Integrand size = 23, antiderivative size = 148 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {x}{a^3}-\frac {\sqrt {a+b} \left (3 a^2-4 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 a^3 b^{5/2} d}-\frac {(a+b) \tanh ^3(c+d x)}{4 a b d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{8 a^2 b^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
x/a^3-1/8*(3*a^2-4*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))*(a+ b)^(1/2)/a^3/b^(5/2)/d-1/4*(a+b)*tanh(d*x+c)^3/a/b/d/(a+b-b*tanh(d*x+c)^2) ^2+1/8*(3*a-4*b)*(a+b)*tanh(d*x+c)/a^2/b^2/d/(a+b-b*tanh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 7.97 (sec) , antiderivative size = 754, normalized size of antiderivative = 5.09 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\left (3 a^3-a^2 b+4 a b^2+8 b^3\right ) (a+2 b+a \cosh (2 c+2 d x))^3 \text {sech}^6(c+d x) \left (\frac {i \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \cosh (2 c)}{64 a^3 b^2 \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}-\frac {i \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \sinh (2 c)}{64 a^3 b^2 \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right )}{\left (a+b \text {sech}^2(c+d x)\right )^3}+\frac {(a+2 b+a \cosh (2 c+2 d x)) \text {sech}(2 c) \text {sech}^6(c+d x) \left (24 a^2 b^2 d x \cosh (2 c)+64 a b^3 d x \cosh (2 c)+64 b^4 d x \cosh (2 c)+16 a^2 b^2 d x \cosh (2 d x)+32 a b^3 d x \cosh (2 d x)+16 a^2 b^2 d x \cosh (4 c+2 d x)+32 a b^3 d x \cosh (4 c+2 d x)+4 a^2 b^2 d x \cosh (2 c+4 d x)+4 a^2 b^2 d x \cosh (6 c+4 d x)-9 a^4 \sinh (2 c)-15 a^3 b \sinh (2 c)+18 a^2 b^2 \sinh (2 c)+72 a b^3 \sinh (2 c)+48 b^4 \sinh (2 c)+9 a^4 \sinh (2 d x)+13 a^3 b \sinh (2 d x)-28 a^2 b^2 \sinh (2 d x)-32 a b^3 \sinh (2 d x)-3 a^4 \sinh (4 c+2 d x)+a^3 b \sinh (4 c+2 d x)+20 a^2 b^2 \sinh (4 c+2 d x)+16 a b^3 \sinh (4 c+2 d x)+3 a^4 \sinh (2 c+4 d x)-3 a^3 b \sinh (2 c+4 d x)-6 a^2 b^2 \sinh (2 c+4 d x)\right )}{128 a^3 b^2 d \left (a+b \text {sech}^2(c+d x)\right )^3} \]
((3*a^3 - a^2*b + 4*a*b^2 + 8*b^3)*(a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[ c + d*x]^6*(((I/64)*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sq rt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cos h[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x ])]*Cosh[2*c])/(a^3*b^2*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) - ( (I/64)*ArcTan[Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c ] - b*Sinh[4*c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Si nh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c] )/(a^3*b^2*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/(a + b*Sech[c + d*x]^2)^3 + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Sech[2*c]*Sech[c + d*x]^6*( 24*a^2*b^2*d*x*Cosh[2*c] + 64*a*b^3*d*x*Cosh[2*c] + 64*b^4*d*x*Cosh[2*c] + 16*a^2*b^2*d*x*Cosh[2*d*x] + 32*a*b^3*d*x*Cosh[2*d*x] + 16*a^2*b^2*d*x*Co sh[4*c + 2*d*x] + 32*a*b^3*d*x*Cosh[4*c + 2*d*x] + 4*a^2*b^2*d*x*Cosh[2*c + 4*d*x] + 4*a^2*b^2*d*x*Cosh[6*c + 4*d*x] - 9*a^4*Sinh[2*c] - 15*a^3*b*Si nh[2*c] + 18*a^2*b^2*Sinh[2*c] + 72*a*b^3*Sinh[2*c] + 48*b^4*Sinh[2*c] + 9 *a^4*Sinh[2*d*x] + 13*a^3*b*Sinh[2*d*x] - 28*a^2*b^2*Sinh[2*d*x] - 32*a*b^ 3*Sinh[2*d*x] - 3*a^4*Sinh[4*c + 2*d*x] + a^3*b*Sinh[4*c + 2*d*x] + 20*a^2 *b^2*Sinh[4*c + 2*d*x] + 16*a*b^3*Sinh[4*c + 2*d*x] + 3*a^4*Sinh[2*c + 4*d *x] - 3*a^3*b*Sinh[2*c + 4*d*x] - 6*a^2*b^2*Sinh[2*c + 4*d*x]))/(128*a^3*b ^2*d*(a + b*Sech[c + d*x]^2)^3)
