3.1.74 \(\int (e x)^{-1+2 n} (a+b \text {sech}(c+d x^n)) \, dx\) [74]

Optimal. Leaf size=135 \[ \frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \text {ArcTan}\left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n} \]

[Out]

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

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Rubi [A]
time = 0.08, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {14, 5548, 5544, 4265, 2317, 2438} \begin {gather*} \frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \text {ArcTan}\left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{d x^n+c}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{d x^n+c}\right )}{d^2 e n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(e*x)^(2*n))/(2*e*n) + (2*b*(e*x)^(2*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x^n) - (I*b*(e*x)^(2*n)*PolyLog[2, (-
I)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (I*b*(e*x)^(2*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*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 2317

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 2438

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 4265

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 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 (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx &=\int \left (a (e x)^{-1+2 n}+b (e x)^{-1+2 n} \text {sech}\left (c+d x^n\right )\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+b \int (e x)^{-1+2 n} \text {sech}\left (c+d x^n\right ) \, dx\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \text {sech}\left (c+d x^n\right ) \, dx}{e}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {\left (b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int x \text {sech}(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \log \left (1+i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}+\frac {\left (i b x^{-2 n} (e x)^{2 n}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}\\ &=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \tan ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (-i e^{c+d x^n}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \text {Li}_2\left (i e^{c+d x^n}\right )}{d^2 e n}\\ \end {align*}

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Mathematica [A]
time = 0.13, size = 260, normalized size = 1.93 \begin {gather*} \frac {x^{-2 n} (e x)^{2 n} \left (a d^2 x^{2 n}+2 i b c \log \left (1-i e^{c+d x^n}\right )-b \pi \log \left (1-i e^{c+d x^n}\right )+2 i b d x^n \log \left (1-i e^{c+d x^n}\right )-2 i b c \log \left (1+i e^{c+d x^n}\right )+b \pi \log \left (1+i e^{c+d x^n}\right )-2 i b d x^n \log \left (1+i e^{c+d x^n}\right )-2 i b c \log \left (\cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )+b \pi \log \left (\cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )-2 i b \text {PolyLog}\left (2,-i e^{c+d x^n}\right )+2 i b \text {PolyLog}\left (2,i e^{c+d x^n}\right )\right )}{2 d^2 e n} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((e*x)^(2*n)*(a*d^2*x^(2*n) + (2*I)*b*c*Log[1 - I*E^(c + d*x^n)] - b*Pi*Log[1 - I*E^(c + d*x^n)] + (2*I)*b*d*x
^n*Log[1 - I*E^(c + d*x^n)] - (2*I)*b*c*Log[1 + I*E^(c + d*x^n)] + b*Pi*Log[1 + I*E^(c + d*x^n)] - (2*I)*b*d*x
^n*Log[1 + I*E^(c + d*x^n)] - (2*I)*b*c*Log[Cot[((2*I)*c + Pi + (2*I)*d*x^n)/4]] + b*Pi*Log[Cot[((2*I)*c + Pi
+ (2*I)*d*x^n)/4]] - (2*I)*b*PolyLog[2, (-I)*E^(c + d*x^n)] + (2*I)*b*PolyLog[2, I*E^(c + d*x^n)]))/(2*d^2*e*n
*x^(2*n))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 3.87, size = 368, normalized size = 2.73

method	result	size
risch	\(\frac {a x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{2 n}+\frac {2 b \,{\mathrm e}^{-i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )} {\mathrm e}^{i \pi n \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}} {\mathrm e}^{i \pi n \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}} {\mathrm e}^{-i \pi n \mathrm {csgn}\left (i e x \right )^{3}} {\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i e x \right )^{3}}{2}} e^{2 n} {\mathrm e}^{c} \left (-\frac {\sqrt {-{\mathrm e}^{2 c}}\, x^{n} d \left (\ln \left (1+{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )-\ln \left (1-{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )\right ) {\mathrm e}^{-2 c}}{2}-\frac {\sqrt {-{\mathrm e}^{2 c}}\, \left (\dilog \left (1+{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )-\dilog \left (1-{\mathrm e}^{d \,x^{n}} \sqrt {-{\mathrm e}^{2 c}}\right )\right ) {\mathrm e}^{-2 c}}{2}\right )}{e n \,d^{2}}\)	\(368\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n)),x,method=_RETURNVERBOSE)

[Out]

