Integrand size = 15, antiderivative size = 38 \[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx=-\frac {a \arctan \left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}}+\frac {\sinh (x)}{b} \] Output:
-a*arctan(b^(1/2)*sinh(x)/(a+b)^(1/2))/b^(3/2)/(a+b)^(1/2)+sinh(x)/b
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx=-\frac {a \arctan \left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}}+\frac {\sinh (x)}{b} \] Input:
Integrate[Cosh[x]^3/(a + b*Cosh[x]^2),x]
Output:
-((a*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sinh[ x]/b
Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 3665, 299, 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.
\(\displaystyle \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )^3}{a+b \sin \left (\frac {\pi }{2}+i x\right )^2}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle \int \frac {\sinh ^2(x)+1}{a+b \sinh ^2(x)+b}d\sinh (x)\) |
\(\Big \downarrow \) 299 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \int \frac {1}{b \sinh ^2(x)+a+b}d\sinh (x)}{b}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\sinh (x)}{b}-\frac {a \arctan \left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{3/2} \sqrt {a+b}}\) |
Input:
Int[Cosh[x]^3/(a + b*Cosh[x]^2),x]
Output:
-((a*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sinh[ x]/b
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[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^ (p_.), x_Symbol] :> With[{ff = FreeFactors[Cos[e + f*x], x]}, Simp[-ff/f Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Leaf count of result is larger than twice the leaf count of optimal. \(100\) vs. \(2(30)=60\).
Time = 0.54 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.66
method | result | size |
default | \(-\frac {2 a \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}\right )}{b}-\frac {1}{b \left (1+\tanh \left (\frac {x}{2}\right )\right )}-\frac {1}{b \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(101\) |
risch | \(\frac {{\mathrm e}^{x}}{2 b}-\frac {{\mathrm e}^{-x}}{2 b}-\frac {a \ln \left ({\mathrm e}^{2 x}+\frac {2 \left (a +b \right ) {\mathrm e}^{x}}{\sqrt {-a b -b^{2}}}-1\right )}{2 \sqrt {-a b -b^{2}}\, b}+\frac {a \ln \left ({\mathrm e}^{2 x}-\frac {2 \left (a +b \right ) {\mathrm e}^{x}}{\sqrt {-a b -b^{2}}}-1\right )}{2 \sqrt {-a b -b^{2}}\, b}\) | \(106\) |
Input:
int(cosh(x)^3/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)
Output:
-2/b*a*(1/2/(a+b)^(1/2)/b^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2*x)+2*a^ (1/2))/b^(1/2))+1/2/(a+b)^(1/2)/b^(1/2)*arctan(1/2*(2*(a+b)^(1/2)*tanh(1/2 *x)-2*a^(1/2))/b^(1/2)))-1/b/(1+tanh(1/2*x))-1/b/(tanh(1/2*x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (30) = 60\).
Time = 0.11 (sec) , antiderivative size = 499, normalized size of antiderivative = 13.13 \[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx =\text {Too large to display} \] Input:
integrate(cosh(x)^3/(a+b*cosh(x)^2),x, algorithm="fricas")
Output:
[1/2*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)* sinh(x)^2 - sqrt(-a*b - b^2)*(a*cosh(x) + a*sinh(x))*log((b*cosh(x)^4 + 4* b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 - 2*(2*a + 3*b)*cosh(x)^2 + 2*(3*b*cosh( x)^2 - 2*a - 3*b)*sinh(x)^2 + 4*(b*cosh(x)^3 - (2*a + 3*b)*cosh(x))*sinh(x ) + 4*(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 - 1)*sin h(x) - cosh(x))*sqrt(-a*b - b^2) + b)/(b*cosh(x)^4 + 4*b*cosh(x)*sinh(x)^3 + b*sinh(x)^4 + 2*(2*a + b)*cosh(x)^2 + 2*(3*b*cosh(x)^2 + 2*a + b)*sinh( x)^2 + 4*(b*cosh(x)^3 + (2*a + b)*cosh(x))*sinh(x) + b)) - a*b - b^2)/((a* b^2 + b^3)*cosh(x) + (a*b^2 + b^3)*sinh(x)), 1/2*((a*b + b^2)*cosh(x)^2 + 2*(a*b + b^2)*cosh(x)*sinh(x) + (a*b + b^2)*sinh(x)^2 - 2*sqrt(a*b + b^2)* (a*cosh(x) + a*sinh(x))*arctan(1/2*(b*cosh(x)^3 + 3*b*cosh(x)*sinh(x)^2 + b*sinh(x)^3 + (4*a + 3*b)*cosh(x) + (3*b*cosh(x)^2 + 4*a + 3*b)*sinh(x))/s qrt(a*b + b^2)) + 2*sqrt(a*b + b^2)*(a*cosh(x) + a*sinh(x))*arctan(2*sqrt( a*b + b^2)/(b*cosh(x) + b*sinh(x))) - a*b - b^2)/((a*b^2 + b^3)*cosh(x) + (a*b^2 + b^3)*sinh(x))]
