Integrand size = 23, antiderivative size = 153 \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=-\frac {(4 a-b) \arctan (\sinh (c+d x))}{2 b^3 d}+\frac {a^{3/2} (4 a+5 b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 b^3 (a+b)^{3/2} d}+\frac {a (2 a+b) \sinh (c+d x)}{2 b^2 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )}+\frac {\text {sech}(c+d x) \tanh (c+d x)}{2 b d \left (a+b+a \sinh ^2(c+d x)\right )} \] Output:
-1/2*(4*a-b)*arctan(sinh(d*x+c))/b^3/d+1/2*a^(3/2)*(4*a+5*b)*arctan(a^(1/2 )*sinh(d*x+c)/(a+b)^(1/2))/b^3/(a+b)^(3/2)/d+1/2*a*(2*a+b)*sinh(d*x+c)/b^2 /(a+b)/d/(a+b+a*sinh(d*x+c)^2)+1/2*sech(d*x+c)*tanh(d*x+c)/b/d/(a+b+a*sinh (d*x+c)^2)
Leaf count is larger than twice the leaf count of optimal. \(489\) vs. \(2(153)=306\).
Time = 3.51 (sec) , antiderivative size = 489, normalized size of antiderivative = 3.20 \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {(a+2 b+a \cosh (2 (c+d x))) \text {sech}(c) \text {sech}^3(c+d x) \left (-a^{3/2} (4 a+5 b) \arctan \left (\frac {\sqrt {a+b} \text {csch}(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} (\cosh (c)+\sinh (c))}{\sqrt {a}}\right ) \cosh ^2(c) (a+2 b+a \cosh (2 (c+d x))) \text {sech}(c+d x)+b (a+b)^{3/2} (a+2 b+a \cosh (2 (c+d x))) \text {sech}^2(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} \sinh (c)-\cosh (c) \text {sech}(c+d x) \left (2 \sqrt {a+b} \left (4 a^2+3 a b-b^2\right ) \arctan \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right ) (a+2 b+a \cosh (2 (c+d x))) \sqrt {(\cosh (c)-\sinh (c))^2}-a^{5/2} b \arctan \left (\frac {\sqrt {a+b} \text {csch}(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} (\cosh (c)+\sinh (c))}{\sqrt {a}}\right ) (13+5 \cosh (2 (c+d x))) \sinh (c)\right )+a^{3/2} \arctan \left (\frac {\sqrt {a+b} \text {csch}(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} (\cosh (c)+\sinh (c))}{\sqrt {a}}\right ) \left (2 a^2+5 b^2+2 a^2 \cosh (2 (c+d x))\right ) \text {sech}(c+d x) \sinh (2 c)+b (a+b)^{3/2} (a+2 b+a \cosh (2 (c+d x))) \text {sech}^3(c+d x) \sqrt {(\cosh (c)-\sinh (c))^2} \sinh (d x)+2 a^2 b \sqrt {a+b} \cosh (c) \sqrt {(\cosh (c)-\sinh (c))^2} \tanh (c+d x)\right )}{8 b^3 (a+b)^{3/2} d \left (a+b \text {sech}^2(c+d x)\right )^2 \sqrt {(\cosh (c)-\sinh (c))^2}} \] Input:
Integrate[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^2,x]
Output:
((a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c]*Sech[c + d*x]^3*(-(a^(3/2)*(4*a + 5*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[ c] + Sinh[c]))/Sqrt[a]]*Cosh[c]^2*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]) + b*(a + b)^(3/2)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^2*S qrt[(Cosh[c] - Sinh[c])^2]*Sinh[c] - Cosh[c]*Sech[c + d*x]*(2*Sqrt[a + b]* (4*a^2 + 3*a*b - b^2)*ArcTan[Tanh[(c + d*x)/2]]*(a + 2*b + a*Cosh[2*(c + d *x)])*Sqrt[(Cosh[c] - Sinh[c])^2] - a^(5/2)*b*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt[(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(13 + 5*Co sh[2*(c + d*x)])*Sinh[c]) + a^(3/2)*ArcTan[(Sqrt[a + b]*Csch[c + d*x]*Sqrt [(Cosh[c] - Sinh[c])^2]*(Cosh[c] + Sinh[c]))/Sqrt[a]]*(2*a^2 + 5*b^2 + 2*a ^2*Cosh[2*(c + d*x)])*Sech[c + d*x]*Sinh[2*c] + b*(a + b)^(3/2)*(a + 2*b + a*Cosh[2*(c + d*x)])*Sech[c + d*x]^3*Sqrt[(Cosh[c] - Sinh[c])^2]*Sinh[d*x ] + 2*a^2*b*Sqrt[a + b]*Cosh[c]*Sqrt[(Cosh[c] - Sinh[c])^2]*Tanh[c + d*x]) )/(8*b^3*(a + b)^(3/2)*d*(a + b*Sech[c + d*x]^2)^2*Sqrt[(Cosh[c] - Sinh[c] )^2])
Time = 0.38 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3042, 4635, 316, 402, 27, 397, 216, 218}
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.
