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

Optimal result

Integrand size = 22, antiderivative size = 135 \[ \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n} \]

[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))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {14, 5548, 5544, 4265, 2317, 2438} \[ \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-i e^{d x^n+c}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,i e^{d x^n+c}\right )}{d^2 e n} \]

[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*} \text {integral}& = \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} \arctan \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} \arctan \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} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.61 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.93 \[ \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {x^{-2 n} (e x)^{2 n} \left (a d^2 x^{2 n}+2 i b c \log \left (1-i e^{c+d x^n}\right )-b \pi \log \left (1-i e^{c+d x^n}\right )+2 i b d x^n \log \left (1-i e^{c+d x^n}\right )-2 i b c \log \left (1+i e^{c+d x^n}\right )+b \pi \log \left (1+i e^{c+d x^n}\right )-2 i b d x^n \log \left (1+i e^{c+d x^n}\right )-2 i b c \log \left (\cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )+b \pi \log \left (\cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )-2 i b \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )+2 i b \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )\right )}{2 d^2 e n} \]

[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))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.70 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.73

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

[In]

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

[Out]

1/2*a/n*x*exp(1/2*(2*n-1)*(-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*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))-l
n(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))-dilo
g(1-exp(d*x^n)*(-exp(2*c))^(1/2)))*exp(-2*c))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 664 vs. \(2 (124) = 248\).

Time = 0.28 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.92 \[ \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\text {Too large to display} \]

[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)*log(e))*cosh(n*log(x))^2 + a*d^2*cosh(n*log(x))^2*sinh((2*n - 1)*log(e)) + (a*d^2*co
sh((2*n - 1)*log(e)) + a*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 - 2*(-I*b*cosh((2*n - 1)*log(e)) - I*b*s
inh((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*sin
h(n*log(x)) + c)) - 2*(I*b*cosh((2*n - 1)*log(e)) + I*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*(I*b*c*cosh((2*n - 1)*log(e))
+ I*b*c*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) - 2*(-I*b*c*cosh((2*n - 1)*log(e)) - I*b*c*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) - 2*(I*b*d*cosh((2*n - 1)
*log(e))*cosh(n*log(x)) + I*b*c*cosh((2*n - 1)*log(e)) + (I*b*d*cosh(n*log(x)) + I*b*c)*sinh((2*n - 1)*log(e))
+ (I*b*d*cosh((2*n - 1)*log(e)) + I*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) - 2*(-I*b*d*cosh((2*n - 1)*log(e
))*cosh(n*log(x)) - I*b*c*cosh((2*n - 1)*log(e)) + (-I*b*d*cosh(n*log(x)) - I*b*c)*sinh((2*n - 1)*log(e)) + (-
I*b*d*cosh((2*n - 1)*log(e)) - I*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) + 2*(a*d^2*cosh((2*n - 1)*log(e))*c
osh(n*log(x)) + a*d^2*cosh(n*log(x))*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))/(d^2*n)

Sympy [F]

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

[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)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int \left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{2\,n-1} \,d x \]

[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)