Integrand size = 10, antiderivative size = 107 \[ \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx=\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{8 a^{5/2} (a+b)^{5/2}}-\frac {b \cosh (x) \sinh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}-\frac {3 b (2 a+b) \cosh (x) \sinh (x)}{8 a^2 (a+b)^2 \left (a+b \cosh ^2(x)\right )} \]
1/8*(8*a^2+8*a*b+3*b^2)*arctanh(a^(1/2)*tanh(x)/(a+b)^(1/2))/a^(5/2)/(a+b) ^(5/2)-1/4*b*cosh(x)*sinh(x)/a/(a+b)/(a+b*cosh(x)^2)^2-3/8*b*(2*a+b)*cosh( x)*sinh(x)/a^2/(a+b)^2/(a+b*cosh(x)^2)
Time = 5.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.99 \[ \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx=\frac {\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}-\frac {\sqrt {a} b \left (16 a^2+16 a b+3 b^2+3 b (2 a+b) \cosh (2 x)\right ) \sinh (2 x)}{(a+b)^2 (2 a+b+b \cosh (2 x))^2}}{8 a^{5/2}} \]
(((8*a^2 + 8*a*b + 3*b^2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(a + b)^ (5/2) - (Sqrt[a]*b*(16*a^2 + 16*a*b + 3*b^2 + 3*b*(2*a + b)*Cosh[2*x])*Sin h[2*x])/((a + b)^2*(2*a + b + b*Cosh[2*x])^2))/(8*a^(5/2))
Time = 0.50 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.12, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3663, 25, 3042, 3652, 27, 3042, 3660, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sin \left (\frac {\pi }{2}+i x\right )^2\right )^3}dx\) |
| \(\Big \downarrow \) 3663 |
| \(\displaystyle -\frac {\int -\frac {-2 b \cosh ^2(x)+4 a+3 b}{\left (b \cosh ^2(x)+a\right )^2}dx}{4 a (a+b)}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-2 b \cosh ^2(x)+4 a+3 b}{\left (b \cosh ^2(x)+a\right )^2}dx}{4 a (a+b)}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}+\frac {\int \frac {-2 b \sin \left (i x+\frac {\pi }{2}\right )^2+4 a+3 b}{\left (b \sin \left (i x+\frac {\pi }{2}\right )^2+a\right )^2}dx}{4 a (a+b)}\) |
| \(\Big \downarrow \) 3652 |
| \(\displaystyle \frac {\frac {\int \frac {8 a^2+8 b a+3 b^2}{b \cosh ^2(x)+a}dx}{2 a (a+b)}-\frac {3 b (2 a+b) \sinh (x) \cosh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}}{4 a (a+b)}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (8 a^2+8 a b+3 b^2\right ) \int \frac {1}{b \cosh ^2(x)+a}dx}{2 a (a+b)}-\frac {3 b (2 a+b) \sinh (x) \cosh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}}{4 a (a+b)}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}+\frac {-\frac {3 b (2 a+b) \sinh (x) \cosh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}+\frac {\left (8 a^2+8 a b+3 b^2\right ) \int \frac {1}{b \sin \left (i x+\frac {\pi }{2}\right )^2+a}dx}{2 a (a+b)}}{4 a (a+b)}\) |
| \(\Big \downarrow \) 3660 |
| \(\displaystyle \frac {\frac {\left (8 a^2+8 a b+3 b^2\right ) \int \frac {1}{a-(a+b) \coth ^2(x)}d\coth (x)}{2 a (a+b)}-\frac {3 b (2 a+b) \sinh (x) \cosh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}}{4 a (a+b)}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\left (8 a^2+8 a b+3 b^2\right ) \text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a}}\right )}{2 a^{3/2} (a+b)^{3/2}}-\frac {3 b (2 a+b) \sinh (x) \cosh (x)}{2 a (a+b) \left (a+b \cosh ^2(x)\right )}}{4 a (a+b)}-\frac {b \sinh (x) \cosh (x)}{4 a (a+b) \left (a+b \cosh ^2(x)\right )^2}\) |
-1/4*(b*Cosh[x]*Sinh[x])/(a*(a + b)*(a + b*Cosh[x]^2)^2) + (((8*a^2 + 8*a* b + 3*b^2)*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a]])/(2*a^(3/2)*(a + b)^(3/2 )) - (3*b*(2*a + b)*Cosh[x]*Sinh[x])/(2*a*(a + b)*(a + b*Cosh[x]^2)))/(4*a *(a + b))
3.1.34.3.1 Defintions of rubi rules used
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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b - a*B))*Cos[e + f*x]*Sin[e + f*x ]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(a + b)*(p + 1))), x] - Simp[1/(2* a*(a + b)*(p + 1)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[a*B - A*(2*a*( p + 1) + b*(2*p + 3)) + 2*(A*b - a*B)*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && LtQ[p, -1] && NeQ[a + b, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(-1), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[1/(a + (a + b)*ff^2*x^ 2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p + 1)/(2*a*f*(p + 1)*(a + b))), x] + Simp[1/(2*a*(p + 1)*(a + b)) Int[(a + b*Sin[e + f*x]^2)^(p + 1)*Simp[2*a*(p + 1) + b*(2*p + 3) - 2*b*(p + 2)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[a + b, 0] && LtQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(263\) vs. \(2(93)=186\).
