\(\int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx\) [113]

Optimal result

Integrand size = 13, antiderivative size = 121 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\left (a^2-b^2\right )^3 \log (a+b \text {sech}(x))}{a b^6}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {sech}(x)}{b^5}-\frac {a \left (a^2-3 b^2\right ) \text {sech}^2(x)}{2 b^4}+\frac {\left (a^2-3 b^2\right ) \text {sech}^3(x)}{3 b^3}-\frac {a \text {sech}^4(x)}{4 b^2}+\frac {\text {sech}^5(x)}{5 b} \]

[Out]

ln(cosh(x))/a-(a^2-b^2)^3*ln(a+b*sech(x))/a/b^6+(a^4-3*a^2*b^2+3*b^4)*sech(x)/b^5-1/2*a*(a^2-3*b^2)*sech(x)^2/
b^4+1/3*(a^2-3*b^2)*sech(x)^3/b^3-1/4*a*sech(x)^4/b^2+1/5*sech(x)^5/b

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 121, 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 = {3970, 908} \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=-\frac {\left (a^2-b^2\right )^3 \log (a+b \text {sech}(x))}{a b^6}-\frac {a \left (a^2-3 b^2\right ) \text {sech}^2(x)}{2 b^4}+\frac {\left (a^2-3 b^2\right ) \text {sech}^3(x)}{3 b^3}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {sech}(x)}{b^5}-\frac {a \text {sech}^4(x)}{4 b^2}+\frac {\log (\cosh (x))}{a}+\frac {\text {sech}^5(x)}{5 b} \]

[In]

Int[Tanh[x]^7/(a + b*Sech[x]),x]

[Out]

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

Rule 908

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 3970

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {\left (b^2-x^2\right )^3}{x (a+x)} \, dx,x,b \text {sech}(x)\right )}{b^6} \\ & = -\frac {\text {Subst}\left (\int \left (-a^4 \left (1+\frac {3 b^2 \left (-a^2+b^2\right )}{a^4}\right )+\frac {b^6}{a x}+a \left (a^2-3 b^2\right ) x-\left (a^2-3 b^2\right ) x^2+a x^3-x^4+\frac {\left (a^2-b^2\right )^3}{a (a+x)}\right ) \, dx,x,b \text {sech}(x)\right )}{b^6} \\ & = \frac {\log (\cosh (x))}{a}-\frac {\left (a^2-b^2\right )^3 \log (a+b \text {sech}(x))}{a b^6}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {sech}(x)}{b^5}-\frac {a \left (a^2-3 b^2\right ) \text {sech}^2(x)}{2 b^4}+\frac {\left (a^2-3 b^2\right ) \text {sech}^3(x)}{3 b^3}-\frac {a \text {sech}^4(x)}{4 b^2}+\frac {\text {sech}^5(x)}{5 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {\log (\cosh (x))}{a}-\frac {\left (a^2-b^2\right )^3 \log (a+b \text {sech}(x))}{a b^6}+\frac {\left (a^4-3 a^2 b^2+3 b^4\right ) \text {sech}(x)}{b^5}-\frac {a \left (a^2-3 b^2\right ) \text {sech}^2(x)}{2 b^4}+\frac {\left (a^2-3 b^2\right ) \text {sech}^3(x)}{3 b^3}-\frac {a \text {sech}^4(x)}{4 b^2}+\frac {\text {sech}^5(x)}{5 b} \]

[In]

Integrate[Tanh[x]^7/(a + b*Sech[x]),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(245\) vs. \(2(113)=226\).