| \(\displaystyle \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (i c+i d x)^7}{\left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {1}{\left (\sinh ^2(c+d x)+1\right )^2 \left (a \sinh ^2(c+d x)+a+b\right )^2}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{2 b \left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {\int \frac {-3 a \sinh ^2(c+d x)+a-b}{\left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )^2}d\sinh (c+d x)}{2 b}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{2 b \left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {\frac {\int \frac {2 \left (2 a^2-(2 a+b) \sinh ^2(c+d x) a+2 b a-b^2\right )}{\left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}d\sinh (c+d x)}{2 b (a+b)}-\frac {a (2 a+b) \sinh (c+d x)}{b (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{2 b \left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {\frac {\int \frac {2 a^2-(2 a+b) \sinh ^2(c+d x) a+2 b a-b^2}{\left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}d\sinh (c+d x)}{b (a+b)}-\frac {a (2 a+b) \sinh (c+d x)}{b (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{2 b \left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {\frac {\frac {(4 a-b) (a+b) \int \frac {1}{\sinh ^2(c+d x)+1}d\sinh (c+d x)}{b}-\frac {a^2 (4 a+5 b) \int \frac {1}{a \sinh ^2(c+d x)+a+b}d\sinh (c+d x)}{b}}{b (a+b)}-\frac {a (2 a+b) \sinh (c+d x)}{b (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{2 b \left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {\frac {\frac {(4 a-b) (a+b) \arctan (\sinh (c+d x))}{b}-\frac {a^2 (4 a+5 b) \int \frac {1}{a \sinh ^2(c+d x)+a+b}d\sinh (c+d x)}{b}}{b (a+b)}-\frac {a (2 a+b) \sinh (c+d x)}{b (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}}{2 b}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\sinh (c+d x)}{2 b \left (\sinh ^2(c+d x)+1\right ) \left (a \sinh ^2(c+d x)+a+b\right )}-\frac {\frac {\frac {(4 a-b) (a+b) \arctan (\sinh (c+d x))}{b}-\frac {a^{3/2} (4 a+5 b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{b \sqrt {a+b}}}{b (a+b)}-\frac {a (2 a+b) \sinh (c+d x)}{b (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}}{2 b}}{d}\) |
Input:
Int[Sech[c + d*x]^7/(a + b*Sech[c + d*x]^2)^2,x]
Output:
(Sinh[c + d*x]/(2*b*(1 + Sinh[c + d*x]^2)*(a + b + a*Sinh[c + d*x]^2)) - ( (((4*a - b)*(a + b)*ArcTan[Sinh[c + d*x]])/b - (a^(3/2)*(4*a + 5*b)*ArcTan [(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(b*Sqrt[a + b]))/(b*(a + b)) - (a*( 2*a + b)*Sinh[c + d*x])/(b*(a + b)*(a + b + a*Sinh[c + d*x]^2)))/(2*b))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[sec[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_ ))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[ff/f Subst[Int[ExpandToSum[b + a*(1 - ff^2*x^2)^(n/2), x]^p/(1 - ff^2*x^2)^((m + n*p + 1)/2), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && In tegerQ[(m - 1)/2] && IntegerQ[n/2] && IntegerQ[p]
Time = 3.05 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.77
| method | result | size |
