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

Optimal. Leaf size=44 \[ \frac{a (e x)^n}{e n}+\frac{b x^{-n} (e x)^n \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )}{d e n} \]

[Out]

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

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Rubi [A]  time = 0.056476, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {14, 5440, 5436, 3770} \[ \frac{a (e x)^n}{e n}+\frac{b x^{-n} (e x)^n \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )}{d e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 3770

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

Rubi steps

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

Mathematica [A]  time = 0.0502048, size = 41, normalized size = 0.93 \[ \frac{x^{-n} (e x)^n \left (a \left (c+d x^n\right )+b \tan ^{-1}\left (\sinh \left (c+d x^n\right )\right )\right )}{d e n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.226, size = 155, normalized size = 3.5 \begin{align*}{\frac{ax}{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{b{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)),x)

[Out]

a/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/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)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.23672, size = 429, normalized size = 9.75 \begin{align*} \frac{a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) \cosh \left (n \log \left (x\right )\right ) + a d \cosh \left (n \log \left (x\right )\right ) \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + 2 \,{\left (b \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + b \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \arctan \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )\right ) +{\left (a d \cosh \left ({\left (n - 1\right )} \log \left (e\right )\right ) + a d \sinh \left ({\left (n - 1\right )} \log \left (e\right )\right )\right )} \sinh \left (n \log \left (x\right )\right )}{d n} \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)),x, algorithm="fricas")

[Out]

(a*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + a*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + 2*(b*cosh((n - 1)*log(e))
+ 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*si
nh(n*log(x)) + c)) + (a*d*cosh((n - 1)*log(e)) + a*d*sinh((n - 1)*log(e)))*sinh(n*log(x)))/(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)),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 )} \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)),x, algorithm="giac")

[Out]

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