Time = 0.49 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.17, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 25, 4629, 25, 2075, 372, 440, 397, 219, 221}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan (i c+i d x)^6}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan (i c+i d x)^6}{\left (b \sec (i c+i d x)^2+a\right )^3}dx\) |
| \(\Big \downarrow \) 4629 |
| \(\displaystyle -\frac {\int -\frac {\tanh ^6(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\tanh ^6(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (a+b \left (1-\tanh ^2(c+d x)\right )\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2075 |
| \(\displaystyle \frac {\int \frac {\tanh ^6(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 372 |
| \(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\int \frac {\tanh ^2(c+d x) \left (3 (a+b)-(3 a-b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a b}}{d}\) |
| \(\Big \downarrow \) 440 |
| \(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{2 a b \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\int \frac {(3 a-4 b) (a+b)-\left (3 a^2-b a+4 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a b}}{4 a b}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{2 a b \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(a+b) \left (3 a^2-4 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}-\frac {8 b^2 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}}{2 a b}}{4 a b}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{2 a b \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {(a+b) \left (3 a^2-4 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}-\frac {8 b^2 \text {arctanh}(\tanh (c+d x))}{a}}{2 a b}}{4 a b}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {\frac {(a+b) \tanh ^3(c+d x)}{4 a b \left (a-b \tanh ^2(c+d x)+b\right )^2}-\frac {\frac {(3 a-4 b) (a+b) \tanh (c+d x)}{2 a b \left (a-b \tanh ^2(c+d x)+b\right )}-\frac {\frac {\sqrt {a+b} \left (3 a^2-4 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {b}}-\frac {8 b^2 \text {arctanh}(\tanh (c+d x))}{a}}{2 a b}}{4 a b}}{d}\) |
-((((a + b)*Tanh[c + d*x]^3)/(4*a*b*(a + b - b*Tanh[c + d*x]^2)^2) - (-1/2 *((-8*b^2*ArcTanh[Tanh[c + d*x]])/a + (Sqrt[a + b]*(3*a^2 - 4*a*b + 8*b^2) *ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sqrt[b]))/(a*b) + ((3*a - 4*b)*(a + b)*Tanh[c + d*x])/(2*a*b*(a + b - b*Tanh[c + d*x]^2)))/(4*a*b) )/d)
3.2.58.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 )^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 )) Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a , b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ )*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[g*(b*e - a*f)*(g*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] - Simp[ g^2/(2*b*(b*c - a*d)*(p + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f)*(m - 1) + (d*(b*e - a*f)*(m + 2*q + 1) - b*2*(c *f - d*e)*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, q}, x] && LtQ[p, -1] && GtQ[m, 1]
Int[(u_)^(p_.)*(v_)^(q_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*Expa ndToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{e, m, p, q}, x] && Binomi alQ[{u, v}, x] && EqQ[BinomialDegree[u, x] - BinomialDegree[v, x], 0] && ! BinomialMatchQ[{u, v}, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f _.)*(x_)])^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[ff/f Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2*x^2 )), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && Inte gerQ[n/2] && (IntegerQ[m/2] || EqQ[n, 2])
Leaf count of result is larger than twice the leaf count of optimal. \(388\) vs. \(2(134)=268\).