1/2*a/n*x*exp(1/2*(-1+2*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*b*exp(-I*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(I*Pi*n
*csgn(I*e)*csgn(I*e*x)^2)*exp(I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*exp(-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)^2/e*exp(c)/n/d^2*(-1/2*(-exp(2*c))^(1/2)*x^n*d*(ln(1+exp(d*x^n)*(-exp(2*c))^(1/2))-
ln(1-exp(d*x^n)*(-exp(2*c))^(1/2)))*exp(-2*c)-1/2*(-exp(2*c))^(1/2)*(dilog(1+exp(d*x^n)*(-exp(2*c))^(1/2))-dil
og(1-exp(d*x^n)*(-exp(2*c))^(1/2)))*exp(-2*c))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(2*n-1>0)', see `assume?` for m
ore details)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (124) = 248\).
time = 0.40, size = 584, normalized size = 4.33 \begin {gather*} \frac {{\left (a d^{2} \cosh \left (2 \, n - 1\right ) + a d^{2} \sinh \left (2 \, n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right )^{2} + 2 \, {\left (a d^{2} \cosh \left (2 \, n - 1\right ) + a d^{2} \sinh \left (2 \, n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) \sinh \left (n \log \left (x\right )\right ) + {\left (a d^{2} \cosh \left (2 \, n - 1\right ) + a d^{2} \sinh \left (2 \, n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )^{2} - 2 \, {\left (-i \, b \cosh \left (2 \, n - 1\right ) - i \, b \sinh \left (2 \, n - 1\right )\right )} {\rm Li}_2\left (i \, \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + i \, \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )\right ) - 2 \, {\left (i \, b \cosh \left (2 \, n - 1\right ) + i \, b \sinh \left (2 \, n - 1\right )\right )} {\rm Li}_2\left (-i \, \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - i \, \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )\right ) - 2 \, {\left (i \, b c \cosh \left (2 \, n - 1\right ) + i \, b c \sinh \left (2 \, n - 1\right )\right )} \log \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 ) + i\right ) - 2 \, {\left (-i \, b c \cosh \left (2 \, n - 1\right ) - i \, b c \sinh \left (2 \, n - 1\right )\right )} \log \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 ) - i\right ) - 2 \, {\left (i \, b c \cosh \left (2 \, n - 1\right ) + i \, b c \sinh \left (2 \, n - 1\right ) + {\left (i \, b d \cosh \left (2 \, n - 1\right ) + i \, b d \sinh \left (2 \, n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (i \, b d \cosh \left (2 \, n - 1\right ) + i \, b d \sinh \left (2 \, n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} \log \left (i \, \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + i \, \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 1\right ) - 2 \, {\left (-i \, b c \cosh \left (2 \, n - 1\right ) - i \, b c \sinh \left (2 \, n - 1\right ) + {\left (-i \, b d \cosh \left (2 \, n - 1\right ) - i \, b d \sinh \left (2 \, n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (-i \, b d \cosh \left (2 \, n - 1\right ) - i \, b d \sinh \left (2 \, n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} \log \left (-i \, \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - i \, \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 1\right )}{2 \, d^{2} n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^(-1+2*n)*(a+b*sech(c+d*x^n)),x, algorithm="fricas")

[Out]

1/2*((a*d^2*cosh(2*n - 1) + a*d^2*sinh(2*n - 1))*cosh(n*log(x))^2 + 2*(a*d^2*cosh(2*n - 1) + a*d^2*sinh(2*n -
1))*cosh(n*log(x))*sinh(n*log(x)) + (a*d^2*cosh(2*n - 1) + a*d^2*sinh(2*n - 1))*sinh(n*log(x))^2 - 2*(-I*b*cos
h(2*n - 1) - I*b*sinh(2*n - 1))*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*(I*b*cosh(2*n - 1) + I*b*sinh(2*n - 1))*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*(I*b*c*cosh(2*n - 1) + I*b*c*sinh(2*n -
1))*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) -
2*(-I*b*c*cosh(2*n - 1) - I*b*c*sinh(2*n - 1))*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) - 2*(I*b*c*cosh(2*n - 1) + I*b*c*sinh(2*n - 1) + (I*b*d*cosh(2*n - 1)
+ I*b*d*sinh(2*n - 1))*cosh(n*log(x)) + (I*b*d*cosh(2*n - 1) + I*b*d*sinh(2*n - 1))*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) - 2*(-I*b*c*c
osh(2*n - 1) - I*b*c*sinh(2*n - 1) + (-I*b*d*cosh(2*n - 1) - I*b*d*sinh(2*n - 1))*cosh(n*log(x)) + (-I*b*d*cos
h(2*n - 1) - I*b*d*sinh(2*n - 1))*sinh(n*log(x)))*log(-I*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - I*sin
h(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1))/(d^2*n)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((e*x)**(2*n - 1)*(a + b*sech(c + d*x**n)), 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+2*n)*(a+b*sech(c+d*x^n)),x, algorithm="giac")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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