\(\int (c+d x)^2 \text {sech}^2(a+b x) \, dx\) [32]

Optimal result

Integrand size = 16, antiderivative size = 73 \[ \int (c+d x)^2 \text {sech}^2(a+b x) \, dx=\frac {(c+d x)^2}{b}-\frac {2 d (c+d x) \log \left (1+e^{2 (a+b x)}\right )}{b^2}-\frac {d^2 \operatorname {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}+\frac {(c+d x)^2 \tanh (a+b x)}{b} \]

[Out]

(d*x+c)^2/b-2*d*(d*x+c)*ln(1+exp(2*b*x+2*a))/b^2-d^2*polylog(2,-exp(2*b*x+2*a))/b^3+(d*x+c)^2*tanh(b*x+a)/b

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4269, 3799, 2221, 2317, 2438} \[ \int (c+d x)^2 \text {sech}^2(a+b x) \, dx=-\frac {d^2 \operatorname {PolyLog}\left (2,-e^{2 (a+b x)}\right )}{b^3}-\frac {2 d (c+d x) \log \left (e^{2 (a+b x)}+1\right )}{b^2}+\frac {(c+d x)^2 \tanh (a+b x)}{b}+\frac {(c+d x)^2}{b} \]

[In]

Int[(c + d*x)^2*Sech[a + b*x]^2,x]

[Out]

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

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 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 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 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]

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

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.25 \[ \int (c+d x)^2 \text {sech}^2(a+b x) \, dx=\frac {-\frac {2 b (c+d x) \left (b (c+d x)+d \left (1+e^{2 a}\right ) \log \left (1+e^{-2 (a+b x)}\right )\right )}{1+e^{2 a}}+d^2 \operatorname {PolyLog}\left (2,-e^{-2 (a+b x)}\right )+b^2 (c+d x)^2 \text {sech}(a) \text {sech}(a+b x) \sinh (b x)}{b^3} \]

[In]

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

[Out]

((-2*b*(c + d*x)*(b*(c + d*x) + d*(1 + E^(2*a))*Log[1 + E^(-2*(a + b*x))]))/(1 + E^(2*a)) + d^2*PolyLog[2, -E^
(-2*(a + b*x))] + b^2*(c + d*x)^2*Sech[a]*Sech[a + b*x]*Sinh[b*x])/b^3

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(73)=146\).

Time = 0.24 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.18

method	result	size
risch	\(-\frac {2 \left (x^{2} d^{2}+2 c d x +c^{2}\right )}{b \left (1+{\mathrm e}^{2 b x +2 a}\right )}-\frac {2 d c \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}+\frac {4 d c \ln \left ({\mathrm e}^{b x +a}\right )}{b^{2}}+\frac {2 d^{2} x^{2}}{b}+\frac {4 d^{2} a x}{b^{2}}+\frac {2 d^{2} a^{2}}{b^{3}}-\frac {2 d^{2} \ln \left (1+{\mathrm e}^{2 b x +2 a}\right ) x}{b^{2}}-\frac {d^{2} \operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}-\frac {4 d^{2} a \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}\)	\(159\)

[In]

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

[Out]

