Integrand size = 23, antiderivative size = 144 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\left (3 a^2+8 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 b^{5/2} (a+b)^{5/2} d}-\frac {a \text {sech}^2(c+d x) \tanh (c+d x)}{4 b (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {3 a (a+2 b) \tanh (c+d x)}{8 b^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
1/8*(3*a^2+8*a*b+8*b^2)*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))/b^(5/2)/( a+b)^(5/2)/d-1/4*a*sech(d*x+c)^2*tanh(d*x+c)/b/(a+b)/d/(a+b-b*tanh(d*x+c)^ 2)^2-3/8*a*(a+2*b)*tanh(d*x+c)/b^2/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)
Time = 0.99 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.87 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {\frac {\left (3 a^2+8 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {a \sqrt {b} \left (3 a^2+16 a b+16 b^2+3 a (a+2 b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{(a+b)^2 (a+2 b+a \cosh (2 (c+d x)))^2}}{8 b^{5/2} d} \]
(((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(5/2) - (a*Sqrt[b]*(3*a^2 + 16*a*b + 16*b^2 + 3*a*(a + 2*b)*Cosh[2*( c + d*x)])*Sinh[2*(c + d*x)])/((a + b)^2*(a + 2*b + a*Cosh[2*(c + d*x)])^2 ))/(8*b^(5/2)*d)
Time = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 4634, 315, 25, 298, 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.
| \(\displaystyle \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (i c+i d x)^6}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4634 |
| \(\displaystyle \frac {\int \frac {\left (1-\tanh ^2(c+d x)\right )^2}{\left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {-\frac {\int -\frac {-\left ((3 a+4 b) \tanh ^2(c+d x)\right )+a+4 b}{\left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 b (a+b)}-\frac {a \tanh (c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {-\left ((3 a+4 b) \tanh ^2(c+d x)\right )+a+4 b}{\left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 b (a+b)}-\frac {a \tanh (c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {\frac {\left (3 a^2+8 a b+8 b^2\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{2 b (a+b)}-\frac {3 a (a+2 b) \tanh (c+d x)}{2 b (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 b (a+b)}-\frac {a \tanh (c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {\left (3 a^2+8 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{2 b^{3/2} (a+b)^{3/2}}-\frac {3 a (a+2 b) \tanh (c+d x)}{2 b (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}}{4 b (a+b)}-\frac {a \tanh (c+d x) \left (1-\tanh ^2(c+d x)\right )}{4 b (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
(-1/4*(a*Tanh[c + d*x]*(1 - Tanh[c + d*x]^2))/(b*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) + (((3*a^2 + 8*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/ Sqrt[a + b]])/(2*b^(3/2)*(a + b)^(3/2)) - (3*a*(a + 2*b)*Tanh[c + d*x])/(2 *b*(a + b)*(a + b - b*Tanh[c + d*x]^2)))/(4*b*(a + b)))/d
3.1.100.3.1 Defintions of rubi rules used
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_) )^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(1 + ff^2*x^2)^(m/2 - 1)*ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p, x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ [m/2] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(327\) vs. \(2(130)=260\).
Time = 1.30 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.28
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right ) b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}-\frac {\left (3 a^{2}+8 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(328\) |
| default | \(\frac {-\frac {2 \left (\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (9 a^{2}+13 a b -8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8 \left (a +b \right )^{2} b^{2}}+\frac {a \left (3 a +8 b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 \left (a +b \right ) b^{2}}\right )}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}-\frac {\left (3 a^{2}+8 a b +8 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{4 b^{2} \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(328\) |
| risch | \(\frac {3 a^{3} {\mathrm e}^{6 d x +6 c}+8 a^{2} b \,{\mathrm e}^{6 d x +6 c}+8 a \,b^{2} {\mathrm e}^{6 d x +6 c}+9 a^{3} {\mathrm e}^{4 d x +4 c}+42 a^{2} b \,{\mathrm e}^{4 d x +4 c}+72 a \,b^{2} {\mathrm e}^{4 d x +4 c}+48 \,{\mathrm e}^{4 d x +4 c} b^{3}+9 a^{3} {\mathrm e}^{2 d x +2 c}+40 a^{2} b \,{\mathrm e}^{2 d x +2 c}+40 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}+3 a^{3}+6 a^{2} b}{4 d \,b^{2} \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a^{2}}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d \,b^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}-2 a b -2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a^{2}}{16 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d \,b^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right ) a}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {a b +b^{2}}+2 b \sqrt {a b +b^{2}}+2 a b +2 b^{2}}{a \sqrt {a b +b^{2}}}\right )}{2 \sqrt {a b +b^{2}}\, \left (a +b \right )^{2} d}\) | \(688\) |
1/d*(-2*(1/8*a*(3*a+8*b)/(a+b)/b^2*tanh(1/2*d*x+1/2*c)^7+1/8*a*(9*a^2+13*a *b-8*b^2)/(a+b)^2/b^2*tanh(1/2*d*x+1/2*c)^5+1/8*a*(9*a^2+13*a*b-8*b^2)/(a+ b)^2/b^2*tanh(1/2*d*x+1/2*c)^3+1/8*a*(3*a+8*b)/(a+b)/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)^2-1/4*(3*a^2+8*a*b+8*b^2)/b^2/(a^2+2*a *b+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*t anh(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. 2823 vs. \(2 (136) = 272\).
Time = 0.33 (sec) , antiderivative size = 5887, normalized size of antiderivative = 40.88 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {sech}^{6}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 395 vs. \(2 (136) = 272\).
Time = 0.41 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.74 \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {{\left (3 \, a^{2} + 8 \, a b + 8 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {3 \, a^{3} + 6 \, a^{2} b + {\left (9 \, a^{3} + 40 \, a^{2} b + 40 \, a b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (3 \, a^{3} + 14 \, a^{2} b + 24 \, a b^{2} + 16 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (3 \, a^{3} + 8 \, a^{2} b + 8 \, a b^{2}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{4 \, {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (3 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 27 \, a^{2} b^{4} + 24 \, a b^{5} + 8 \, b^{6}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (a^{4} b^{2} + 4 \, a^{3} b^{3} + 5 \, a^{2} b^{4} + 2 \, a b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} \]
-1/16*(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) - 1/4*(3*a^3 + 6*a^2*b + (9*a^3 + 40*a^ 2*b + 40*a*b^2)*e^(-2*d*x - 2*c) + 3*(3*a^3 + 14*a^2*b + 24*a*b^2 + 16*b^3 )*e^(-4*d*x - 4*c) + (3*a^3 + 8*a^2*b + 8*a*b^2)*e^(-6*d*x - 6*c))/((a^4*b ^2 + 2*a^3*b^3 + a^2*b^4 + 4*(a^4*b^2 + 4*a^3*b^3 + 5*a^2*b^4 + 2*a*b^5)*e ^(-2*d*x - 2*c) + 2*(3*a^4*b^2 + 14*a^3*b^3 + 27*a^2*b^4 + 24*a*b^5 + 8*b^ 6)*e^(-4*d*x - 4*c) + 4*(a^4*b^2 + 4*a^3*b^3 + 5*a^2*b^4 + 2*a*b^5)*e^(-6* d*x - 6*c) + (a^4*b^2 + 2*a^3*b^3 + a^2*b^4)*e^(-8*d*x - 8*c))*d)
\[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{6}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {sech}^6(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^6\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \]