\(\int (e x)^{-1+3 n} (a+b \text {sech}(c+d x^n))^2 \, dx\) [78]

Optimal result

Integrand size = 24, antiderivative size = 363 \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^{3 n}}{3 e n}+\frac {b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac {4 a b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}-\frac {b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{c+d x^n}\right )}{d^3 e n}-\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{c+d x^n}\right )}{d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n} \]

[Out]

1/3*a^2*(e*x)^(3*n)/e/n+b^2*(e*x)^(3*n)/d/e/n/(x^n)+4*a*b*(e*x)^(3*n)*arctan(exp(c+d*x^n))/d/e/n/(x^n)-2*b^2*(
e*x)^(3*n)*ln(1+exp(2*c+2*d*x^n))/d^2/e/n/(x^(2*n))-4*I*a*b*(e*x)^(3*n)*polylog(2,-I*exp(c+d*x^n))/d^2/e/n/(x^
(2*n))+4*I*a*b*(e*x)^(3*n)*polylog(2,I*exp(c+d*x^n))/d^2/e/n/(x^(2*n))-b^2*(e*x)^(3*n)*polylog(2,-exp(2*c+2*d*
x^n))/d^3/e/n/(x^(3*n))+4*I*a*b*(e*x)^(3*n)*polylog(3,-I*exp(c+d*x^n))/d^3/e/n/(x^(3*n))-4*I*a*b*(e*x)^(3*n)*p
olylog(3,I*exp(c+d*x^n))/d^3/e/n/(x^(3*n))+b^2*(e*x)^(3*n)*tanh(c+d*x^n)/d/e/n/(x^n)

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5548, 5544, 4275, 4265, 2611, 2320, 6724, 4269, 3799, 2221, 2317, 2438} \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\frac {a^2 (e x)^{3 n}}{3 e n}+\frac {4 a b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}+\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{d x^n+c}\right )}{d^3 e n}-\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{d x^n+c}\right )}{d^3 e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{d x^n+c}\right )}{d^2 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{d x^n+c}\right )}{d^2 e n}-\frac {b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{2 \left (d x^n+c\right )}\right )}{d^3 e n}-\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (e^{2 \left (c+d x^n\right )}+1\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac {b^2 x^{-n} (e x)^{3 n}}{d e n} \]

[In]

Int[(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n])^2,x]

[Out]

(a^2*(e*x)^(3*n))/(3*e*n) + (b^2*(e*x)^(3*n))/(d*e*n*x^n) + (4*a*b*(e*x)^(3*n)*ArcTan[E^(c + d*x^n)])/(d*e*n*x
^n) - (2*b^2*(e*x)^(3*n)*Log[1 + E^(2*(c + d*x^n))])/(d^2*e*n*x^(2*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, (-I
)*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[2, I*E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) -
(b^2*(e*x)^(3*n)*PolyLog[2, -E^(2*(c + d*x^n))])/(d^3*e*n*x^(3*n)) + ((4*I)*a*b*(e*x)^(3*n)*PolyLog[3, (-I)*E^
(c + d*x^n)])/(d^3*e*n*x^(3*n)) - ((4*I)*a*b*(e*x)^(3*n)*PolyLog[3, I*E^(c + d*x^n)])/(d^3*e*n*x^(3*n)) + (b^2
*(e*x)^(3*n)*Tanh[c + d*x^n])/(d*e*n*x^n)

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

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 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

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 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3799

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> Simp[(-I)*((c + d*x)^(m
+ 1)/(d*(m + 1))), x] + Dist[2*I, Int[(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x]
, x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

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 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4275

