\(\int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx\) [103]

Optimal result

Integrand size = 13, antiderivative size = 83 \[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=\frac {\left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{b^5}-\frac {a \left (a^2-2 b^2\right ) \tanh (x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \tanh ^2(x)}{2 b^3}-\frac {a \tanh ^3(x)}{3 b^2}+\frac {\tanh ^4(x)}{4 b} \]

[Out]

(a^2-b^2)^2*ln(a+b*tanh(x))/b^5-a*(a^2-2*b^2)*tanh(x)/b^4+1/2*(a^2-2*b^2)*tanh(x)^2/b^3-1/3*a*tanh(x)^3/b^2+1/
4*tanh(x)^4/b

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3587, 711} \[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=\frac {\left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{b^5}-\frac {a \left (a^2-2 b^2\right ) \tanh (x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \tanh ^2(x)}{2 b^3}-\frac {a \tanh ^3(x)}{3 b^2}+\frac {\tanh ^4(x)}{4 b} \]

[In]

Int[Sech[x]^6/(a + b*Tanh[x]),x]

[Out]

((a^2 - b^2)^2*Log[a + b*Tanh[x]])/b^5 - (a*(a^2 - 2*b^2)*Tanh[x])/b^4 + ((a^2 - 2*b^2)*Tanh[x]^2)/(2*b^3) - (
a*Tanh[x]^3)/(3*b^2) + Tanh[x]^4/(4*b)

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rule 3587

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(b*f), Subst
[Int[(a + x)^n*(1 + x^2/b^2)^(m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && NeQ[a^2 + b
^2, 0] && IntegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1-\frac {x^2}{b^2}\right )^2}{a+x} \, dx,x,b \tanh (x)\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (\frac {-a^3+2 a b^2}{b^4}-\frac {\left (-a^2+2 b^2\right ) x}{b^4}-\frac {a x^2}{b^4}+\frac {x^3}{b^4}+\frac {\left (-a^2+b^2\right )^2}{b^4 (a+x)}\right ) \, dx,x,b \tanh (x)\right )}{b} \\ & = \frac {\left (a^2-b^2\right )^2 \log (a+b \tanh (x))}{b^5}-\frac {a \left (a^2-2 b^2\right ) \tanh (x)}{b^4}+\frac {\left (a^2-2 b^2\right ) \tanh ^2(x)}{2 b^3}-\frac {a \tanh ^3(x)}{3 b^2}+\frac {\tanh ^4(x)}{4 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.24 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.96 \[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=\frac {12 \left (a^2-b^2\right )^2 \log (a+b \tanh (x))+3 b^4 \text {sech}^4(x)-12 a b \left (a^2-2 b^2\right ) \tanh (x)+6 b^2 \left (a^2-b^2\right ) \tanh ^2(x)-4 a b^3 \tanh ^3(x)}{12 b^5} \]

[In]

Integrate[Sech[x]^6/(a + b*Tanh[x]),x]

[Out]

(12*(a^2 - b^2)^2*Log[a + b*Tanh[x]] + 3*b^4*Sech[x]^4 - 12*a*b*(a^2 - 2*b^2)*Tanh[x] + 6*b^2*(a^2 - b^2)*Tanh
[x]^2 - 4*a*b^3*Tanh[x]^3)/(12*b^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(221\) vs. \(2(77)=154\).

