n*x^n) + (b^2*(e*x)^n*Tanh[c + d*x^n])/(d*e
*n*x^n)

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Rubi [A]  time = 0.102068, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {5440, 5436, 3773, 3770, 3767, 8} \[ \frac{a^2 (e x)^n}{e n}+\frac{2 a b x^{-n} (e x)^n \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )}{d e n}+\frac{b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{d e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^2*(e*x)^n)/(e*n) + (2*a*b*(e*x)^n*ArcTan[Sinh[c + d*x^n]])/(d*e*n*x^n) + (b^2*(e*x)^n*Tanh[c + d*x^n])/(d*e
*n*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 3773

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^2, x_Symbol] :> Simp[a^2*x, x] + (Dist[2*a*b, Int[Csc[c + d*x], x],
x] + Dist[b^2, Int[Csc[c + d*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int (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 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 (a+b \text{sech}(c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^n}{e n}+\frac{\left (2 a b x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int \text{sech}(c+d x) \, dx,x,x^n\right )}{e n}+\frac{\left (b^2 x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int \text{sech}^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^n}{e n}+\frac{2 a b x^{-n} (e x)^n \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )}{d e n}+\frac{\left (i b^2 x^{-n} (e x)^n\right ) \operatorname{Subst}\left (\int 1 \, dx,x,-i \tanh \left (c+d x^n\right )\right )}{d e n}\\ &=\frac{a^2 (e x)^n}{e n}+\frac{2 a b x^{-n} (e x)^n \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )}{d e n}+\frac{b^2 x^{-n} (e x)^n \tanh \left (c+d x^n\right )}{d e n}\\ \end{align*}

Mathematica [A]  time = 0.201293, size = 57, normalized size = 0.72 \[ \frac{x^{-n} (e x)^n \left (a \left (a \left (c+d x^n\right )+2 b \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )\right )+b^2 \tanh \left (c+d x^n\right )\right )}{d e n} \]

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*(c + d*x^n) + 2*b*ArcTan[Sinh[c + d*x^n]]) + b^2*Tanh[c + d*x^n]))/(d*e*n*x^n)

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Maple [C]  time = 0.112, size = 271, normalized size = 3.4 \begin{align*}{\frac{{a}^{2}x}{n}{{\rm e}^{-{\frac{ \left ( -1+n \right ) \left ( i{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) \pi -i{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}\pi -i{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}\pi +i \left ({\it csgn} \left ( iex \right ) \right ) ^{3}\pi -2\,\ln \left ( e \right ) -2\,\ln \left ( x \right ) \right ) }{2}}}}}-2\,{\frac{x{b}^{2}{{\rm e}^{-1/2\, \left ( -1+n \right ) \left ( i{\it csgn} \left ( ie \right ){\it csgn} \left ( ix \right ){\it csgn} \left ( iex \right ) \pi -i{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}\pi -i{\it csgn} \left ( ix \right ) \left ({\it csgn} \left ( iex \right ) \right ) ^{2}\pi +i \left ({\it csgn} \left ( iex \right ) \right ) ^{3}\pi -2\,\ln \left ( e \right ) -2\,\ln \left ( x \right ) \right ) }}}{d{x}^{n}n \left ( 1+{{\rm e}^{2\,c+2\,d{x}^{n}}} \right ) }}+4\,{\frac{ba{e}^{n}\arctan \left ({{\rm e}^{c+d{x}^{n}}} \right ){{\rm e}^{-i/2\pi \,{\it csgn} \left ( iex \right ) \left ( -1+n \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ix \right ) \right ) \left ( -{\it csgn} \left ( iex \right ) +{\it csgn} \left ( ie \right ) \right ) }}}{ned}} \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]

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/d/n*x/(x^n)*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)))/(1+exp
(2*c+2*d*x^n))+4*b*a/n*e^n/e/d*arctan(exp(c+d*x^n))*exp(-1/2*I*Pi*csgn(I*e*x)*(-1+n)*(-csgn(I*e*x)+csgn(I*x))*
(-csgn(I*e*x)+csgn(I*e)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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]

Exception raised: ValueError

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Fricas [B]  time = 2.5153, size = 2187, normalized size = 27.68 \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^2*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^2*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + a^2*d*cosh(n*log(x))
*sinh((n - 1)*log(e)) + (a^2*d*cosh((n - 1)*log(e)) + a^2*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(
n*log(x)) + d*sinh(n*log(x)) + c)^2 - 2*b^2*cosh((n - 1)*log(e)) + 2*(a^2*d*cosh((n - 1)*log(e))*cosh(n*log(x)
) + a^2*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + (a^2*d*cosh((n - 1)*log(e)) + a^2*d*sinh((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) + (a^2
*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + a^2*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + (a^2*d*cosh((n - 1)*log(e
)) + a^2*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 4*((a*b*cos
h((n - 1)*log(e)) + a*b*sinh((n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + a*b*cosh((n -
1)*log(e)) + 2*(a*b*cosh((n - 1)*log(e)) + a*b*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) + (a*b*cosh((n - 1)*log(e)) + a*b*sinh((n - 1)*log(e)))*sin
h(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + a*b*sinh((n - 1)*log(e)))*arctan(cosh(d*cosh(n*log(x)) + d*sinh
(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) + (a^2*d*cosh(n*log(x)) - 2*b^2)*sinh((n - 1)
*log(e)) + (a^2*d*cosh((n - 1)*log(e)) + a^2*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))/(d*n*cosh(d*cosh(n*log(x)
) + d*sinh(n*log(x)) + c)^2 + 2*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c) + d*n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + d*n)

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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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{sech}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{n - 1}\,{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((b*sech(d*x^n + c) + a)^2*(e*x)^(n - 1), x)