\(\int (e x)^{-1+3 n} (a+b \text {sech}(c+d x^n)) \, dx\) [82]

Optimal result

Integrand size = 22, antiderivative size = 217 \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{3 n}}{3 e n}+\frac {2 b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{c+d x^n}\right )}{d^3 e n}-\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{c+d x^n}\right )}{d^3 e n} \] Output:

1/3*a*(e*x)^(3*n)/e/n+2*b*(e*x)^(3*n)*arctan(exp(c+d*x^n))/d/e/n/(x^n)-2*I 
*b*(e*x)^(3*n)*polylog(2,-I*exp(c+d*x^n))/d^2/e/n/(x^(2*n))+2*I*b*(e*x)^(3 
*n)*polylog(2,I*exp(c+d*x^n))/d^2/e/n/(x^(2*n))+2*I*b*(e*x)^(3*n)*polylog( 
3,-I*exp(c+d*x^n))/d^3/e/n/(x^(3*n))-2*I*b*(e*x)^(3*n)*polylog(3,I*exp(c+d 
*x^n))/d^3/e/n/(x^(3*n))
 

Mathematica [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx \] Input:

Integrate[(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n]),x]
 

Output:

Integrate[(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n]), x]
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 217, 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.

\(\displaystyle \int (e x)^{3 n-1} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2010

\(\displaystyle \int \left (a (e x)^{3 n-1}+b (e x)^{3 n-1} \text {sech}\left (c+d x^n\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a (e x)^{3 n}}{3 e n}+\frac {2 b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}+\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{d x^n+c}\right )}{d^3 e n}-\frac {2 i b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{d x^n+c}\right )}{d^3 e n}-\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{d x^n+c}\right )}{d^2 e n}+\frac {2 i b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{d x^n+c}\right )}{d^2 e n}\)

Input:

Int[(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n]),x]
 

Output:

(a*(e*x)^(3*n))/(3*e*n) + (2*b*(e*x)^(3*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x 
^n) - ((2*I)*b*(e*x)^(3*n)*PolyLog[2, (-I)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n 
)) + ((2*I)*b*(e*x)^(3*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + 
 ((2*I)*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) - 
((2*I)*b*(e*x)^(3*n)*PolyLog[3, I*E^(c + d*x^n)])/(d^3*e*n*x^(3*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 [F]

\[\int \left (e x \right )^{-1+3 n} \left (a +b \,\operatorname {sech}\left (c +d \,x^{n}\right )\right )d x\]

Input:

int((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n)),x)
 

Output:

int((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n)),x)
 

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. 1082 vs. \(2 (200) = 400\).

Time = 0.13 (sec) , antiderivative size = 1082, normalized size of antiderivative = 4.99 \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n)),x, algorithm="fricas")
 

Output:

1/3*(a*d^3*cosh((3*n - 1)*log(e))*cosh(n*log(x))^3 + a*d^3*cosh(n*log(x))^ 
3*sinh((3*n - 1)*log(e)) + (a*d^3*cosh((3*n - 1)*log(e)) + a*d^3*sinh((3*n 
 - 1)*log(e)))*sinh(n*log(x))^3 + 3*(a*d^3*cosh((3*n - 1)*log(e))*cosh(n*l 
og(x)) + a*d^3*cosh(n*log(x))*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 - 6 
*(-I*b*d*cosh((3*n - 1)*log(e))*cosh(n*log(x)) - I*b*d*cosh(n*log(x))*sinh 
((3*n - 1)*log(e)) + (-I*b*d*cosh((3*n - 1)*log(e)) - I*b*d*sinh((3*n - 1) 
*log(e)))*sinh(n*log(x)))*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)) - 6*(I*b*d*cosh(( 
3*n - 1)*log(e))*cosh(n*log(x)) + I*b*d*cosh(n*log(x))*sinh((3*n - 1)*log( 
e)) + (I*b*d*cosh((3*n - 1)*log(e)) + I*b*d*sinh((3*n - 1)*log(e)))*sinh(n 
*log(x)))*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)) - 3*(-I*b*c^2*cosh((3*n - 1)*log 
(e)) - I*b*c^2*sinh((3*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) - 3*(I 
*b*c^2*cosh((3*n - 1)*log(e)) + I*b*c^2*sinh((3*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) - 3*(I*b*d^2*cosh((3*n - 1)*log(e))*cosh(n*log(x))^2 - 
I*b*c^2*cosh((3*n - 1)*log(e)) + (I*b*d^2*cosh((3*n - 1)*log(e)) + I*b*d^2 
*sinh((3*n - 1)*log(e)))*sinh(n*log(x))^2 + (I*b*d^2*cosh(n*log(x))^2 - I* 
b*c^2)*sinh((3*n - 1)*log(e)) + 2*(I*b*d^2*cosh((3*n - 1)*log(e))*cosh(...
 

Sympy [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int \left (e x\right )^{3 n - 1} \left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )\, dx \] Input:

integrate((e*x)**(-1+3*n)*(a+b*sech(c+d*x**n)),x)
 

Output:

Integral((e*x)**(3*n - 1)*(a + b*sech(c + d*x**n)), x)
 

Maxima [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n)),x, algorithm="maxima")
 

Output:

2*b*integrate((e*x)^(3*n - 1)/(e^(d*x^n + c) + e^(-d*x^n - c)), x) + 1/3*( 
e*x)^(3*n)*a/(e*n)
 

Giac [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int { {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )} \left (e x\right )^{3 \, n - 1} \,d x } \] Input:

integrate((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n)),x, algorithm="giac")
 

Output:

integrate((b*sech(d*x^n + c) + a)*(e*x)^(3*n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\int \left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{3\,n-1} \,d x \] Input:

int((a + b/cosh(c + d*x^n))*(e*x)^(3*n - 1),x)
 

Output:

int((a + b/cosh(c + d*x^n))*(e*x)^(3*n - 1), x)
 

Reduce [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right ) \, dx=\frac {e^{3 n} \left (x^{3 n} a +3 \left (\int \frac {x^{3 n} \mathrm {sech}\left (x^{n} d +c \right )}{x}d x \right ) b n \right )}{3 e n} \] Input:

int((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n)),x)
 

Output:

(e**(3*n)*(x**(3*n)*a + 3*int((x**(3*n)*sech(x**n*d + c))/x,x)*b*n))/(3*e* 
n)