Time = 2.31 (sec) , antiderivative size = 246, normalized size of antiderivative = 2.03

method	result	size
default	\(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\left (a -b \right )^{3} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )}{a \,b^{6}}+\frac {\frac {8 b^{3} \left (a^{2}+3 a b +3 b^{2}\right )}{3 \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{3}}+a \left (a^{4}-3 a^{2} b^{2}+3 b^{4}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\frac {32 b^{5}}{5 \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{5}}-\frac {2 b^{2} \left (a^{3}+2 a^{2} b -2 b^{3}\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\frac {2 b \left (a^{4}+a^{3} b -2 a^{2} b^{2}-2 a \,b^{3}+b^{4}\right )}{1+\tanh \left (\frac {x}{2}\right )^{2}}-\frac {4 b^{4} \left (a +4 b \right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{4}}}{b^{6}}\)	\(246\)
risch	\(-\frac {x}{a}+\frac {2 \,{\mathrm e}^{x} \left (15 a^{4} {\mathrm e}^{8 x}-45 a^{2} b^{2} {\mathrm e}^{8 x}+45 b^{4} {\mathrm e}^{8 x}-15 a^{3} b \,{\mathrm e}^{7 x}+45 a \,b^{3} {\mathrm e}^{7 x}+60 a^{4} {\mathrm e}^{6 x}-160 a^{2} b^{2} {\mathrm e}^{6 x}+120 b^{4} {\mathrm e}^{6 x}-45 a^{3} b \,{\mathrm e}^{5 x}+105 a \,b^{3} {\mathrm e}^{5 x}+90 a^{4} {\mathrm e}^{4 x}-230 a^{2} b^{2} {\mathrm e}^{4 x}+198 b^{4} {\mathrm e}^{4 x}-45 a^{3} b \,{\mathrm e}^{3 x}+105 a \,b^{3} {\mathrm e}^{3 x}+60 a^{4} {\mathrm e}^{2 x}-160 a^{2} b^{2} {\mathrm e}^{2 x}+120 b^{4} {\mathrm e}^{2 x}-15 a^{3} b \,{\mathrm e}^{x}+45 a \,b^{3} {\mathrm e}^{x}+15 a^{4}-45 a^{2} b^{2}+45 b^{4}\right )}{15 b^{5} \left (1+{\mathrm e}^{2 x}\right )^{5}}+\frac {a^{5} \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{6}}-\frac {3 a^{3} \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{4}}+\frac {3 a \ln \left (1+{\mathrm e}^{2 x}\right )}{b^{2}}-\frac {a^{5} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{6}}+\frac {3 a^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{4}}-\frac {3 a \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{b^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}+1\right )}{a}\)	\(366\)

[In]

int(tanh(x)^7/(a+b*sech(x)),x,method=_RETURNVERBOSE)

[Out]

-1/a*ln(tanh(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)-(a-b)^3*(a^3+3*a^2*b+3*a*b^2+b^3)/a/b^6*ln(tanh(1/2*x)^2*a-tanh(1
/2*x)^2*b+a+b)+1/b^6*(8/3*b^3*(a^2+3*a*b+3*b^2)/(1+tanh(1/2*x)^2)^3+a*(a^4-3*a^2*b^2+3*b^4)*ln(1+tanh(1/2*x)^2
)+32/5*b^5/(1+tanh(1/2*x)^2)^5-2*b^2*(a^3+2*a^2*b-2*b^3)/(1+tanh(1/2*x)^2)^2+2*b*(a^4+a^3*b-2*a^2*b^2-2*a*b^3+
b^4)/(1+tanh(1/2*x)^2)-4*b^4*(a+4*b)/(1+tanh(1/2*x)^2)^4)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4077 vs. \(2 (113) = 226\).

Time = 0.31 (sec) , antiderivative size = 4077, normalized size of antiderivative = 33.69 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\text {Too large to display} \]

[In]

integrate(tanh(x)^7/(a+b*sech(x)),x, algorithm="fricas")

[Out]

