3.144 \(\int \frac{x \text{sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\)

Optimal. Leaf size=231 \[ \frac{\text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^2}-\frac{\text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^2}+\frac{x \log \left (\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{x \log \left (\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d} \]

[Out]

(x*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) - (x*Log[1 + ((a +
b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) + PolyLog[2, -(((a + b)*E^(2*c + 2*d
*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2) - PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*S
qrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2)

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Rubi [A]  time = 0.540997, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {5632, 3320, 2264, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^2}-\frac{\text{PolyLog}\left (2,-\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}\right )}{4 \sqrt{-a} \sqrt{b} d^2}+\frac{x \log \left (\frac{(a+b) e^{2 c+2 d x}}{-2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{x \log \left (\frac{(a+b) e^{2 c+2 d x}}{2 \sqrt{-a} \sqrt{b}+a-b}+1\right )}{2 \sqrt{-a} \sqrt{b} d} \]

Antiderivative was successfully verified.

[In]

Int[(x*Sech[c + d*x]^2)/(a + b*Tanh[c + d*x]^2),x]

[Out]

(x*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) - (x*Log[1 + ((a +
b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) + PolyLog[2, -(((a + b)*E^(2*c + 2*d
*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2) - PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*S
qrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2)

Rule 5632

Int[(((f_.) + (g_.)*(x_))^(m_.)*Sech[(d_.) + (e_.)*(x_)]^2)/((b_) + (c_.)*Tanh[(d_.) + (e_.)*(x_)]^2), x_Symbo
l] :> Dist[2, Int[(f + g*x)^m/(b - c + (b + c)*Cosh[2*d + 2*e*x]), x], x] /; FreeQ[{b, c, d, e, f, g}, x] && I
GtQ[m, 0]

Rule 3320

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]), x_Symbol]
:> Dist[2, Int[((c + d*x)^m*E^(-(I*e) + f*fz*x))/(E^(I*Pi*(k - 1/2))*(b + (2*a*E^(-(I*e) + f*fz*x))/E^(I*Pi*(k
- 1/2)) - (b*E^(2*(-(I*e) + f*fz*x)))/E^(2*I*k*Pi))), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && IGtQ[m, 0]

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

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 2391

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]

Rubi steps

\begin{align*} \int \frac{x \text{sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=2 \int \frac{x}{a-b+(a+b) \cosh (2 c+2 d x)} \, dx\\ &=4 \int \frac{e^{2 c+2 d x} x}{a+b+2 (a-b) e^{2 c+2 d x}+(a+b) e^{2 (2 c+2 d x)}} \, dx\\ &=\frac{(2 (a+b)) \int \frac{e^{2 c+2 d x} x}{2 (a-b)-4 \sqrt{-a} \sqrt{b}+2 (a+b) e^{2 c+2 d x}} \, dx}{\sqrt{-a} \sqrt{b}}-\frac{(2 (a+b)) \int \frac{e^{2 c+2 d x} x}{2 (a-b)+4 \sqrt{-a} \sqrt{b}+2 (a+b) e^{2 c+2 d x}} \, dx}{\sqrt{-a} \sqrt{b}}\\ &=\frac{x \log \left (1+\frac{(a+b) e^{2 c+2 d x}}{a-2 \sqrt{-a} \sqrt{b}-b}\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{x \log \left (1+\frac{(a+b) e^{2 c+2 d x}}{a+2 \sqrt{-a} \sqrt{b}-b}\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{\int \log \left (1+\frac{2 (a+b) e^{2 c+2 d x}}{2 (a-b)-4 \sqrt{-a} \sqrt{b}}\right ) \, dx}{2 \sqrt{-a} \sqrt{b} d}+\frac{\int \log \left (1+\frac{2 (a+b) e^{2 c+2 d x}}{2 (a-b)+4 \sqrt{-a} \sqrt{b}}\right ) \, dx}{2 \sqrt{-a} \sqrt{b} d}\\ &=\frac{x \log \left (1+\frac{(a+b) e^{2 c+2 d x}}{a-2 \sqrt{-a} \sqrt{b}-b}\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{x \log \left (1+\frac{(a+b) e^{2 c+2 d x}}{a+2 \sqrt{-a} \sqrt{b}-b}\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 (a+b) x}{2 (a-b)-4 \sqrt{-a} \sqrt{b}}\right )}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 \sqrt{-a} \sqrt{b} d^2}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 (a+b) x}{2 (a-b)+4 \sqrt{-a} \sqrt{b}}\right )}{x} \, dx,x,e^{2 c+2 d x}\right )}{4 \sqrt{-a} \sqrt{b} d^2}\\ &=\frac{x \log \left (1+\frac{(a+b) e^{2 c+2 d x}}{a-2 \sqrt{-a} \sqrt{b}-b}\right )}{2 \sqrt{-a} \sqrt{b} d}-\frac{x \log \left (1+\frac{(a+b) e^{2 c+2 d x}}{a+2 \sqrt{-a} \sqrt{b}-b}\right )}{2 \sqrt{-a} \sqrt{b} d}+\frac{\text{Li}_2\left (-\frac{(a+b) e^{2 c+2 d x}}{a-2 \sqrt{-a} \sqrt{b}-b}\right )}{4 \sqrt{-a} \sqrt{b} d^2}-\frac{\text{Li}_2\left (-\frac{(a+b) e^{2 c+2 d x}}{a+2 \sqrt{-a} \sqrt{b}-b}\right )}{4 \sqrt{-a} \sqrt{b} d^2}\\ \end{align*}