Time = 191.71 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.63
| method | result | size |
| derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {\frac {2 \left (\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 b^{2}}+\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (3 a^{3}-a^{2} b +4 a \,b^{2}+8 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2}}}{a^{3}}}{d}\) | \(389\) |
| default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{3}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{3}}+\frac {\frac {2 \left (\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 b^{2}}+\frac {\left (9 a^{3}-14 a^{2} b -19 a \,b^{2}+4 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 b^{2}}+\frac {a \left (3 a^{3}+2 a^{2} b -5 a \,b^{2}-4 b^{3}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (3 a^{3}-a^{2} b +4 a \,b^{2}+8 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2}}}{a^{3}}}{d}\) | \(389\) |
| risch | \(\frac {x}{a^{3}}-\frac {3 a^{4} {\mathrm e}^{6 d x +6 c}-a^{3} b \,{\mathrm e}^{6 d x +6 c}-20 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}-16 a \,b^{3} {\mathrm e}^{6 d x +6 c}+9 a^{4} {\mathrm e}^{4 d x +4 c}+15 a^{3} b \,{\mathrm e}^{4 d x +4 c}-18 a^{2} b^{2} {\mathrm e}^{4 d x +4 c}-72 a \,b^{3} {\mathrm e}^{4 d x +4 c}-48 b^{4} {\mathrm e}^{4 d x +4 c}+9 a^{4} {\mathrm e}^{2 d x +2 c}+13 a^{3} b \,{\mathrm e}^{2 d x +2 c}-28 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}-32 a \,b^{3} {\mathrm e}^{2 d x +2 c}+3 a^{4}-3 a^{3} b -6 a^{2} b^{2}}{4 a^{3} d \,b^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{16 b^{3} d a}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{4 b^{2} d \,a^{2}}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{2 b d \,a^{3}}-\frac {3 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{16 b^{3} d a}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{4 b^{2} d \,a^{2}}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{2 b d \,a^{3}}\) | \(565\) |
1/d*(-1/a^3*ln(tanh(1/2*d*x+1/2*c)-1)+1/a^3*ln(1+tanh(1/2*d*x+1/2*c))+2/a^ 3*((1/8*a*(3*a^3+2*a^2*b-5*a*b^2-4*b^3)/b^2*tanh(1/2*d*x+1/2*c)^7+1/8*(9*a ^3-14*a^2*b-19*a*b^2+4*b^3)*a/b^2*tanh(1/2*d*x+1/2*c)^5+1/8*(9*a^3-14*a^2* b-19*a*b^2+4*b^3)*a/b^2*tanh(1/2*d*x+1/2*c)^3+1/8*a*(3*a^3+2*a^2*b-5*a*b^2 -4*b^3)/b^2*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2 *c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2+1/8*(3* a^3-a^2*b+4*a*b^2+8*b^3)/b^2*(-1/4/b^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh (1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))+1/4/b^(1/2)/( a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2-2*tanh(1/2*d*x+1/2*c)*b^(1 /2)+(a+b)^(1/2)))))
Leaf count of result is larger than twice the leaf count of optimal. 2604 vs. \(2 (140) = 280\).
Time = 0.34 (sec) , antiderivative size = 5463, normalized size of antiderivative = 36.91 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\tanh ^{6}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 3239 vs. \(2 (140) = 280\).
Time = 1.02 (sec) , antiderivative size = 3239, normalized size of antiderivative = 21.89 \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
-45/1024*(a + 2*b)*a*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b)) /(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) - 9/512*a^2*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*s qrt((a + b)*b))/(a*e^(2*d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b ^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 45/1024*(a + 2*b)*a*log((a*e^(-2* d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) + 9/512* a^2*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) - 1/1024*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*log((a*e^(2*d*x + 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(2* d*x + 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5*b^2 + 2*a^4*b^3 + a^3*b^4 )*sqrt((a + b)*b)*d) + 1/1024*(3*a^5 - 10*a^4*b + 80*a^3*b^2 + 480*a^2*b^3 + 640*a*b^4 + 256*b^5)*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b) *b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^5*b^2 + 2*a^4 *b^3 + a^3*b^4)*sqrt((a + b)*b)*d) + 5/256*(3*a^2 + 8*a*b + 8*b^2)*log((a* e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d*x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^2*b^2 + 2*a*b^3 + b^4)*sqrt((a + b)*b)*d) - 1/256*(3*a^6 - 12*a^5*b - 204*a^4*b^2 - 384*a^3*b^3 - 192*a^2*b^4 + (3*a^6 - 10*a^5*b - 560*a^4*b^2 - 2080*a^3*b^3 - 2560*a^2*b^4 - 1024*a*b^5)*e...
\[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\tanh \left (d x + c\right )^{6}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\tanh ^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\left ({\mathrm {cosh}\left (c+d\,x\right )}^2-1\right )}^3}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]