-1/15*(15*b^6*x*cosh(x)^10 + 15*b^6*x*sinh(x)^10 - 30*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x)^9 + 30*(5*b^6*x*co
sh(x) - a^5*b + 3*a^3*b^3 - 3*a*b^5)*sinh(x)^9 + 15*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^8 + 15*(45*b^6*x
*cosh(x)^2 + 5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4 - 18*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x))*sinh(x)^8 - 40*(3*a^5
*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^7 + 40*(45*b^6*x*cosh(x)^3 - 3*a^5*b + 8*a^3*b^3 - 6*a*b^5 - 27*(a^5*b - 3*a
^3*b^3 + 3*a*b^5)*cosh(x)^2 + 3*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x))*sinh(x)^7 + 15*b^6*x + 30*(5*b^6*x
+ 3*a^4*b^2 - 7*a^2*b^4)*cosh(x)^6 + 10*(315*b^6*x*cosh(x)^4 + 15*b^6*x + 9*a^4*b^2 - 21*a^2*b^4 - 252*(a^5*b
- 3*a^3*b^3 + 3*a*b^5)*cosh(x)^3 + 42*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^2 - 28*(3*a^5*b - 8*a^3*b^3 +
6*a*b^5)*cosh(x))*sinh(x)^6 - 4*(45*a^5*b - 115*a^3*b^3 + 99*a*b^5)*cosh(x)^5 + 4*(945*b^6*x*cosh(x)^5 - 45*a^
5*b + 115*a^3*b^3 - 99*a*b^5 - 945*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x)^4 + 210*(5*b^6*x + 2*a^4*b^2 - 6*a^2*
b^4)*cosh(x)^3 - 210*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^2 + 45*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x))
*sinh(x)^5 + 30*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x)^4 + 10*(315*b^6*x*cosh(x)^6 + 15*b^6*x + 9*a^4*b^2 -
21*a^2*b^4 - 378*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x)^5 + 105*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^4 -
140*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^3 + 45*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x)^2 - 2*(45*a^5*b -
115*a^3*b^3 + 99*a*b^5)*cosh(x))*sinh(x)^4 - 40*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^3 + 40*(45*b^6*x*cosh
(x)^7 - 63*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x)^6 - 3*a^5*b + 8*a^3*b^3 - 6*a*b^5 + 21*(5*b^6*x + 2*a^4*b^2 -
6*a^2*b^4)*cosh(x)^5 - 35*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^4 + 15*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*co
sh(x)^3 - (45*a^5*b - 115*a^3*b^3 + 99*a*b^5)*cosh(x)^2 + 3*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x))*sinh(x)
^3 + 15*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^2 + 5*(135*b^6*x*cosh(x)^8 - 216*(a^5*b - 3*a^3*b^3 + 3*a*b^
5)*cosh(x)^7 + 15*b^6*x + 84*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x)^6 + 6*a^4*b^2 - 18*a^2*b^4 - 168*(3*a^5
*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^5 + 90*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x)^4 - 8*(45*a^5*b - 115*a^3*b
^3 + 99*a*b^5)*cosh(x)^3 + 36*(5*b^6*x + 3*a^4*b^2 - 7*a^2*b^4)*cosh(x)^2 - 24*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)
*cosh(x))*sinh(x)^2 - 30*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x) + 15*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(
x)^10 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)*sinh(x)^9 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)
^10 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^8 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 9*(a^6 - 3*a^4*
b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + (a^6 - 3*a
^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^7 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + 10*(a^6 - 3*
a^4*b^2 + 3*a^2*b^4 - b^6 + 21*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 14*(a^6 - 3*a^4*b^2 + 3*a^2*b^4
- b^6)*cosh(x)^2)*sinh(x)^6 + a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 4*(63*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*c
osh(x)^5 + 70*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*
sinh(x)^5 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 10*(21*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh
(x)^6 + a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 35*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 15*(a^6 - 3*a^4
*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 + 7*(a^6 -
3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + (a^6 - 3*a^4*b^2 +