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 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]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-3 n} (e x)^{3 n}\right ) \int x^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx}{e} \\ & = \frac {\left (x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x^2 (a+b \text {sech}(c+d x))^2 \, dx,x,x^n\right )}{e n} \\ & = \frac {\left (x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \left (a^2 x^2+2 a b x^2 \text {sech}(c+d x)+b^2 x^2 \text {sech}^2(c+d x)\right ) \, dx,x,x^n\right )}{e n} \\ & = \frac {a^2 (e x)^{3 n}}{3 e n}+\frac {\left (2 a b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x^2 \text {sech}(c+d x) \, dx,x,x^n\right )}{e n}+\frac {\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x^2 \text {sech}^2(c+d x) \, dx,x,x^n\right )}{e n} \\ & = \frac {a^2 (e x)^{3 n}}{3 e n}+\frac {4 a b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}+\frac {b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}-\frac {\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac {\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \log \left (1+i e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}-\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int x \tanh (c+d x) \, dx,x,x^n\right )}{d e n} \\ & = \frac {a^2 (e x)^{3 n}}{3 e n}+\frac {b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac {4 a b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac {\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx,x,x^n\right )}{d^2 e n}-\frac {\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx,x,x^n\right )}{d^2 e n}-\frac {\left (4 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x}{1+e^{2 (c+d x)}} \, dx,x,x^n\right )}{d e n} \\ & = \frac {a^2 (e x)^{3 n}}{3 e n}+\frac {b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac {4 a b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac {\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}-\frac {\left (4 i a b x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^3 e n}+\frac {\left (2 b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \log \left (1+e^{2 (c+d x)}\right ) \, dx,x,x^n\right )}{d^2 e n} \\ & = \frac {a^2 (e x)^{3 n}}{3 e n}+\frac {b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac {4 a b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}+\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{c+d x^n}\right )}{d^3 e n}-\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{c+d x^n}\right )}{d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n}+\frac {\left (b^2 x^{-3 n} (e x)^{3 n}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \left (c+d x^n\right )}\right )}{d^3 e n} \\ & = \frac {a^2 (e x)^{3 n}}{3 e n}+\frac {b^2 x^{-n} (e x)^{3 n}}{d e n}+\frac {4 a b x^{-n} (e x)^{3 n} \arctan \left (e^{c+d x^n}\right )}{d e n}-\frac {2 b^2 x^{-2 n} (e x)^{3 n} \log \left (1+e^{2 \left (c+d x^n\right )}\right )}{d^2 e n}-\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-i e^{c+d x^n}\right )}{d^2 e n}+\frac {4 i a b x^{-2 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,i e^{c+d x^n}\right )}{d^2 e n}-\frac {b^2 x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (2,-e^{2 \left (c+d x^n\right )}\right )}{d^3 e n}+\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,-i e^{c+d x^n}\right )}{d^3 e n}-\frac {4 i a b x^{-3 n} (e x)^{3 n} \operatorname {PolyLog}\left (3,i e^{c+d x^n}\right )}{d^3 e n}+\frac {b^2 x^{-n} (e x)^{3 n} \tanh \left (c+d x^n\right )}{d e n} \\ \end{align*}

Mathematica [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx \]

[In]

Integrate[(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n])^2,x]

[Out]

Integrate[(e*x)^(-1 + 3*n)*(a + b*Sech[c + d*x^n])^2, x]

Maple [F]

\[\int \left (e x \right )^{-1+3 n} {\left (a +b \,\operatorname {sech}\left (c +d \,x^{n}\right )\right )}^{2}d x\]

[In]

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

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\text {Timed out} \]

[In]

integrate((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n))^2,x, algorithm="fricas")

[Out]

Timed out

Sympy [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\int \left (e x\right )^{3 n - 1} \left (a + b \operatorname {sech}{\left (c + d x^{n} \right )}\right )^{2}\, dx \]

[In]

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

[Out]

Integral((e*x)**(3*n - 1)*(a + b*sech(c + d*x**n))**2, x)

Maxima [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{3 \, n - 1} \,d x } \]

[In]

integrate((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n))^2,x, algorithm="maxima")

[Out]

-2*b^2*e^(3*n)*x^(2*n)/(d*e*n*e^(2*d*x^n + 2*c) + d*e*n) + 1/3*(e*x)^(3*n)*a^2/(e*n) + integrate(4*(a*b*d*e^(3
*n)*e^(d*x^n + 3*n*log(x) + c) + b^2*e^(3*n)*x^(2*n))/(d*e*x*e^(2*d*x^n + 2*c) + d*e*x), x)

Giac [F]

\[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{3 \, n - 1} \,d x } \]

[In]

integrate((e*x)^(-1+3*n)*(a+b*sech(c+d*x^n))^2,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+3 n} \left (a+b \text {sech}\left (c+d x^n\right )\right )^2 \, dx=\int {\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x^n\right )}\right )}^2\,{\left (e\,x\right )}^{3\,n-1} \,d x \]

[In]

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

[Out]

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