Mathematica [A]  time = 2.44641, size = 250, normalized size = 1.08 \[ \frac{\sqrt{a} \text{PolyLog}\left (2,-\frac{(a+b) e^{2 (c+d x)}}{-2 \sqrt{-a} \sqrt{b}+a-b}\right )-\sqrt{a} \text{PolyLog}\left (2,-\frac{(a+b) e^{2 (c+d x)}}{2 \sqrt{-a} \sqrt{b}+a-b}\right )+2 \sqrt{a} (c+d x) \log \left (\frac{(a+b) e^{2 (c+d x)}}{-2 \sqrt{-a} \sqrt{b}+a-b}+1\right )-2 \sqrt{a} (c+d x) \log \left (\frac{(a+b) e^{2 (c+d x)}}{2 \sqrt{-a} \sqrt{b}+a-b}+1\right )-4 \sqrt{-a} c \tan ^{-1}\left (\frac{(a+b) e^{2 (c+d x)}+a-b}{2 \sqrt{a} \sqrt{b}}\right )}{4 \sqrt{-a^2} \sqrt{b} d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.155, size = 953, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sech(d*x+c)^2/(a+b*tanh(d*x+c)^2),x)

[Out]

1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*c^2+1/4/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*polylog(2,(a+b)*ex
p(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*a+1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2
*(-a*b)^(1/2)-a+b))*a*c-1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-
a+b))*b*c-1/d/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*c*x+1/d/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*b*c*x+1/2/d/(-a*
b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*a*x-1/2/d/(-a*b)^(1/2)/(-2*(-a
*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*b*x-1/d/(-a*b)^(1/2)*c*x+1/d/(-2*(-a*b)^(1/2)-
a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*x+1/2/d/(-a*b)^(1/2)*ln(1-(a+b)*exp(2*d*x+2*c)/(2*(-a*b)
^(1/2)-a+b))*x-2/d/(-2*(-a*b)^(1/2)-a+b)*c*x+1/d^2/(-2*(-a*b)^(1/2)-a+b)*ln(1-(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^
(1/2)-a+b))*c+1/2/d^2/(-a*b)^(1/2)*ln(1-(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))*c-1/d^2*c/(a*b)^(1/2)*arcta
n(1/4*(2*(a+b)*exp(2*d*x+2*c)+2*a-2*b)/(a*b)^(1/2))-1/2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*x^2+1/2/(-a*b)^(1
/2)/(-2*(-a*b)^(1/2)-a+b)*b*x^2-1/(-2*(-a*b)^(1/2)-a+b)*x^2-1/2/(-a*b)^(1/2)*x^2-1/d^2/(-2*(-a*b)^(1/2)-a+b)*c
^2-1/2/d^2/(-a*b)^(1/2)*c^2+1/2/d^2/(-2*(-a*b)^(1/2)-a+b)*polylog(2,(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b)
)+1/4/d^2/(-a*b)^(1/2)*polylog(2,(a+b)*exp(2*d*x+2*c)/(2*(-a*b)^(1/2)-a+b))-1/4/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1
/2)-a+b)*polylog(2,(a+b)*exp(2*d*x+2*c)/(-2*(-a*b)^(1/2)-a+b))*b-1/2/d^2/(-a*b)^(1/2)/(-2*(-a*b)^(1/2)-a+b)*a*
c^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{sech}\left (d x + c\right )^{2}}{b \tanh \left (d x + c\right )^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.82195, size = 3791, normalized size = 16.41 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c*log(2*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/
(a + b)) + 2*cosh(d*x + c) + 2*sinh(d*x + c)) + (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c*log(-2*sqrt(-(2*(a +
b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) + 2*cosh(d*x + c) + 2*sinh(d*x + c)) - (a + b)*sqrt(-a*b/(
a^2 + 2*a*b + b^2))*c*log(2*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) + 2*cosh(d*x + c)
+ 2*sinh(d*x + c)) - (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*c*log(-2*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b +
b^2)) - a + b)/(a + b)) + 2*cosh(d*x + c) + 2*sinh(d*x + c)) - (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*dilog(-
(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/
(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) + a + b)/(a + b) + 1)
- (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*dilog((((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*co
sh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b +
b^2)) + a - b)/(a + b)) - a - b)/(a + b) + 1) + (a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*dilog(-(((a - b)*cosh(
d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b +
b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) + a + b)/(a + b) + 1) + (a + b)*sqrt(-
a*b/(a^2 + 2*a*b + b^2))*dilog((((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x + c) + (a
+ b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(
a + b)) - a - b)/(a + b) + 1) - ((a + b)*d*x + (a + b)*c)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*log((((a - b)*cosh(d*
x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^
2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) + a - b)/(a + b)) + a + b)/(a + b)) - ((a + b)*d*x + (a +
b)*c)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*log(-(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) - 2*((a + b)*cosh(d
*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt(-(2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2
)) + a - b)/(a + b)) - a - b)/(a + b)) + ((a + b)*d*x + (a + b)*c)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*log((((a - b
)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2
*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*b + b^2)) - a + b)/(a + b)) + a + b)/(a + b)) + ((a + b)*d*
x + (a + b)*c)*sqrt(-a*b/(a^2 + 2*a*b + b^2))*log(-(((a - b)*cosh(d*x + c) + (a - b)*sinh(d*x + c) + 2*((a + b
)*cosh(d*x + c) + (a + b)*sinh(d*x + c))*sqrt(-a*b/(a^2 + 2*a*b + b^2)))*sqrt((2*(a + b)*sqrt(-a*b/(a^2 + 2*a*
b + b^2)) - a + b)/(a + b)) - a - b)/(a + b)))/(a*b*d^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{sech}^{2}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{sech}\left (d x + c\right )^{2}}{b \tanh \left (d x + c\right )^{2} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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