Integrand size = 15, antiderivative size = 135 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx=\frac {\left (2 a^2 A+A b^2-3 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}-\frac {(A b-a B) \sinh (x)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}-\frac {\left (3 a A b-a^2 B-2 b^2 B\right ) \sinh (x)}{2 \left (a^2-b^2\right )^2 (a+b \cosh (x))} \]
(2*A*a^2+A*b^2-3*B*a*b)*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2))/(a-b) ^(5/2)/(a+b)^(5/2)-1/2*(A*b-B*a)*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^2-1/2*(3* A*a*b-B*a^2-2*B*b^2)*sinh(x)/(a^2-b^2)^2/(a+b*cosh(x))
Time = 0.30 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx=\frac {1}{2} \left (-\frac {2 \left (2 a^2 A+A b^2-3 a b B\right ) \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {(-A b+a B) \sinh (x)}{(a-b) (a+b) (a+b \cosh (x))^2}+\frac {\left (-3 a A b+a^2 B+2 b^2 B\right ) \sinh (x)}{(a-b)^2 (a+b)^2 (a+b \cosh (x))}\right ) \]
((-2*(2*a^2*A + A*b^2 - 3*a*b*B)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^ 2]])/(-a^2 + b^2)^(5/2) + ((-(A*b) + a*B)*Sinh[x])/((a - b)*(a + b)*(a + b *Cosh[x])^2) + ((-3*a*A*b + a^2*B + 2*b^2*B)*Sinh[x])/((a - b)^2*(a + b)^2 *(a + b*Cosh[x])))/2
Time = 0.54 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3233, 25, 3042, 3233, 25, 27, 3042, 3138, 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.
| \(\displaystyle \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (\frac {\pi }{2}+i x\right )}{\left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {\int -\frac {2 (a A-b B)-(A b-a B) \cosh (x)}{(a+b \cosh (x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 (a A-b B)-(A b-a B) \cosh (x)}{(a+b \cosh (x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}+\frac {\int \frac {2 (a A-b B)+(a B-A b) \sin \left (i x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )^2}dx}{2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 A a^2-3 b B a+A b^2}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 A a^2-3 b B a+A b^2}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (2 a^2 A-3 a b B+A b^2\right ) \int \frac {1}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}+\frac {-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}+\frac {\left (2 a^2 A-3 a b B+A b^2\right ) \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {2 \left (2 a^2 A-3 a b B+A b^2\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {2 \left (2 a^2 A-3 a b B+A b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}\) |
-1/2*((A*b - a*B)*Sinh[x])/((a^2 - b^2)*(a + b*Cosh[x])^2) + ((2*(2*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sinh[x])/((a^2 - b^2)*(a + b*Cosh[x])))/(2*(a^2 - b^2))
3.2.12.3.1 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[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Time = 0.27 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.53
| method | result | size |
| default | \(-\frac {2 \left (-\frac {\left (4 b A a +b^{2} A -2 a^{2} B -B a b -2 B \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (4 b A a -b^{2} A -2 a^{2} B +B a b -2 B \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b -a -b \right )^{2}}+\frac {\left (2 A \,a^{2}+b^{2} A -3 B a b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(207\) |
| risch | \(\frac {2 A \,a^{2} b^{2} {\mathrm e}^{3 x}+A \,b^{4} {\mathrm e}^{3 x}-3 B a \,b^{3} {\mathrm e}^{3 x}+6 A \,a^{3} b \,{\mathrm e}^{2 x}+3 A a \,b^{3} {\mathrm e}^{2 x}-2 B \,a^{4} {\mathrm e}^{2 x}-5 B \,a^{2} b^{2} {\mathrm e}^{2 x}-2 B \,b^{4} {\mathrm e}^{2 x}+10 A \,a^{2} b^{2} {\mathrm e}^{x}-A \,b^{4} {\mathrm e}^{x}-4 B \,a^{3} b \,{\mathrm e}^{x}-5 B a \,b^{3} {\mathrm e}^{x}+3 A a \,b^{3}-B \,a^{2} b^{2}-2 B \,b^{4}}{b \left (a^{2}-b^{2}\right )^{2} \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) b^{2} A}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B a b}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) b^{2} A}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B a b}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2}}\) | \(597\) |
-2*(-1/2*(4*A*a*b+A*b^2-2*B*a^2-B*a*b-2*B*b^2)/(a-b)/(a^2+2*a*b+b^2)*tanh( 1/2*x)^3+1/2*(4*A*a*b-A*b^2-2*B*a^2+B*a*b-2*B*b^2)/(a+b)/(a^2-2*a*b+b^2)*t anh(1/2*x))/(tanh(1/2*x)^2*a-tanh(1/2*x)^2*b-a-b)^2+(2*A*a^2+A*b^2-3*B*a*b )/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b) *(a-b))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 1545 vs. \(2 (120) = 240\).