| derivativedivides | \(\frac {\frac {2 a^{2} \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a +b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a +2 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (4 a +5 b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{2 a +2 b}\right )}{b^{3}}-\frac {2 \left (\frac {\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (4 a -b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{3}}}{d}\) | \(271\) |
| default | \(\frac {\frac {2 a^{2} \left (\frac {-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 \left (a +b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a +2 b}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (4 a +5 b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{2 a +2 b}\right )}{b^{3}}-\frac {2 \left (\frac {\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}}{\left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{2}}+\frac {\left (4 a -b \right ) \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2}\right )}{b^{3}}}{d}\) | \(271\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (2 a^{2} {\mathrm e}^{6 d x +6 c}+a b \,{\mathrm e}^{6 d x +6 c}+2 a^{2} {\mathrm e}^{4 d x +4 c}+5 a b \,{\mathrm e}^{4 d x +4 c}+4 b^{2} {\mathrm e}^{4 d x +4 c}-2 a^{2} {\mathrm e}^{2 d x +2 c}-5 a b \,{\mathrm e}^{2 d x +2 c}-4 b^{2} {\mathrm e}^{2 d x +2 c}-2 a^{2}-a b \right )}{d \,b^{2} \left ({\mathrm e}^{2 d x +2 c}+1\right )^{2} \left (a +b \right ) \left ({\mathrm e}^{4 d x +4 c} a +2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {2 i \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{d \,b^{3}}-\frac {i \ln \left ({\mathrm e}^{d x +c}-i\right )}{2 d \,b^{2}}-\frac {2 i \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{d \,b^{3}}+\frac {i \ln \left ({\mathrm e}^{d x +c}+i\right )}{2 d \,b^{2}}+\frac {\sqrt {-\left (a +b \right ) a}\, a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-\left (a +b \right ) a}\, {\mathrm e}^{d x +c}}{a}-1\right )}{\left (a +b \right )^{2} d \,b^{3}}+\frac {5 \sqrt {-\left (a +b \right ) a}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-\left (a +b \right ) a}\, {\mathrm e}^{d x +c}}{a}-1\right ) a}{4 \left (a +b \right )^{2} d \,b^{2}}-\frac {\sqrt {-\left (a +b \right ) a}\, a^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-\left (a +b \right ) a}\, {\mathrm e}^{d x +c}}{a}-1\right )}{\left (a +b \right )^{2} d \,b^{3}}-\frac {5 \sqrt {-\left (a +b \right ) a}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-\left (a +b \right ) a}\, {\mathrm e}^{d x +c}}{a}-1\right ) a}{4 \left (a +b \right )^{2} d \,b^{2}}\) | \(483\) |
Input:
int(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x,method=_RETURNVERBOSE)
Output:
1/d*(2/b^3*a^2*((-1/2*b/(a+b)*tanh(1/2*d*x+1/2*c)^3+1/2*b/(a+b)*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)+1/2*(4*a+5*b)/(a+b)*(1/2/(a+b)^ (1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)-2*b^(1/2))/a^( 1/2))+1/2/(a+b)^(1/2)/a^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*d*x+1/2*c )+2*b^(1/2))/a^(1/2))))-2/b^3*((1/2*b*tanh(1/2*d*x+1/2*c)^3-1/2*b*tanh(1/2 *d*x+1/2*c))/(1+tanh(1/2*d*x+1/2*c)^2)^2+1/2*(4*a-b)*arctan(tanh(1/2*d*x+1 /2*c))))
Leaf count of result is larger than twice the leaf count of optimal. 3476 vs. \(2 (137) = 274\).
