Integrand size = 23, antiderivative size = 125 \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {b^2 (6 a+5 b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{7/2} (a+b)^{3/2} d}+\frac {(a-2 b) \sinh (c+d x)}{a^3 d}+\frac {\sinh ^3(c+d x)}{3 a^2 d}-\frac {b^3 \sinh (c+d x)}{2 a^3 (a+b) d \left (a+b+a \sinh ^2(c+d x)\right )} \]
1/2*b^2*(6*a+5*b)*arctan(sinh(d*x+c)*a^(1/2)/(a+b)^(1/2))/a^(7/2)/(a+b)^(3 /2)/d+(a-2*b)*sinh(d*x+c)/a^3/d+1/3*sinh(d*x+c)^3/a^2/d-1/2*b^3*sinh(d*x+c )/a^3/(a+b)/d/(a+b+a*sinh(d*x+c)^2)
Time = 0.96 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\frac {-\frac {6 b^2 (6 a+5 b) \arctan \left (\frac {\sqrt {a+b} \text {csch}(c+d x)}{\sqrt {a}}\right )}{(a+b)^{3/2}}+3 \sqrt {a} \left (3 a-8 b-\frac {4 b^3}{(a+b) (a+2 b+a \cosh (2 (c+d x)))}\right ) \sinh (c+d x)+a^{3/2} \sinh (3 (c+d x))}{12 a^{7/2} d} \]
((-6*b^2*(6*a + 5*b)*ArcTan[(Sqrt[a + b]*Csch[c + d*x])/Sqrt[a]])/(a + b)^ (3/2) + 3*Sqrt[a]*(3*a - 8*b - (4*b^3)/((a + b)*(a + 2*b + a*Cosh[2*(c + d *x)])))*Sinh[c + d*x] + a^(3/2)*Sinh[3*(c + d*x)])/(12*a^(7/2)*d)
Time = 0.35 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4635, 300, 2009}
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.
| \(\displaystyle \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (i c+i d x)^3 \left (a+b \sec (i c+i d x)^2\right )^2}dx\) |
| \(\Big \downarrow \) 4635 |
| \(\displaystyle \frac {\int \frac {\left (\sinh ^2(c+d x)+1\right )^3}{\left (a \sinh ^2(c+d x)+a+b\right )^2}d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 300 |
| \(\displaystyle \frac {\int \left (\frac {\sinh ^2(c+d x)}{a^2}+\frac {a-2 b}{a^3}+\frac {3 a \sinh ^2(c+d x) b^2+(3 a+2 b) b^2}{a^3 \left (a \sinh ^2(c+d x)+a+b\right )^2}\right )d\sinh (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^2 (6 a+5 b) \arctan \left (\frac {\sqrt {a} \sinh (c+d x)}{\sqrt {a+b}}\right )}{2 a^{7/2} (a+b)^{3/2}}-\frac {b^3 \sinh (c+d x)}{2 a^3 (a+b) \left (a \sinh ^2(c+d x)+a+b\right )}+\frac {(a-2 b) \sinh (c+d x)}{a^3}+\frac {\sinh ^3(c+d x)}{3 a^2}}{d}\) |
((b^2*(6*a + 5*b)*ArcTan[(Sqrt[a]*Sinh[c + d*x])/Sqrt[a + b]])/(2*a^(7/2)* (a + b)^(3/2)) + ((a - 2*b)*Sinh[c + d*x])/a^3 + Sinh[c + d*x]^3/(3*a^2) - (b^3*Sinh[c + d*x])/(2*a^3*(a + b)*(a + b + a*Sinh[c + d*x]^2)))/d
3.1.83.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Int [PolynomialDivide[(a + b*x^2)^p, (c + d*x^2)^(-q), x], x] /; FreeQ[{a, b, c , d}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
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. \(320\) vs. \(2(111)=222\).
