\(\int (e x)^{-1+2 n} (a+b \text {sech}(c+d x^n)) \, dx\) [81]

Optimal result

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

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))
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.93 \[ \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {x^{-2 n} (e x)^{2 n} \left (a d^2 x^{2 n}+2 i b c \log \left (1-i e^{c+d x^n}\right )-b \pi \log \left (1-i e^{c+d x^n}\right )+2 i b d x^n \log \left (1-i e^{c+d x^n}\right )-2 i b c \log \left (1+i e^{c+d x^n}\right )+b \pi \log \left (1+i e^{c+d x^n}\right )-2 i b d x^n \log \left (1+i e^{c+d x^n}\right )-2 i b c \log \left (\cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )+b \pi \log \left (\cot \left (\frac {1}{4} \left (2 i c+\pi +2 i d x^n\right )\right )\right )-2 i b \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )+2 i b \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )\right )}{2 d^2 e n} \] Input:

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

Output:

((e*x)^(2*n)*(a*d^2*x^(2*n) + (2*I)*b*c*Log[1 - I*E^(c + d*x^n)] - b*Pi*Lo 
g[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))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2010, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a (e x)^{2 n}}{2 e n}+\frac {2 b x^{-n} (e x)^{2 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-i e^{d x^n+c}\right )}{d^2 e n}+\frac {i b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,i e^{d x^n+c}\right )}{d^2 e n}\)

Input:

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

Output:

(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))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

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]]
 

Maple [C] (warning: unable to verify)

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

Time = 1.51 (sec) , antiderivative size = 368, normalized size of antiderivative = 2.73

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

Input:

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

Output:

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*csg 
n(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*c 
sgn(I*e)*csgn(I*e*x)^2)*exp(I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*exp(-I*Pi*n*cs 
gn(I*e*x)^3)*exp(1/2*I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(-1/2*I*Pi*c 
sgn(I*e)*csgn(I*e*x)^2)*exp(-1/2*I*Pi*csgn(I*x)*csgn(I*e*x)^2)*exp(1/2*I*P 
i*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))-dilog(1-e 
xp(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 = 0.13 (sec) , antiderivative size = 664, normalized size of antiderivative = 4.92 \[ \int (e x)^{-1+2 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

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

Output:

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*cosh((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*sin 
h((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*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*cos 
h(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*co 
sh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - I*sinh(d*cosh(n*log(x)) +...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**(2*n)*(x**(2*n)*a + 2*int((x**(2*n)*sech(x**n*d + c))/x,x)*b*n))/(2*e* 
n)