\(\int (c+d x)^3 \text {sech}(a+b x) \, dx\) [26]

Optimal result

Integrand size = 14, antiderivative size = 179 \[ \int (c+d x)^3 \text {sech}(a+b x) \, dx=\frac {2 (c+d x)^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{a+b x}\right )}{b^4} \]

[Out]

2*(d*x+c)^3*arctan(exp(b*x+a))/b-3*I*d*(d*x+c)^2*polylog(2,-I*exp(b*x+a))/b^2+3*I*d*(d*x+c)^2*polylog(2,I*exp(
b*x+a))/b^2+6*I*d^2*(d*x+c)*polylog(3,-I*exp(b*x+a))/b^3-6*I*d^2*(d*x+c)*polylog(3,I*exp(b*x+a))/b^3-6*I*d^3*p
olylog(4,-I*exp(b*x+a))/b^4+6*I*d^3*polylog(4,I*exp(b*x+a))/b^4

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4265, 2611, 6744, 2320, 6724} \[ \int (c+d x)^3 \text {sech}(a+b x) \, dx=\frac {2 (c+d x)^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{a+b x}\right )}{b^4}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2} \]

[In]

Int[(c + d*x)^3*Sech[a + b*x],x]

[Out]

(2*(c + d*x)^3*ArcTan[E^(a + b*x)])/b - ((3*I)*d*(c + d*x)^2*PolyLog[2, (-I)*E^(a + b*x)])/b^2 + ((3*I)*d*(c +
d*x)^2*PolyLog[2, I*E^(a + b*x)])/b^2 + ((6*I)*d^2*(c + d*x)*PolyLog[3, (-I)*E^(a + b*x)])/b^3 - ((6*I)*d^2*(
c + d*x)*PolyLog[3, I*E^(a + b*x)])/b^3 - ((6*I)*d^3*PolyLog[4, (-I)*E^(a + b*x)])/b^4 + ((6*I)*d^3*PolyLog[4,
I*E^(a + b*x)])/b^4

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 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 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 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]

Rule 6744

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 (c+d x)^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {(3 i d) \int (c+d x)^2 \log \left (1-i e^{a+b x}\right ) \, dx}{b}+\frac {(3 i d) \int (c+d x)^2 \log \left (1+i e^{a+b x}\right ) \, dx}{b} \\ & = \frac {2 (c+d x)^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {\left (6 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,-i e^{a+b x}\right ) \, dx}{b^2}-\frac {\left (6 i d^2\right ) \int (c+d x) \operatorname {PolyLog}\left (2,i e^{a+b x}\right ) \, dx}{b^2} \\ & = \frac {2 (c+d x)^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {\left (6 i d^3\right ) \int \operatorname {PolyLog}\left (3,-i e^{a+b x}\right ) \, dx}{b^3}+\frac {\left (6 i d^3\right ) \int \operatorname {PolyLog}\left (3,i e^{a+b x}\right ) \, dx}{b^3} \\ & = \frac {2 (c+d x)^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {\left (6 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {\left (6 i d^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{a+b x}\right )}{b^4} \\ & = \frac {2 (c+d x)^3 \arctan \left (e^{a+b x}\right )}{b}-\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )}{b^2}+\frac {3 i d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )}{b^2}+\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )}{b^3}-\frac {6 i d^2 (c+d x) \operatorname {PolyLog}\left (3,i e^{a+b x}\right )}{b^3}-\frac {6 i d^3 \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )}{b^4}+\frac {6 i d^3 \operatorname {PolyLog}\left (4,i e^{a+b x}\right )}{b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.92 \[ \int (c+d x)^3 \text {sech}(a+b x) \, dx=\frac {i \left (-2 i b^3 c^3 \arctan \left (e^{a+b x}\right )+3 b^3 c^2 d x \log \left (1-i e^{a+b x}\right )+3 b^3 c d^2 x^2 \log \left (1-i e^{a+b x}\right )+b^3 d^3 x^3 \log \left (1-i e^{a+b x}\right )-3 b^3 c^2 d x \log \left (1+i e^{a+b x}\right )-3 b^3 c d^2 x^2 \log \left (1+i e^{a+b x}\right )-b^3 d^3 x^3 \log \left (1+i e^{a+b x}\right )-3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,-i e^{a+b x}\right )+3 b^2 d (c+d x)^2 \operatorname {PolyLog}\left (2,i e^{a+b x}\right )+6 b c d^2 \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )+6 b d^3 x \operatorname {PolyLog}\left (3,-i e^{a+b x}\right )-6 b c d^2 \operatorname {PolyLog}\left (3,i e^{a+b x}\right )-6 b d^3 x \operatorname {PolyLog}\left (3,i e^{a+b x}\right )-6 d^3 \operatorname {PolyLog}\left (4,-i e^{a+b x}\right )+6 d^3 \operatorname {PolyLog}\left (4,i e^{a+b x}\right )\right )}{b^4} \]

[In]

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

[Out]