Time = 67.23 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.67

method	result	size
default	\(\frac {\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a +2 b \tanh \left (\frac {x}{2}\right )+a \right )}{b^{5}}-\frac {2 \left (\frac {\left (a^{3} b -2 a \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{7}+\left (-a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{6}+\left (3 a^{3} b -\frac {14}{3} a \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{5}+\left (-2 a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{4}+\left (3 a^{3} b -\frac {14}{3} a \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{2} b^{2}+2 b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (a^{3} b -2 a \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{4}}+\frac {\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{2}\right )}{b^{5}}\)	\(222\)
risch	\(\frac {2 a^{3} {\mathrm e}^{6 x}-2 \,{\mathrm e}^{6 x} a^{2} b -2 \,{\mathrm e}^{6 x} a \,b^{2}+2 b^{3} {\mathrm e}^{6 x}+6 a^{3} {\mathrm e}^{4 x}-4 a^{2} b \,{\mathrm e}^{4 x}-10 a \,b^{2} {\mathrm e}^{4 x}+8 b^{3} {\mathrm e}^{4 x}+6 a^{3} {\mathrm e}^{2 x}-2 \,{\mathrm e}^{2 x} a^{2} b -\frac {34 \,{\mathrm e}^{2 x} a \,b^{2}}{3}+2 b^{3} {\mathrm e}^{2 x}+2 a^{3}-\frac {10 a \,b^{2}}{3}}{b^{4} \left (1+{\mathrm e}^{2 x}\right )^{4}}-\frac {\ln \left (1+{\mathrm e}^{2 x}\right ) a^{4}}{b^{5}}+\frac {2 \ln \left (1+{\mathrm e}^{2 x}\right ) a^{2}}{b^{3}}-\frac {\ln \left (1+{\mathrm e}^{2 x}\right )}{b}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{4}}{b^{5}}-\frac {2 \ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right ) a^{2}}{b^{3}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {a -b}{a +b}\right )}{b}\)	\(253\)

[In]

int(sech(x)^6/(a+b*tanh(x)),x,method=_RETURNVERBOSE)

[Out]

(a^4-2*a^2*b^2+b^4)/b^5*ln(tanh(1/2*x)^2*a+2*b*tanh(1/2*x)+a)-2/b^5*(((a^3*b-2*a*b^3)*tanh(1/2*x)^7+(-a^2*b^2+
2*b^4)*tanh(1/2*x)^6+(3*a^3*b-14/3*a*b^3)*tanh(1/2*x)^5+(-2*a^2*b^2+2*b^4)*tanh(1/2*x)^4+(3*a^3*b-14/3*a*b^3)*
tanh(1/2*x)^3+(-a^2*b^2+2*b^4)*tanh(1/2*x)^2+(a^3*b-2*a*b^3)*tanh(1/2*x))/(1+tanh(1/2*x)^2)^4+1/2*(a^4-2*a^2*b
^2+b^4)*ln(1+tanh(1/2*x)^2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1827 vs. \(2 (77) = 154\).

Time = 0.29 (sec) , antiderivative size = 1827, normalized size of antiderivative = 22.01 \[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=\text {Too large to display} \]

[In]

integrate(sech(x)^6/(a+b*tanh(x)),x, algorithm="fricas")

[Out]