3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^3 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2 + 5*(9*(a^6 - 3*a^4*b^2
+ 3*a^2*b^4 - b^6)*cosh(x)^8 + 28*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + a^6 - 3*a^4*b^2 + 3*a^2*b^4
- b^6 + 30*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 12*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*s
inh(x)^2 + 10*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^9 + 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7
+ 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 + 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + (a^6 -
3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x))*log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))) - 15*((a^6 - 3*a^4*b
^2 + 3*a^2*b^4)*cosh(x)^10 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)*sinh(x)^9 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4
)*sinh(x)^10 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^8 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 + 9*(a^6 - 3*a^4*b^2
+ 3*a^2*b^4)*cosh(x)^2)*sinh(x)^8 + 40*(3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 + (a^6 - 3*a^4*b^2 + 3*a^2*
b^4)*cosh(x))*sinh(x)^7 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^6 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 + 21*(a
^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4 + 14*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2)*sinh(x)^6 + a^6 - 3*a^4*b^
2 + 3*a^2*b^4 + 4*(63*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^5 + 70*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 + 1
5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x))*sinh(x)^5 + 10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4 + 10*(21*(a^6
- 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^6 + a^6 - 3*a^4*b^2 + 3*a^2*b^4 + 35*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^4
+ 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2)*sinh(x)^4 + 40*(3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^7 + 7*(a
^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^5 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^3 + (a^6 - 3*a^4*b^2 + 3*a^2*b
^4)*cosh(x))*sinh(x)^3 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2 + 5*(9*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x
)^8 + 28*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^6 + a^6 - 3*a^4*b^2 + 3*a^2*b^4 + 30*(a^6 - 3*a^4*b^2 + 3*a^2*b
^4)*cosh(x)^4 + 12*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^2)*sinh(x)^2 + 10*((a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh
(x)^9 + 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^7 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x)^5 + 4*(a^6 - 3*a^4
*b^2 + 3*a^2*b^4)*cosh(x)^3 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x)
)) + 10*(15*b^6*x*cosh(x)^9 - 27*(a^5*b - 3*a^3*b^3 + 3*a*b^5)*cosh(x)^8 + 12*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4
)*cosh(x)^7 - 28*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^6 - 3*a^5*b + 9*a^3*b^3 - 9*a*b^5 + 18*(5*b^6*x + 3*a
^4*b^2 - 7*a^2*b^4)*cosh(x)^5 - 2*(45*a^5*b - 115*a^3*b^3 + 99*a*b^5)*cosh(x)^4 + 12*(5*b^6*x + 3*a^4*b^2 - 7*
a^2*b^4)*cosh(x)^3 - 12*(3*a^5*b - 8*a^3*b^3 + 6*a*b^5)*cosh(x)^2 + 3*(5*b^6*x + 2*a^4*b^2 - 6*a^2*b^4)*cosh(x
))*sinh(x))/(a*b^6*cosh(x)^10 + 10*a*b^6*cosh(x)*sinh(x)^9 + a*b^6*sinh(x)^10 + 5*a*b^6*cosh(x)^8 + 10*a*b^6*c
osh(x)^6 + 10*a*b^6*cosh(x)^4 + 5*a*b^6*cosh(x)^2 + 5*(9*a*b^6*cosh(x)^2 + a*b^6)*sinh(x)^8 + 40*(3*a*b^6*cosh
(x)^3 + a*b^6*cosh(x))*sinh(x)^7 + a*b^6 + 10*(21*a*b^6*cosh(x)^4 + 14*a*b^6*cosh(x)^2 + a*b^6)*sinh(x)^6 + 4*
(63*a*b^6*cosh(x)^5 + 70*a*b^6*cosh(x)^3 + 15*a*b^6*cosh(x))*sinh(x)^5 + 10*(21*a*b^6*cosh(x)^6 + 35*a*b^6*cos
h(x)^4 + 15*a*b^6*cosh(x)^2 + a*b^6)*sinh(x)^4 + 40*(3*a*b^6*cosh(x)^7 + 7*a*b^6*cosh(x)^5 + 5*a*b^6*cosh(x)^3
+ a*b^6*cosh(x))*sinh(x)^3 + 5*(9*a*b^6*cosh(x)^8 + 28*a*b^6*cosh(x)^6 + 30*a*b^6*cosh(x)^4 + 12*a*b^6*cosh(x
)^2 + a*b^6)*sinh(x)^2 + 10*(a*b^6*cosh(x)^9 + 4*a*b^6*cosh(x)^7 + 6*a*b^6*cosh(x)^5 + 4*a*b^6*cosh(x)^3 + a*b
^6*cosh(x))*sinh(x))