Time = 0.85 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.47
| method | result | size |
| default | \(-\frac {2 \left (\frac {b \left (8 a +3 b \right ) \tanh \left (\frac {x}{2}\right )^{7}}{8 a^{2} \left (a +b \right )}-\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{5}}{8 \left (a +b \right )^{2} a^{2}}-\frac {b \left (8 a^{2}-13 a b -9 b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{8 \left (a +b \right )^{2} a^{2}}+\frac {b \left (8 a +3 b \right ) \tanh \left (\frac {x}{2}\right )}{8 a^{2} \left (a +b \right )}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{4} a +\tanh \left (\frac {x}{2}\right )^{4} b -2 \tanh \left (\frac {x}{2}\right )^{2} a +2 \tanh \left (\frac {x}{2}\right )^{2} b +a +b \right )^{2}}-\frac {\left (8 a^{2}+8 a b +3 b^{2}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}+2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )^{2}-2 \tanh \left (\frac {x}{2}\right ) \sqrt {a}+\sqrt {a +b}\right )}{4 \sqrt {a}\, \sqrt {a +b}}\right )}{4 a^{2} \left (a^{2}+2 a b +b^{2}\right )}\) | \(264\) |
| risch | \(\frac {8 \,{\mathrm e}^{6 x} a^{2} b +8 \,{\mathrm e}^{6 x} a \,b^{2}+3 b^{3} {\mathrm e}^{6 x}+48 a^{3} {\mathrm e}^{4 x}+72 a^{2} b \,{\mathrm e}^{4 x}+42 a \,b^{2} {\mathrm e}^{4 x}+9 b^{3} {\mathrm e}^{4 x}+40 \,{\mathrm e}^{2 x} a^{2} b +40 \,{\mathrm e}^{2 x} a \,b^{2}+9 b^{3} {\mathrm e}^{2 x}+6 a \,b^{2}+3 b^{3}}{4 a^{2} \left (a +b \right )^{2} \left (b \,{\mathrm e}^{4 x}+4 a \,{\mathrm e}^{2 x}+2 b \,{\mathrm e}^{2 x}+b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right ) b}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} a}+\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}-2 a^{2}-2 a b}{b \sqrt {a^{2}+a b}}\right ) b^{2}}{16 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} a^{2}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right )}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right ) b}{2 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} a}-\frac {3 \ln \left ({\mathrm e}^{2 x}+\frac {2 a \sqrt {a^{2}+a b}+b \sqrt {a^{2}+a b}+2 a^{2}+2 a b}{b \sqrt {a^{2}+a b}}\right ) b^{2}}{16 \sqrt {a^{2}+a b}\, \left (a +b \right )^{2} a^{2}}\) | \(572\) |
-2*(1/8*b*(8*a+3*b)/a^2/(a+b)*tanh(1/2*x)^7-1/8*b*(8*a^2-13*a*b-9*b^2)/(a+ b)^2/a^2*tanh(1/2*x)^5-1/8*b*(8*a^2-13*a*b-9*b^2)/(a+b)^2/a^2*tanh(1/2*x)^ 3+1/8*b*(8*a+3*b)/a^2/(a+b)*tanh(1/2*x))/(tanh(1/2*x)^4*a+tanh(1/2*x)^4*b- 2*tanh(1/2*x)^2*a+2*tanh(1/2*x)^2*b+a+b)^2-1/4*(8*a^2+8*a*b+3*b^2)/a^2/(a^ 2+2*a*b+b^2)*(-1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1/2*x)^2+2*tanh (1/2*x)*a^(1/2)+(a+b)^(1/2))+1/4/a^(1/2)/(a+b)^(1/2)*ln((a+b)^(1/2)*tanh(1 /2*x)^2-2*tanh(1/2*x)*a^(1/2)+(a+b)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 2478 vs. \(2 (93) = 186\).