1/3*(6*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^6 + 36*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)*sinh(x)^5 + 6*(a
^3*b - a^2*b^2 - a*b^3 + b^4)*sinh(x)^6 + 6*(3*a^3*b - 2*a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(x)^4 + 6*(3*a^3*b - 2
*a^2*b^2 - 5*a*b^3 + 4*b^4 + 15*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^2)*sinh(x)^4 + 6*a^3*b - 10*a*b^3 + 24
*(5*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^3 + (3*a^3*b - 2*a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(x))*sinh(x)^3 + 2
*(9*a^3*b - 3*a^2*b^2 - 17*a*b^3 + 3*b^4)*cosh(x)^2 + 2*(45*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^4 + 9*a^3*
b - 3*a^2*b^2 - 17*a*b^3 + 3*b^4 + 18*(3*a^3*b - 2*a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(x)^2)*sinh(x)^2 + 3*((a^4 -
2*a^2*b^2 + b^4)*cosh(x)^8 + 8*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^7 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^8
+ 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 4*(a^4 - 2*a^2*b^2 + b^4 + 7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x
)^6 + 8*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^5 + 6*(a^4 - 2*a^2*b
^2 + b^4)*cosh(x)^4 + 2*(35*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 3*a^4 - 6*a^2*b^2 + 3*b^4 + 30*(a^4 - 2*a^2*b^
2 + b^4)*cosh(x)^2)*sinh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 8*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 10*(a^4 - 2*a
^2*b^2 + b^4)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x)^3 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 +
4*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 15*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 9*(a^
4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^7 + 3*(a^4 - 2*a^2*b^2 + b^4)*c
osh(x)^5 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + (a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x))*log(2*(a*cosh(x) +
b*sinh(x))/(cosh(x) - sinh(x))) - 3*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^8 + 8*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sin
h(x)^7 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^8 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 4*(a^4 - 2*a^2*b^2 + b^4 +
7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2 +
b^4)*cosh(x))*sinh(x)^5 + 6*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 2*(35*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4 + 3*a
^4 - 6*a^2*b^2 + 3*b^4 + 30*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 8*(7*(a^4 -
2*a^2*b^2 + b^4)*cosh(x)^5 + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x))*sinh(x
)^3 + 4*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2 + 4*(7*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^6 + 15*(a^4 - 2*a^2*b^2 + b^4
)*cosh(x)^4 + a^4 - 2*a^2*b^2 + b^4 + 9*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2 + 8*((a^4 - 2*a^2*b^2 + b
^4)*cosh(x)^7 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 3*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3 + (a^4 - 2*a^2*b^2 +
b^4)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))) + 4*(9*(a^3*b - a^2*b^2 - a*b^3 + b^4)*cosh(x)^5 +
6*(3*a^3*b - 2*a^2*b^2 - 5*a*b^3 + 4*b^4)*cosh(x)^3 + (9*a^3*b - 3*a^2*b^2 - 17*a*b^3 + 3*b^4)*cosh(x))*sinh(x
))/(b^5*cosh(x)^8 + 8*b^5*cosh(x)*sinh(x)^7 + b^5*sinh(x)^8 + 4*b^5*cosh(x)^6 + 6*b^5*cosh(x)^4 + 4*b^5*cosh(x
)^2 + 4*(7*b^5*cosh(x)^2 + b^5)*sinh(x)^6 + 8*(7*b^5*cosh(x)^3 + 3*b^5*cosh(x))*sinh(x)^5 + b^5 + 2*(35*b^5*co
sh(x)^4 + 30*b^5*cosh(x)^2 + 3*b^5)*sinh(x)^4 + 8*(7*b^5*cosh(x)^5 + 10*b^5*cosh(x)^3 + 3*b^5*cosh(x))*sinh(x)
^3 + 4*(7*b^5*cosh(x)^6 + 15*b^5*cosh(x)^4 + 9*b^5*cosh(x)^2 + b^5)*sinh(x)^2 + 8*(b^5*cosh(x)^7 + 3*b^5*cosh(
x)^5 + 3*b^5*cosh(x)^3 + b^5*cosh(x))*sinh(x))

Sympy [F]

\[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=\int \frac {\operatorname {sech}^{6}{\left (x \right )}}{a + b \tanh {\left (x \right )}}\, dx \]

[In]

integrate(sech(x)**6/(a+b*tanh(x)),x)

[Out]

Integral(sech(x)**6/(a + b*tanh(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (77) = 154\).

Time = 0.28 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.46 \[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=-\frac {2 \, {\left (3 \, a^{3} - 5 \, a b^{2} + {\left (9 \, a^{3} + 3 \, a^{2} b - 17 \, a b^{2} - 3 \, b^{3}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (3 \, a^{3} + 2 \, a^{2} b - 5 \, a b^{2} - 4 \, b^{3}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (a^{3} + a^{2} b - a b^{2} - b^{3}\right )} e^{\left (-6 \, x\right )}\right )}}{3 \, {\left (4 \, b^{4} e^{\left (-2 \, x\right )} + 6 \, b^{4} e^{\left (-4 \, x\right )} + 4 \, b^{4} e^{\left (-6 \, x\right )} + b^{4} e^{\left (-8 \, x\right )} + b^{4}\right )}} + \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{b^{5}} - \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{5}} \]

[In]

integrate(sech(x)^6/(a+b*tanh(x)),x, algorithm="maxima")

[Out]

-2/3*(3*a^3 - 5*a*b^2 + (9*a^3 + 3*a^2*b - 17*a*b^2 - 3*b^3)*e^(-2*x) + 3*(3*a^3 + 2*a^2*b - 5*a*b^2 - 4*b^3)*
e^(-4*x) + 3*(a^3 + a^2*b - a*b^2 - b^3)*e^(-6*x))/(4*b^4*e^(-2*x) + 6*b^4*e^(-4*x) + 4*b^4*e^(-6*x) + b^4*e^(
-8*x) + b^4) + (a^4 - 2*a^2*b^2 + b^4)*log(-(a - b)*e^(-2*x) - a - b)/b^5 - (a^4 - 2*a^2*b^2 + b^4)*log(e^(-2*
x) + 1)/b^5

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (77) = 154\).