(I*((-2*I)*b^3*c^3*ArcTan[E^(a + b*x)] + 3*b^3*c^2*d*x*Log[1 - I*E^(a + b*x)] + 3*b^3*c*d^2*x^2*Log[1 - I*E^(a
+ b*x)] + b^3*d^3*x^3*Log[1 - I*E^(a + b*x)] - 3*b^3*c^2*d*x*Log[1 + I*E^(a + b*x)] - 3*b^3*c*d^2*x^2*Log[1 +
I*E^(a + b*x)] - b^3*d^3*x^3*Log[1 + I*E^(a + b*x)] - 3*b^2*d*(c + d*x)^2*PolyLog[2, (-I)*E^(a + b*x)] + 3*b^
2*d*(c + d*x)^2*PolyLog[2, I*E^(a + b*x)] + 6*b*c*d^2*PolyLog[3, (-I)*E^(a + b*x)] + 6*b*d^3*x*PolyLog[3, (-I)
*E^(a + b*x)] - 6*b*c*d^2*PolyLog[3, I*E^(a + b*x)] - 6*b*d^3*x*PolyLog[3, I*E^(a + b*x)] - 6*d^3*PolyLog[4, (
-I)*E^(a + b*x)] + 6*d^3*PolyLog[4, I*E^(a + b*x)]))/b^4

Maple [F]

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

[In]

int((d*x+c)^3*sech(b*x+a),x)

[Out]

int((d*x+c)^3*sech(b*x+a),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. 497 vs. \(2 (146) = 292\).

Time = 0.27 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.78 \[ \int (c+d x)^3 \text {sech}(a+b x) \, dx=\frac {6 i \, d^{3} {\rm polylog}\left (4, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 6 i \, d^{3} {\rm polylog}\left (4, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) - 3 \, {\left (-i \, b^{2} d^{3} x^{2} - 2 i \, b^{2} c d^{2} x - i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 3 \, {\left (i \, b^{2} d^{3} x^{2} + 2 i \, b^{2} c d^{2} x + i \, b^{2} c^{2} d\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\left (i \, b^{3} c^{3} - 3 i \, a b^{2} c^{2} d + 3 i \, a^{2} b c d^{2} - i \, a^{3} d^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + {\left (-i \, b^{3} c^{3} + 3 i \, a b^{2} c^{2} d - 3 i \, a^{2} b c d^{2} + i \, a^{3} d^{3}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + {\left (-i \, b^{3} d^{3} x^{3} - 3 i \, b^{3} c d^{2} x^{2} - 3 i \, b^{3} c^{2} d x - 3 i \, a b^{2} c^{2} d + 3 i \, a^{2} b c d^{2} - i \, a^{3} d^{3}\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + {\left (i \, b^{3} d^{3} x^{3} + 3 i \, b^{3} c d^{2} x^{2} + 3 i \, b^{3} c^{2} d x + 3 i \, a b^{2} c^{2} d - 3 i \, a^{2} b c d^{2} + i \, a^{3} d^{3}\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right ) - 6 \, {\left (i \, b d^{3} x + i \, b c d^{2}\right )} {\rm polylog}\left (3, i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) - 6 \, {\left (-i \, b d^{3} x - i \, b c d^{2}\right )} {\rm polylog}\left (3, -i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right )}{b^{4}} \]

[In]

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

[Out]

(6*I*d^3*polylog(4, I*cosh(b*x + a) + I*sinh(b*x + a)) - 6*I*d^3*polylog(4, -I*cosh(b*x + a) - I*sinh(b*x + a)
) - 3*(-I*b^2*d^3*x^2 - 2*I*b^2*c*d^2*x - I*b^2*c^2*d)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 3*(I*b^2*d^3
*x^2 + 2*I*b^2*c*d^2*x + I*b^2*c^2*d)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + (I*b^3*c^3 - 3*I*a*b^2*c^2*d
+ 3*I*a^2*b*c*d^2 - I*a^3*d^3)*log(cosh(b*x + a) + sinh(b*x + a) + I) + (-I*b^3*c^3 + 3*I*a*b^2*c^2*d - 3*I*a
^2*b*c*d^2 + I*a^3*d^3)*log(cosh(b*x + a) + sinh(b*x + a) - I) + (-I*b^3*d^3*x^3 - 3*I*b^3*c*d^2*x^2 - 3*I*b^3
*c^2*d*x - 3*I*a*b^2*c^2*d + 3*I*a^2*b*c*d^2 - I*a^3*d^3)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) + (I*b^3*
d^3*x^3 + 3*I*b^3*c*d^2*x^2 + 3*I*b^3*c^2*d*x + 3*I*a*b^2*c^2*d - 3*I*a^2*b*c*d^2 + I*a^3*d^3)*log(-I*cosh(b*x
+ a) - I*sinh(b*x + a) + 1) - 6*(I*b*d^3*x + I*b*c*d^2)*polylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) - 6*(-I
*b*d^3*x - I*b*c*d^2)*polylog(3, -I*cosh(b*x + a) - I*sinh(b*x + a)))/b^4

Sympy [F]

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

[In]

integrate((d*x+c)**3*sech(b*x+a),x)

[Out]

Integral((c + d*x)**3*sech(a + b*x), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((d*x + c)^3*sech(b*x + a), x)

Mupad [F(-1)]

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

[In]

int((c + d*x)^3/cosh(a + b*x),x)

[Out]

int((c + d*x)^3/cosh(a + b*x), x)