Sympy [F]

\[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\int \frac {\tanh ^{7}{\left (x \right )}}{a + b \operatorname {sech}{\left (x \right )}}\, dx \]

[In]

integrate(tanh(x)**7/(a+b*sech(x)),x)

[Out]

Integral(tanh(x)**7/(a + b*sech(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 332 vs. \(2 (113) = 226\).

Time = 0.29 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.74 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {2 \, {\left (15 \, {\left (a^{4} - 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-x\right )} - 15 \, {\left (a^{3} b - 3 \, a b^{3}\right )} e^{\left (-2 \, x\right )} + 20 \, {\left (3 \, a^{4} - 8 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-3 \, x\right )} - 15 \, {\left (3 \, a^{3} b - 7 \, a b^{3}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (45 \, a^{4} - 115 \, a^{2} b^{2} + 99 \, b^{4}\right )} e^{\left (-5 \, x\right )} - 15 \, {\left (3 \, a^{3} b - 7 \, a b^{3}\right )} e^{\left (-6 \, x\right )} + 20 \, {\left (3 \, a^{4} - 8 \, a^{2} b^{2} + 6 \, b^{4}\right )} e^{\left (-7 \, x\right )} - 15 \, {\left (a^{3} b - 3 \, a b^{3}\right )} e^{\left (-8 \, x\right )} + 15 \, {\left (a^{4} - 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} e^{\left (-9 \, x\right )}\right )}}{15 \, {\left (5 \, b^{5} e^{\left (-2 \, x\right )} + 10 \, b^{5} e^{\left (-4 \, x\right )} + 10 \, b^{5} e^{\left (-6 \, x\right )} + 5 \, b^{5} e^{\left (-8 \, x\right )} + b^{5} e^{\left (-10 \, x\right )} + b^{5}\right )}} + \frac {x}{a} + \frac {{\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{b^{6}} - \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a b^{6}} \]

[In]

integrate(tanh(x)^7/(a+b*sech(x)),x, algorithm="maxima")

[Out]

2/15*(15*(a^4 - 3*a^2*b^2 + 3*b^4)*e^(-x) - 15*(a^3*b - 3*a*b^3)*e^(-2*x) + 20*(3*a^4 - 8*a^2*b^2 + 6*b^4)*e^(
-3*x) - 15*(3*a^3*b - 7*a*b^3)*e^(-4*x) + 2*(45*a^4 - 115*a^2*b^2 + 99*b^4)*e^(-5*x) - 15*(3*a^3*b - 7*a*b^3)*
e^(-6*x) + 20*(3*a^4 - 8*a^2*b^2 + 6*b^4)*e^(-7*x) - 15*(a^3*b - 3*a*b^3)*e^(-8*x) + 15*(a^4 - 3*a^2*b^2 + 3*b
^4)*e^(-9*x))/(5*b^5*e^(-2*x) + 10*b^5*e^(-4*x) + 10*b^5*e^(-6*x) + 5*b^5*e^(-8*x) + b^5*e^(-10*x) + b^5) + x/
a + (a^5 - 3*a^3*b^2 + 3*a*b^4)*log(e^(-2*x) + 1)/b^6 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(2*b*e^(-x) + a
*e^(-2*x) + a)/(a*b^6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (113) = 226\).

Time = 0.31 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.21 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {{\left (a^{5} - 3 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \log \left (e^{\left (-x\right )} + e^{x}\right )}{b^{6}} - \frac {{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a b^{6}} - \frac {137 \, a^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5} - 411 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5} + 411 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5} - 120 \, a^{4} b {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 360 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 360 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 120 \, a^{3} b^{2} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 360 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 160 \, a^{2} b^{3} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 480 \, b^{5} {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 240 \, a b^{4} {\left (e^{\left (-x\right )} + e^{x}\right )} - 384 \, b^{5}}{60 \, b^{6} {\left (e^{\left (-x\right )} + e^{x}\right )}^{5}} \]