Time = 1.80 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.57
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{3 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a -2 b}{a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b^{2} \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{2 a +2 b}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (6 a +5 b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{2 a +2 b}\right )}{a^{3}}-\frac {1}{3 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {1}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a -2 b}{a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\) | \(321\) |
| default | \(\frac {-\frac {1}{3 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {a -2 b}{a^{3} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 b^{2} \left (\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} b}{2 a +2 b}-\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right )}}{\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b}+\frac {\left (6 a +5 b \right ) \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 \sqrt {b}}{2 \sqrt {a}}\right )}{2 \sqrt {a +b}\, \sqrt {a}}\right )}{2 a +2 b}\right )}{a^{3}}-\frac {1}{3 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {1}{2 a^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {a -2 b}{a^{3} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{d}\) | \(321\) |
| risch | \(\frac {{\mathrm e}^{3 d x +3 c}}{24 a^{2} d}+\frac {3 \,{\mathrm e}^{d x +c}}{8 a^{2} d}-\frac {{\mathrm e}^{d x +c} b}{a^{3} d}-\frac {3 \,{\mathrm e}^{-d x -c}}{8 a^{2} d}+\frac {{\mathrm e}^{-d x -c} b}{a^{3} d}-\frac {{\mathrm e}^{-3 d x -3 c}}{24 a^{2} d}-\frac {b^{3} {\mathrm e}^{d x +c} \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{3} \left (a +b \right ) d \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}-\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}-\frac {5 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{3}}+\frac {3 b^{2} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{2 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{2}}+\frac {5 b^{3} \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \left (a +b \right ) {\mathrm e}^{d x +c}}{\sqrt {-a^{2}-a b}}-1\right )}{4 \sqrt {-a^{2}-a b}\, \left (a +b \right ) d \,a^{3}}\) | \(423\) |
1/d*(-1/3/a^2/(tanh(1/2*d*x+1/2*c)-1)^3-1/2/a^2/(tanh(1/2*d*x+1/2*c)-1)^2- (a-2*b)/a^3/(tanh(1/2*d*x+1/2*c)-1)+2/a^3*b^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*(6*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))))-1/3/a^2/(1+tanh(1/2* d*x+1/2*c))^3+1/2/a^2/(1+tanh(1/2*d*x+1/2*c))^2-(a-2*b)/a^3/(1+tanh(1/2*d* x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 3084 vs. \(2 (111) = 222\).
Time = 0.33 (sec) , antiderivative size = 5842, normalized size of antiderivative = 46.74 \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\text {Too large to display} \]
\[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {\cosh ^{3}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
\[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
-1/24*(a^3 + a^2*b - (a^3*e^(10*c) + a^2*b*e^(10*c))*e^(10*d*x) - (11*a^3* e^(8*c) - 9*a^2*b*e^(8*c) - 20*a*b^2*e^(8*c))*e^(8*d*x) - 2*(5*a^3*e^(6*c) + 11*a^2*b*e^(6*c) - 42*a*b^2*e^(6*c) - 60*b^3*e^(6*c))*e^(6*d*x) + 2*(5* a^3*e^(4*c) + 11*a^2*b*e^(4*c) - 42*a*b^2*e^(4*c) - 60*b^3*e^(4*c))*e^(4*d *x) + (11*a^3*e^(2*c) - 9*a^2*b*e^(2*c) - 20*a*b^2*e^(2*c))*e^(2*d*x))/((a ^5*d*e^(7*c) + a^4*b*d*e^(7*c))*e^(7*d*x) + 2*(a^5*d*e^(5*c) + 3*a^4*b*d*e ^(5*c) + 2*a^3*b^2*d*e^(5*c))*e^(5*d*x) + (a^5*d*e^(3*c) + a^4*b*d*e^(3*c) )*e^(3*d*x)) + 1/8*integrate(8*((6*a*b^2*e^(3*c) + 5*b^3*e^(3*c))*e^(3*d*x ) + (6*a*b^2*e^c + 5*b^3*e^c)*e^(d*x))/(a^5 + a^4*b + (a^5*e^(4*c) + a^4*b *e^(4*c))*e^(4*d*x) + 2*(a^5*e^(2*c) + 3*a^4*b*e^(2*c) + 2*a^3*b^2*e^(2*c) )*e^(2*d*x)), x)
\[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int { \frac {\cosh \left (d x + c\right )^{3}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\cosh ^3(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^3}{{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \]