Integrand size = 23, antiderivative size = 123 \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(4 a+b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{8 a^{3/2} (a+b)^{5/2} d}-\frac {b \sinh (c+d x)}{4 a (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )^2}+\frac {(4 a+b) \sinh (c+d x)}{8 a (a+b)^2 d \left (a+b+a \sinh ^2(c+d x)\right )} \]
1/8*(4*a+b)*arctan(sinh(d*x+c)*a^(1/2)/(a+b)^(1/2))/a^(3/2)/(a+b)^(5/2)/d- 1/4*b*sinh(d*x+c)/a/(a+b)/d/(a+b+a*sinh(d*x+c)^2)^2+1/8*(4*a+b)*sinh(d*x+c )/a/(a+b)^2/d/(a+b+a*sinh(d*x+c)^2)
Time = 0.85 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.29 \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {(a+2 b+a \cosh (2 (c+d x)))^3 \text {sech}^6(c+d x) \left (\frac {8 \sinh (c+d x)}{\left (a+b+a \sinh ^2(c+d x)\right )^2}-(4 a+b) \left (\frac {3 \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {a} (a+b)^{5/2}}+\frac {5 (a+b) \sinh (c+d x)+3 a \sinh ^3(c+d x)}{(a+b)^2 \left (a+b+a \sinh ^2(c+d x)\right )^2}\right )\right )}{192 a d \left (a+b \text {sech}^2(c+d x)\right )^3} \]
-1/192*((a + 2*b + a*Cosh[2*(c + d*x)])^3*Sech[c + d*x]^6*((8*Sinh[c + d*x ])/(a + b + a*Sinh[c + d*x]^2)^2 - (4*a + b)*((3*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(Sqrt[a]*(a + b)^(5/2)) + (5*(a + b)*Sinh[c + d*x] + 3 *a*Sinh[c + d*x]^3)/((a + b)^2*(a + b + a*Sinh[c + d*x]^2)^2))))/(a*d*(a + b*Sech[c + d*x]^2)^3)
Time = 0.29 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3042, 4635, 298, 215, 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.
| \(\displaystyle \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (i c+i d x)^3}{\left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\sinh ^2(c+d x)+1}{\left (a \sinh ^2(c+d x)+a+b\right )^3}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 298 |
| \(\displaystyle \frac {\frac {(4 a+b) \int \frac {1}{\left (a \sinh ^2(c+d x)+a+b\right )^2}d\sinh (c+d x)}{4 a (a+b)}-\frac {b \sinh (c+d x)}{4 a (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 215 |
| \(\displaystyle \frac {\frac {(4 a+b) \left (\frac {\int \frac {1}{a \sinh ^2(c+d x)+a+b}d\sinh (c+d x)}{2 (a+b)}+\frac {\sinh (c+d x)}{2 (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}\right )}{4 a (a+b)}-\frac {b \sinh (c+d x)}{4 a (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {(4 a+b) \left (\frac {\arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 \sqrt {a} (a+b)^{3/2}}+\frac {\sinh (c+d x)}{2 (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}\right )}{4 a (a+b)}-\frac {b \sinh (c+d x)}{4 a (a+b) \left (a \sinh ^2(c+d x)+a+b\right )^2}}{d}\) |
(-1/4*(b*Sinh[c + d*x])/(a*(a + b)*(a + b + a*Sinh[c + d*x]^2)^2) + ((4*a + b)*(ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]]/(2*Sqrt[a]*(a + b)^(3/2) ) + Sinh[c + d*x]/(2*(a + b)*(a + b + a*Sinh[c + d*x]^2))))/(4*a*(a + b))) /d
3.1.97.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 *p])
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), 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[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]
Leaf count of result is larger than twice the leaf count of optimal. \(293\) vs. \(2(109)=218\).