[In]

integrate(tanh(x)^7/(a+b*sech(x)),x, algorithm="giac")

[Out]

(a^5 - 3*a^3*b^2 + 3*a*b^4)*log(e^(-x) + e^x)/b^6 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*log(abs(a*(e^(-x) + e^
x) + 2*b))/(a*b^6) - 1/60*(137*a^5*(e^(-x) + e^x)^5 - 411*a^3*b^2*(e^(-x) + e^x)^5 + 411*a*b^4*(e^(-x) + e^x)^
5 - 120*a^4*b*(e^(-x) + e^x)^4 + 360*a^2*b^3*(e^(-x) + e^x)^4 - 360*b^5*(e^(-x) + e^x)^4 + 120*a^3*b^2*(e^(-x)
+ e^x)^3 - 360*a*b^4*(e^(-x) + e^x)^3 - 160*a^2*b^3*(e^(-x) + e^x)^2 + 480*b^5*(e^(-x) + e^x)^2 + 240*a*b^4*(
e^(-x) + e^x) - 384*b^5)/(b^6*(e^(-x) + e^x)^5)

Mupad [B] (verification not implemented)

Time = 2.59 (sec) , antiderivative size = 316, normalized size of antiderivative = 2.61 \[ \int \frac {\tanh ^7(x)}{a+b \text {sech}(x)} \, dx=\frac {\frac {8\,a}{b^2}-\frac {8\,{\mathrm {e}}^x\,\left (5\,a^2-27\,b^2\right )}{15\,b^3}}{3\,{\mathrm {e}}^{2\,x}+3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}+1}-\frac {\frac {4\,a}{b^2}+\frac {64\,{\mathrm {e}}^x}{5\,b}}{4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1}+\frac {\frac {8\,{\mathrm {e}}^x\,\left (a^2-3\,b^2\right )}{3\,b^3}+\frac {2\,\left (a^4-5\,a^2\,b^2\right )}{a\,b^4}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {x}{a}+\frac {\frac {2\,{\mathrm {e}}^x\,\left (a^4-3\,a^2\,b^2+3\,b^4\right )}{b^5}-\frac {2\,\left (a^4-3\,a^2\,b^2\right )}{a\,b^4}}{{\mathrm {e}}^{2\,x}+1}+\frac {32\,{\mathrm {e}}^x}{5\,b\,\left (5\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{4\,x}+10\,{\mathrm {e}}^{6\,x}+5\,{\mathrm {e}}^{8\,x}+{\mathrm {e}}^{10\,x}+1\right )}+\frac {\ln \left ({\mathrm {e}}^{2\,x}+1\right )\,\left (a^5-3\,a^3\,b^2+3\,a\,b^4\right )}{b^6}-\frac {\ln \left (a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}{a\,b^6} \]

[In]

int(tanh(x)^7/(a + b/cosh(x)),x)

[Out]

((8*a)/b^2 - (8*exp(x)*(5*a^2 - 27*b^2))/(15*b^3))/(3*exp(2*x) + 3*exp(4*x) + exp(6*x) + 1) - ((4*a)/b^2 + (64
*exp(x))/(5*b))/(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1) + ((8*exp(x)*(a^2 - 3*b^2))/(3*b^3) + (2
*(a^4 - 5*a^2*b^2))/(a*b^4))/(2*exp(2*x) + exp(4*x) + 1) - x/a + ((2*exp(x)*(a^4 + 3*b^4 - 3*a^2*b^2))/b^5 - (
2*(a^4 - 3*a^2*b^2))/(a*b^4))/(exp(2*x) + 1) + (32*exp(x))/(5*b*(5*exp(2*x) + 10*exp(4*x) + 10*exp(6*x) + 5*ex
p(8*x) + exp(10*x) + 1)) + (log(exp(2*x) + 1)*(3*a*b^4 + a^5 - 3*a^3*b^2))/b^6 - (log(a + 2*b*exp(x) + a*exp(2
*x))*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2))/(a*b^6)