Timed out. \[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(x)**3/(a+b*cosh(x)**2),x)
Output:
Timed out
\[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\cosh \left (x\right )^{3}}{b \cosh \left (x\right )^{2} + a} \,d x } \] Input:
integrate(cosh(x)^3/(a+b*cosh(x)^2),x, algorithm="maxima")
Output:
1/2*(e^(2*x) - 1)*e^(-x)/b - 1/8*integrate(16*(a*e^(3*x) + a*e^x)/(b^2*e^( 4*x) + b^2 + 2*(2*a*b + b^2)*e^(2*x)), x)
Exception generated. \[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cosh(x)^3/(a+b*cosh(x)^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
Time = 2.59 (sec) , antiderivative size = 204, normalized size of antiderivative = 5.37 \[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {{\mathrm {e}}^x}{2\,b}-\frac {{\mathrm {e}}^{-x}}{2\,b}-\frac {\left (2\,\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^x\,\sqrt {b^3\,\left (a+b\right )}}{2\,b\,\left (a+b\right )\,{\left (a^2\right )}^{3/2}}\right )-2\,\mathrm {atan}\left (\left (\frac {b^5\,\sqrt {b^4+a\,b^3}}{4}+\frac {a\,b^4\,\sqrt {b^4+a\,b^3}}{4}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^3}{b^5\,{\left (a+b\right )}^2\,{\left (a^2\right )}^{3/2}}-\frac {4\,\left (2\,b^2\,{\left (a^2\right )}^{3/2}+2\,a\,b\,{\left (a^2\right )}^{3/2}\right )}{a^3\,b^4\,\left (a+b\right )\,\sqrt {b^3\,\left (a+b\right )}\,\sqrt {b^4+a\,b^3}}\right )-\frac {2\,a^3\,{\mathrm {e}}^{3\,x}}{b^5\,{\left (a+b\right )}^2\,{\left (a^2\right )}^{3/2}}\right )\right )\right )\,\sqrt {a^2}}{2\,\sqrt {b^4+a\,b^3}} \] Input:
int(cosh(x)^3/(a + b*cosh(x)^2),x)
Output:
exp(x)/(2*b) - exp(-x)/(2*b) - ((2*atan((a^3*exp(x)*(b^3*(a + b))^(1/2))/( 2*b*(a + b)*(a^2)^(3/2))) - 2*atan(((b^5*(a*b^3 + b^4)^(1/2))/4 + (a*b^4*( a*b^3 + b^4)^(1/2))/4)*(exp(x)*((2*a^3)/(b^5*(a + b)^2*(a^2)^(3/2)) - (4*( 2*b^2*(a^2)^(3/2) + 2*a*b*(a^2)^(3/2)))/(a^3*b^4*(a + b)*(b^3*(a + b))^(1/ 2)*(a*b^3 + b^4)^(1/2))) - (2*a^3*exp(3*x))/(b^5*(a + b)^2*(a^2)^(3/2))))) *(a^2)^(1/2))/(2*(a*b^3 + b^4)^(1/2))
Time = 0.24 (sec) , antiderivative size = 461, normalized size of antiderivative = 12.13 \[ \int \frac {\cosh ^3(x)}{a+b \cosh ^2(x)} \, dx=\frac {2 \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a -2 \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a^{2}-2 \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a b +\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a -\sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a +\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2}+\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a b -\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2}-\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a b +2 \sinh \left (x \right ) a \,b^{2}+2 \sinh \left (x \right ) b^{3}}{2 b^{3} \left (a +b \right )} \] Input:
int(cosh(x)^3/(a+b*cosh(x)^2),x)
Output:
(2*sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan( (e**x*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*a - 2*sqrt(b)*sq rt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sqrt(a)* sqrt(a + b) + 2*a + b)))*a**2 - 2*sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)))*a*b + sqrt(b)*sqrt(a)*sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*a - sqrt(b)*sqrt(a)* sqrt(a + b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)*sqrt( a + b) - 2*a - b) + e**x*sqrt(b))*a + sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*a** 2 + sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log( - sqrt(2*sqrt(a)*sq rt(a + b) - 2*a - b) + e**x*sqrt(b))*a*b - sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b) + e**x*sqrt(b))*a **2 - sqrt(b)*sqrt(2*sqrt(a)*sqrt(a + b) - 2*a - b)*log(sqrt(2*sqrt(a)*sqr t(a + b) - 2*a - b) + e**x*sqrt(b))*a*b + 2*sinh(x)*a*b**2 + 2*sinh(x)*b** 3)/(2*b**3*(a + b))