Time = 0.27 (sec) , antiderivative size = 316, normalized size of antiderivative = 3.81 \[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=\frac {{\left (a^{5} + a^{4} b - 2 \, a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4} + b^{5}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b^{5} + b^{6}} - \frac {{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (e^{\left (2 \, x\right )} + 1\right )}{b^{5}} + \frac {25 \, a^{4} e^{\left (8 \, x\right )} - 50 \, a^{2} b^{2} e^{\left (8 \, x\right )} + 25 \, b^{4} e^{\left (8 \, x\right )} + 100 \, a^{4} e^{\left (6 \, x\right )} + 24 \, a^{3} b e^{\left (6 \, x\right )} - 224 \, a^{2} b^{2} e^{\left (6 \, x\right )} - 24 \, a b^{3} e^{\left (6 \, x\right )} + 124 \, b^{4} e^{\left (6 \, x\right )} + 150 \, a^{4} e^{\left (4 \, x\right )} + 72 \, a^{3} b e^{\left (4 \, x\right )} - 348 \, a^{2} b^{2} e^{\left (4 \, x\right )} - 120 \, a b^{3} e^{\left (4 \, x\right )} + 246 \, b^{4} e^{\left (4 \, x\right )} + 100 \, a^{4} e^{\left (2 \, x\right )} + 72 \, a^{3} b e^{\left (2 \, x\right )} - 224 \, a^{2} b^{2} e^{\left (2 \, x\right )} - 136 \, a b^{3} e^{\left (2 \, x\right )} + 124 \, b^{4} e^{\left (2 \, x\right )} + 25 \, a^{4} + 24 \, a^{3} b - 50 \, a^{2} b^{2} - 40 \, a b^{3} + 25 \, b^{4}}{12 \, b^{5} {\left (e^{\left (2 \, x\right )} + 1\right )}^{4}} \]

[In]

integrate(sech(x)^6/(a+b*tanh(x)),x, algorithm="giac")

[Out]

(a^5 + a^4*b - 2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a*b^5 + b^6) - (a
^4 - 2*a^2*b^2 + b^4)*log(e^(2*x) + 1)/b^5 + 1/12*(25*a^4*e^(8*x) - 50*a^2*b^2*e^(8*x) + 25*b^4*e^(8*x) + 100*
a^4*e^(6*x) + 24*a^3*b*e^(6*x) - 224*a^2*b^2*e^(6*x) - 24*a*b^3*e^(6*x) + 124*b^4*e^(6*x) + 150*a^4*e^(4*x) +
72*a^3*b*e^(4*x) - 348*a^2*b^2*e^(4*x) - 120*a*b^3*e^(4*x) + 246*b^4*e^(4*x) + 100*a^4*e^(2*x) + 72*a^3*b*e^(2
*x) - 224*a^2*b^2*e^(2*x) - 136*a*b^3*e^(2*x) + 124*b^4*e^(2*x) + 25*a^4 + 24*a^3*b - 50*a^2*b^2 - 40*a*b^3 +
25*b^4)/(b^5*(e^(2*x) + 1)^4)

Mupad [B] (verification not implemented)

Time = 1.88 (sec) , antiderivative size = 169, normalized size of antiderivative = 2.04 \[ \int \frac {\text {sech}^6(x)}{a+b \tanh (x)} \, dx=\frac {4}{b\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}+\frac {2\,{\left (a-b\right )}^2}{b^3\,\left (2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1\right )}+\frac {8\,\left (a-3\,b\right )}{3\,b^2\,\left (3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1\right )}+\frac {2\,\left (a+b\right )\,{\left (a-b\right )}^2}{b^4\,\left ({\mathrm {e}}^{2\,x}+1\right )}+\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{b^5}-\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,{\left (a+b\right )}^2\,{\left (a-b\right )}^2}{b^5} \]

[In]

int(1/(cosh(x)^6*(a + b*tanh(x))),x)

[Out]

4/(b*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1)) + (2*(a - b)^2)/(b^3*(2*exp(2*x) + exp(4*x) + 1))
+ (8*(a - 3*b))/(3*b^2*(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1)) + (2*(a + b)*(a - b)^2)/(b^4*(exp(2*x) + 1))
+ (log(a - b + a*exp(2*x) + b*exp(2*x))*(a + b)^2*(a - b)^2)/b^5 - (log(exp(2*x) + 1)*(a + b)^2*(a - b)^2)/b^5