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 = 0.06, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 3770

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

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 0.41, size = 122, normalized size = 2.77 \[ \frac {a d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) \cosh \left (n \log \relax (x)\right ) + a d \cosh \left (n \log \relax (x)\right ) \sinh \left ({\left (n - 1\right )} \log \relax (e)\right ) + 2 \, {\left (b \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + b \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \arctan \left (\cosh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \relax (x)\right ) + d \sinh \left (n \log \relax (x)\right ) + c\right )\right ) + {\left (a d \cosh \left ({\left (n - 1\right )} \log \relax (e)\right ) + a d \sinh \left ({\left (n - 1\right )} \log \relax (e)\right )\right )} \sinh \left (n \log \relax (x)\right )}{d n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{n - 1}\,{d x} \]

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)

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maple [C]  time = 0.73, size = 155, normalized size = 3.52 \[ \frac {a x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \relax (x )+2 \ln \relax (e )\right )}{2}}}{n}+\frac {2 \arctan \left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} b \,{\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d e n} \]

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*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)))+2*arctan(exp(c+d*x^n))/d/e*e^n/n*b*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]  time = 0.00, size = 0, normalized size = 0.00 \[ 2 \, b \int \frac {\left (e x\right )^{n - 1}}{e^{\left (d x^{n} + c\right )} + e^{\left (-d x^{n} - c\right )}}\,{d x} + \frac {\left (e x\right )^{n} a}{e n} \]

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]

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

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mupad [B]  time = 1.40, size = 110, normalized size = 2.50 \[ \frac {2\,\mathrm {atan}\left (\frac {b\,x\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,{\left (e\,x\right )}^{n-1}\,\sqrt {d^2\,n^2\,x^{2\,n}}}{d\,n\,x^n\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {d^2\,n^2\,x^{2\,n}}}+\frac {a\,x\,{\left (e\,x\right )}^{n-1}}{n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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]

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

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