Time = 0.34 (sec) , antiderivative size = 3166, normalized size of antiderivative = 23.45 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx=\text {Too large to display} \]
[-1/2*(2*B*a^4*b^2 - 6*A*a^3*b^3 + 2*B*a^2*b^4 + 6*A*a*b^5 - 4*B*b^6 - 2*( 2*A*a^4*b^2 - 3*B*a^3*b^3 - A*a^2*b^4 + 3*B*a*b^5 - A*b^6)*cosh(x)^3 - 2*( 2*A*a^4*b^2 - 3*B*a^3*b^3 - A*a^2*b^4 + 3*B*a*b^5 - A*b^6)*sinh(x)^3 + 2*( 2*B*a^6 - 6*A*a^5*b + 3*B*a^4*b^2 + 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 - 2*B*b^6)*cosh(x)^2 + 2*(2*B*a^6 - 6*A*a^5*b + 3*B*a^4*b^2 + 3*A*a^3*b^3 - 3*B*a^2*b^4 + 3*A*a*b^5 - 2*B*b^6 - 3*(2*A*a^4*b^2 - 3*B*a^3*b^3 - A*a^2 *b^4 + 3*B*a*b^5 - A*b^6)*cosh(x))*sinh(x)^2 - (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5 + (2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cosh(x)^4 + (2*A*a^2*b^3 - 3*B*a *b^4 + A*b^5)*sinh(x)^4 + 4*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4)*cosh(x)^ 3 + 4*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4 + (2*A*a^2*b^3 - 3*B*a*b^4 + A* b^5)*cosh(x))*sinh(x)^3 + 2*(4*A*a^4*b - 6*B*a^3*b^2 + 4*A*a^2*b^3 - 3*B*a *b^4 + A*b^5)*cosh(x)^2 + 2*(4*A*a^4*b - 6*B*a^3*b^2 + 4*A*a^2*b^3 - 3*B*a *b^4 + A*b^5 + 3*(2*A*a^2*b^3 - 3*B*a*b^4 + A*b^5)*cosh(x)^2 + 6*(2*A*a^3* b^2 - 3*B*a^2*b^3 + A*a*b^4)*cosh(x))*sinh(x)^2 + 4*(2*A*a^3*b^2 - 3*B*a^2 *b^3 + A*a*b^4)*cosh(x) + 4*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a*b^4 + (2*A*a^ 2*b^3 - 3*B*a*b^4 + A*b^5)*cosh(x)^3 + 3*(2*A*a^3*b^2 - 3*B*a^2*b^3 + A*a* b^4)*cosh(x)^2 + (4*A*a^4*b - 6*B*a^3*b^2 + 4*A*a^2*b^3 - 3*B*a*b^4 + A*b^ 5)*cosh(x))*sinh(x))*sqrt(a^2 - b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cos...
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 249 vs. \(2 (120) = 240\).
Time = 0.26 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.84 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx=\frac {{\left (2 \, A a^{2} - 3 \, B a b + A b^{2}\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, A a^{2} b^{2} e^{\left (3 \, x\right )} - 3 \, B a b^{3} e^{\left (3 \, x\right )} + A b^{4} e^{\left (3 \, x\right )} - 2 \, B a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b e^{\left (2 \, x\right )} - 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} + 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, B b^{4} e^{\left (2 \, x\right )} - 4 \, B a^{3} b e^{x} + 10 \, A a^{2} b^{2} e^{x} - 5 \, B a b^{3} e^{x} - A b^{4} e^{x} - B a^{2} b^{2} + 3 \, A a b^{3} - 2 \, B b^{4}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}^{2}} \]
(2*A*a^2 - 3*B*a*b + A*b^2)*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/((a^4 - 2 *a^2*b^2 + b^4)*sqrt(-a^2 + b^2)) + (2*A*a^2*b^2*e^(3*x) - 3*B*a*b^3*e^(3* x) + A*b^4*e^(3*x) - 2*B*a^4*e^(2*x) + 6*A*a^3*b*e^(2*x) - 5*B*a^2*b^2*e^( 2*x) + 3*A*a*b^3*e^(2*x) - 2*B*b^4*e^(2*x) - 4*B*a^3*b*e^x + 10*A*a^2*b^2* e^x - 5*B*a*b^3*e^x - A*b^4*e^x - B*a^2*b^2 + 3*A*a*b^3 - 2*B*b^4)/((a^4*b - 2*a^2*b^3 + b^5)*(b*e^(2*x) + 2*a*e^x + b)^2)
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^3} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^3} \,d x \]