Time = 0.43 (sec) , antiderivative size = 6499, normalized size of antiderivative = 42.48 \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x, algorithm="fricas")
Output:
Too large to include
\[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\operatorname {sech}^{7}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \] Input:
integrate(sech(d*x+c)**7/(a+b*sech(d*x+c)**2)**2,x)
Output:
Integral(sech(c + d*x)**7/(a + b*sech(c + d*x)**2)**2, x)
\[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{7}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \] Input:
integrate(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x, algorithm="maxima")
Output:
((2*a^2*e^(7*c) + a*b*e^(7*c))*e^(7*d*x) + (2*a^2*e^(5*c) + 5*a*b*e^(5*c) + 4*b^2*e^(5*c))*e^(5*d*x) - (2*a^2*e^(3*c) + 5*a*b*e^(3*c) + 4*b^2*e^(3*c ))*e^(3*d*x) - (2*a^2*e^c + a*b*e^c)*e^(d*x))/(a^2*b^2*d + a*b^3*d + (a^2* b^2*d*e^(8*c) + a*b^3*d*e^(8*c))*e^(8*d*x) + 4*(a^2*b^2*d*e^(6*c) + 2*a*b^ 3*d*e^(6*c) + b^4*d*e^(6*c))*e^(6*d*x) + 2*(3*a^2*b^2*d*e^(4*c) + 7*a*b^3* d*e^(4*c) + 4*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^2*b^2*d*e^(2*c) + 2*a*b^3*d* e^(2*c) + b^4*d*e^(2*c))*e^(2*d*x)) - (4*a*e^c - b*e^c)*arctan(e^(d*x + c) )*e^(-c)/(b^3*d) + 128*integrate(1/128*((4*a^3*e^(3*c) + 5*a^2*b*e^(3*c))* e^(3*d*x) + (4*a^3*e^c + 5*a^2*b*e^c)*e^(d*x))/(a^2*b^3 + a*b^4 + (a^2*b^3 *e^(4*c) + a*b^4*e^(4*c))*e^(4*d*x) + 2*(a^2*b^3*e^(2*c) + 3*a*b^4*e^(2*c) + 2*b^5*e^(2*c))*e^(2*d*x)), x)
Exception generated. \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^7\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \] Input:
int(1/(cosh(c + d*x)^7*(a + b/cosh(c + d*x)^2)^2),x)
Output:
int(1/(cosh(c + d*x)^7*(a + b/cosh(c + d*x)^2)^2), x)
Time = 0.55 (sec) , antiderivative size = 6826, normalized size of antiderivative = 44.61 \[ \int \frac {\text {sech}^7(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx =\text {Too large to display} \] Input:
int(sech(d*x+c)^7/(a+b*sech(d*x+c)^2)^2,x)
Output:
( - 16*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**4 - 28*e**(8*c + 8*d*x)*atan (e**(c + d*x))*a**3*b - 8*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a**2*b**2 + 4*e**(8*c + 8*d*x)*atan(e**(c + d*x))*a*b**3 - 64*e**(6*c + 6*d*x)*atan(e* *(c + d*x))*a**4 - 176*e**(6*c + 6*d*x)*atan(e**(c + d*x))*a**3*b - 144*e* *(6*c + 6*d*x)*atan(e**(c + d*x))*a**2*b**2 - 16*e**(6*c + 6*d*x)*atan(e** (c + d*x))*a*b**3 + 16*e**(6*c + 6*d*x)*atan(e**(c + d*x))*b**4 - 96*e**(4 *c + 4*d*x)*atan(e**(c + d*x))*a**4 - 296*e**(4*c + 4*d*x)*atan(e**(c + d* x))*a**3*b - 272*e**(4*c + 4*d*x)*atan(e**(c + d*x))*a**2*b**2 - 40*e**(4* c + 4*d*x)*atan(e**(c + d*x))*a*b**3 + 32*e**(4*c + 4*d*x)*atan(e**(c + d* x))*b**4 - 64*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a**4 - 176*e**(2*c + 2*d *x)*atan(e**(c + d*x))*a**3*b - 144*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a* *2*b**2 - 16*e**(2*c + 2*d*x)*atan(e**(c + d*x))*a*b**3 + 16*e**(2*c + 2*d *x)*atan(e**(c + d*x))*b**4 - 16*atan(e**(c + d*x))*a**4 - 28*atan(e**(c + d*x))*a**3*b - 8*atan(e**(c + d*x))*a**2*b**2 + 4*atan(e**(c + d*x))*a*b* *3 - 8*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)))*a**2 - 10*e**(8*c + 8*d*x)*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2 *sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(sqrt(a)*sqrt(2*sqrt (b)*sqrt(a + b) + a + 2*b)))*a*b - 32*e**(6*c + 6*d*x)*sqrt(b)*sqrt(a)*sqr t(a + b)*sqrt(2*sqrt(b)*sqrt(a + b) + a + 2*b)*atan((e**(c + d*x)*a)/(s...