Time = 1.28 (sec) , antiderivative size = 294, normalized size of antiderivative = 2.39
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a \left (a +b \right )}-\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (4 a +b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{4 a \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(294\) |
| default | \(\frac {\frac {-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 a \left (a +b \right )}-\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a^{2}-5 a b +3 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{4 \left (a +b \right )^{2} a}+\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{2}}+\frac {\left (4 a +b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{4 a \left (a^{2}+2 a b +b^{2}\right )}}{d}\) | \(294\) |
| risch | \(\frac {{\mathrm e}^{d x +c} \left (4 a^{2} {\mathrm e}^{6 d x +6 c}+a b \,{\mathrm e}^{6 d x +6 c}+4 a^{2} {\mathrm e}^{4 d x +4 c}+9 a b \,{\mathrm e}^{4 d x +4 c}-4 \,{\mathrm e}^{4 d x +4 c} b^{2}-4 a^{2} {\mathrm e}^{2 d x +2 c}-9 a b \,{\mathrm e}^{2 d x +2 c}+4 \,{\mathrm e}^{2 d x +2 c} b^{2}-4 a^{2}-a b \right )}{4 a \left (a +b \right )^{2} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}-\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d}+\frac {b \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{16 \sqrt {-a^{2}-a b}\, \left (a +b \right )^{2} d a}\) | \(414\) |
1/d*(2*(-1/8*(4*a-b)/a/(a+b)*tanh(1/2*d*x+1/2*c)^7-1/8*(4*a^2-5*a*b+3*b^2) /(a+b)^2/a*tanh(1/2*d*x+1/2*c)^5+1/8*(4*a^2-5*a*b+3*b^2)/(a+b)^2/a*tanh(1/ 2*d*x+1/2*c)^3+1/8*(4*a-b)/a/(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)^2+1/4*(4*a+b)/a/(a^2+2*a*b+b^2)*(1/2/(a+b)^(1/2)/a^(1/2)*arc tan(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))))
Leaf count of result is larger than twice the leaf count of optimal. 3262 vs. \(2 (109) = 218\).
Time = 0.32 (sec) , antiderivative size = 6037, normalized size of antiderivative = 49.08 \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
\[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
1/4*((4*a^2*e^(7*c) + a*b*e^(7*c))*e^(7*d*x) + (4*a^2*e^(5*c) + 9*a*b*e^(5 *c) - 4*b^2*e^(5*c))*e^(5*d*x) - (4*a^2*e^(3*c) + 9*a*b*e^(3*c) - 4*b^2*e^ (3*c))*e^(3*d*x) - (4*a^2*e^c + a*b*e^c)*e^(d*x))/(a^5*d + 2*a^4*b*d + a^3 *b^2*d + (a^5*d*e^(8*c) + 2*a^4*b*d*e^(8*c) + a^3*b^2*d*e^(8*c))*e^(8*d*x) + 4*(a^5*d*e^(6*c) + 4*a^4*b*d*e^(6*c) + 5*a^3*b^2*d*e^(6*c) + 2*a^2*b^3* d*e^(6*c))*e^(6*d*x) + 2*(3*a^5*d*e^(4*c) + 14*a^4*b*d*e^(4*c) + 27*a^3*b^ 2*d*e^(4*c) + 24*a^2*b^3*d*e^(4*c) + 8*a*b^4*d*e^(4*c))*e^(4*d*x) + 4*(a^5 *d*e^(2*c) + 4*a^4*b*d*e^(2*c) + 5*a^3*b^2*d*e^(2*c) + 2*a^2*b^3*d*e^(2*c) )*e^(2*d*x)) + 8*integrate(1/32*((4*a*e^(3*c) + b*e^(3*c))*e^(3*d*x) + (4* a*e^c + b*e^c)*e^(d*x))/(a^4 + 2*a^3*b + a^2*b^2 + (a^4*e^(4*c) + 2*a^3*b* e^(4*c) + a^2*b^2*e^(4*c))*e^(4*d*x) + 2*(a^4*e^(2*c) + 4*a^3*b*e^(2*c) + 5*a^2*b^2*e^(2*c) + 2*a*b^3*e^(2*c))*e^(2*d*x)), x)
\[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {sech}\left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {sech}^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {1}{{\mathrm {cosh}\left (c+d\,x\right )}^3\,{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^3} \,d x \]