3.2.44 \(\int \frac {x \text {sech}^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) [144]

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

[Out]

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

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Rubi [A]
time = 0.40, antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5751, 3401, 2296, 2221, 2317, 2438} \begin {gather*} \frac {\text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a-b-2 \sqrt {-a} \sqrt {b}}\right )}{4 \sqrt {-a} \sqrt {b} d^2}-\frac {\text {Li}_2\left (-\frac {(a+b) e^{2 c+2 d x}}{a-b+2 \sqrt {-a} \sqrt {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} \end {gather*}

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 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 2296

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 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 3401

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)/(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))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c, d, e, f, fz}, x] && IntegerQ[
2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 5751

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]

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 {\text {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 {\text {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*}

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Mathematica [A]
time = 1.65, size = 223, normalized size = 0.97 \begin {gather*} \frac {-\frac {4 c \text {ArcTan}\left (\frac {a-b+(a+b) e^{2 (c+d x)}}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 (c+d x) \left (\log \left (1+\frac {(a+b) e^{2 (c+d x)}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )-\log \left (1+\frac {(a+b) e^{2 (c+d x)}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )\right )+\text {PolyLog}\left (2,-\frac {(a+b) e^{2 (c+d x)}}{a-2 \sqrt {-a} \sqrt {b}-b}\right )-\text {PolyLog}\left (2,-\frac {(a+b) e^{2 (c+d x)}}{a+2 \sqrt {-a} \sqrt {b}-b}\right )}{\sqrt {-a}}}{4 \sqrt {b} d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(952\) vs. \(2(187)=374\).
time = 2.93, size = 953, normalized size = 4.13

method	result	size
risch	\(-\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) b c}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) a c}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {c^{2}}{d^{2} \left (-2 \sqrt {-a b}-a +b \right )}-\frac {c^{2}}{2 d^{2} \sqrt {-a b}}-\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) b x}{2 d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right )}{2 d^{2} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right )}{4 d^{2} \sqrt {-a b}}+\frac {\polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) a}{4 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {\polylog \left (2, \frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) b}{4 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {a \,c^{2}}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {b \,c^{2}}{2 d^{2} \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {a c x}{d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) a x}{2 d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}+\frac {b c x}{d \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {2 c x}{d \left (-2 \sqrt {-a b}-a +b \right )}+\frac {b \,x^{2}}{2 \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {a \,x^{2}}{2 \sqrt {-a b}\, \left (-2 \sqrt {-a b}-a +b \right )}-\frac {c \arctan \left (\frac {2 \left (a +b \right ) {\mathrm e}^{2 d x +2 c}+2 a -2 b}{4 \sqrt {a b}}\right )}{d^{2} \sqrt {a b}}-\frac {c x}{d \sqrt {-a b}}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) c}{d^{2} \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right ) c}{2 d^{2} \sqrt {-a b}}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{-2 \sqrt {-a b}-a +b}\right ) x}{d \left (-2 \sqrt {-a b}-a +b \right )}+\frac {\ln \left (1-\frac {\left (a +b \right ) {\mathrm e}^{2 d x +2 c}}{2 \sqrt {-a b}-a +b}\right ) x}{2 d \sqrt {-a b}}-\frac {x^{2}}{-2 \sqrt {-a b}-a +b}-\frac {x^{2}}{2 \sqrt {-a b}}\)	\(953\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 1516 vs. \(2 (185) = 370\).
time = 0.42, size = 1516, normalized size = 6.56 \begin {gather*} \text {Too large to display} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \operatorname {sech}^{2}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x}{{\mathrm {cosh}\left (c+d\,x\right )}^2\,\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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