Time = 0.33 (sec) , antiderivative size = 5117, normalized size of antiderivative = 47.82 \[ \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 344 vs. \(2 (93) = 186\).
Time = 0.30 (sec) , antiderivative size = 344, normalized size of antiderivative = 3.21 \[ \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx=-\frac {{\left (8 \, a^{2} + 8 \, a b + 3 \, b^{2}\right )} \log \left (\frac {b e^{\left (-2 \, x\right )} + 2 \, a + b - 2 \, \sqrt {{\left (a + b\right )} a}}{b e^{\left (-2 \, x\right )} + 2 \, a + b + 2 \, \sqrt {{\left (a + b\right )} a}}\right )}{16 \, {\left (a^{4} + 2 \, a^{3} b + a^{2} b^{2}\right )} \sqrt {{\left (a + b\right )} a}} - \frac {6 \, a b^{2} + 3 \, b^{3} + {\left (40 \, a^{2} b + 40 \, a b^{2} + 9 \, b^{3}\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (16 \, a^{3} + 24 \, a^{2} b + 14 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (-4 \, x\right )} + {\left (8 \, a^{2} b + 8 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (-6 \, x\right )}}{4 \, {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4} + 4 \, {\left (2 \, a^{5} b + 5 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (8 \, a^{6} + 24 \, a^{5} b + 27 \, a^{4} b^{2} + 14 \, a^{3} b^{3} + 3 \, a^{2} b^{4}\right )} e^{\left (-4 \, x\right )} + 4 \, {\left (2 \, a^{5} b + 5 \, a^{4} b^{2} + 4 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-6 \, x\right )} + {\left (a^{4} b^{2} + 2 \, a^{3} b^{3} + a^{2} b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \]
-1/16*(8*a^2 + 8*a*b + 3*b^2)*log((b*e^(-2*x) + 2*a + b - 2*sqrt((a + b)*a ))/(b*e^(-2*x) + 2*a + b + 2*sqrt((a + b)*a)))/((a^4 + 2*a^3*b + a^2*b^2)* sqrt((a + b)*a)) - 1/4*(6*a*b^2 + 3*b^3 + (40*a^2*b + 40*a*b^2 + 9*b^3)*e^ (-2*x) + 3*(16*a^3 + 24*a^2*b + 14*a*b^2 + 3*b^3)*e^(-4*x) + (8*a^2*b + 8* a*b^2 + 3*b^3)*e^(-6*x))/(a^4*b^2 + 2*a^3*b^3 + a^2*b^4 + 4*(2*a^5*b + 5*a ^4*b^2 + 4*a^3*b^3 + a^2*b^4)*e^(-2*x) + 2*(8*a^6 + 24*a^5*b + 27*a^4*b^2 + 14*a^3*b^3 + 3*a^2*b^4)*e^(-4*x) + 4*(2*a^5*b + 5*a^4*b^2 + 4*a^3*b^3 + a^2*b^4)*e^(-6*x) + (a^4*b^2 + 2*a^3*b^3 + a^2*b^4)*e^(-8*x))
\[ \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx=\int { \frac {1}{{\left (b \cosh \left (x\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \cosh ^2(x)\right )^3} \, dx=\int \frac {1}{{\left (b\,{\mathrm {cosh}\left (x\right )}^2+a\right )}^3} \,d x \]