-2*(d^2*x^2+2*c*d*x+c^2)/b/(1+exp(2*b*x+2*a))-2/b^2*d*c*ln(1+exp(2*b*x+2*a))+4/b^2*d*c*ln(exp(b*x+a))+2/b*d^2*
x^2+4/b^2*d^2*a*x+2/b^3*d^2*a^2-2/b^2*d^2*ln(1+exp(2*b*x+2*a))*x-d^2*polylog(2,-exp(2*b*x+2*a))/b^3-4/b^3*d^2*
a*ln(exp(b*x+a))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 715, normalized size of antiderivative = 9.79 \[ \int (c+d x)^2 \text {sech}^2(a+b x) \, dx=-\frac {2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2} - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + 2 \, a b c d - a^{2} d^{2}\right )} \sinh \left (b x + a\right )^{2} + {\left (d^{2} \cosh \left (b x + a\right )^{2} + 2 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} \sinh \left (b x + a\right )^{2} + d^{2}\right )} {\rm Li}_2\left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right )\right ) + {\left (d^{2} \cosh \left (b x + a\right )^{2} + 2 \, d^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} \sinh \left (b x + a\right )^{2} + d^{2}\right )} {\rm Li}_2\left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right )\right ) + {\left (b c d - a d^{2} + {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b c d - a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + i\right ) + {\left (b c d - a d^{2} + {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b c d - a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b c d - a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - i\right ) + {\left (b d^{2} x + a d^{2} + {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b d^{2} x + a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (i \, \cosh \left (b x + a\right ) + i \, \sinh \left (b x + a\right ) + 1\right ) + {\left (b d^{2} x + a d^{2} + {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b d^{2} x + a d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b d^{2} x + a d^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-i \, \cosh \left (b x + a\right ) - i \, \sinh \left (b x + a\right ) + 1\right )\right )}}{b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2} + b^{3}} \]

[In]

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

[Out]

-2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2 - (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*cosh(b*x + a)^2 - 2*(b^2
*d^2*x^2 + 2*b^2*c*d*x + 2*a*b*c*d - a^2*d^2)*cosh(b*x + a)*sinh(b*x + a) - (b^2*d^2*x^2 + 2*b^2*c*d*x + 2*a*b
*c*d - a^2*d^2)*sinh(b*x + a)^2 + (d^2*cosh(b*x + a)^2 + 2*d^2*cosh(b*x + a)*sinh(b*x + a) + d^2*sinh(b*x + a)
^2 + d^2)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) + (d^2*cosh(b*x + a)^2 + 2*d^2*cosh(b*x + a)*sinh(b*x + a)
+ d^2*sinh(b*x + a)^2 + d^2)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) + (b*c*d - a*d^2 + (b*c*d - a*d^2)*cosh
(b*x + a)^2 + 2*(b*c*d - a*d^2)*cosh(b*x + a)*sinh(b*x + a) + (b*c*d - a*d^2)*sinh(b*x + a)^2)*log(cosh(b*x +
a) + sinh(b*x + a) + I) + (b*c*d - a*d^2 + (b*c*d - a*d^2)*cosh(b*x + a)^2 + 2*(b*c*d - a*d^2)*cosh(b*x + a)*s
inh(b*x + a) + (b*c*d - a*d^2)*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) - I) + (b*d^2*x + a*d^2 + (b
*d^2*x + a*d^2)*cosh(b*x + a)^2 + 2*(b*d^2*x + a*d^2)*cosh(b*x + a)*sinh(b*x + a) + (b*d^2*x + a*d^2)*sinh(b*x
+ a)^2)*log(I*cosh(b*x + a) + I*sinh(b*x + a) + 1) + (b*d^2*x + a*d^2 + (b*d^2*x + a*d^2)*cosh(b*x + a)^2 + 2
*(b*d^2*x + a*d^2)*cosh(b*x + a)*sinh(b*x + a) + (b*d^2*x + a*d^2)*sinh(b*x + a)^2)*log(-I*cosh(b*x + a) - I*s
inh(b*x + a) + 1))/(b^3*cosh(b*x + a)^2 + 2*b^3*cosh(b*x + a)*sinh(b*x + a) + b^3*sinh(b*x + a)^2 + b^3)

Sympy [F]

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

[In]

integrate((d*x+c)**2*sech(b*x+a)**2,x)

[Out]

Integral((c + d*x)**2*sech(a + b*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

integrate((d*x + c)^2*sech(b*x + a)^2, x)

Mupad [F(-1)]

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

[In]

int((c + d*x)^2/cosh(a + b*x)^2,x)

[Out]

int((c + d*x)^2/cosh(a + b*x)^2, x)