Integrand size = 15, antiderivative size = 56 \[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=\frac {a^2 \arctan \left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sinh (x)}{b^2}+\frac {\sinh ^3(x)}{3 b} \] Output:
a^2*arctan(b^(1/2)*sinh(x)/(a+b)^(1/2))/b^(5/2)/(a+b)^(1/2)-(a-b)*sinh(x)/ b^2+1/3*sinh(x)^3/b
Time = 0.13 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09 \[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=-\frac {a^2 \arctan \left (\frac {\sqrt {a+b} \text {csch}(x)}{\sqrt {b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(4 a-3 b) \sinh (x)}{4 b^2}+\frac {\sinh (3 x)}{12 b} \] Input:
Integrate[Cosh[x]^5/(a + b*Cosh[x]^2),x]
Output:
-((a^2*ArcTan[(Sqrt[a + b]*Csch[x])/Sqrt[b]])/(b^(5/2)*Sqrt[a + b])) - ((4 *a - 3*b)*Sinh[x])/(4*b^2) + Sinh[3*x]/(12*b)
Time = 0.27 (sec) , antiderivative size = 56, 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, 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.
\(\displaystyle \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )^5}{a+b \sin \left (\frac {\pi }{2}+i x\right )^2}dx\) |
\(\Big \downarrow \) 3665 |
\(\displaystyle \int \frac {\left (\sinh ^2(x)+1\right )^2}{a+b \sinh ^2(x)+b}d\sinh (x)\) |
\(\Big \downarrow \) 300 |
\(\displaystyle \int \left (\frac {a^2}{b^2 \left (a+b \sinh ^2(x)+b\right )}-\frac {a-b}{b^2}+\frac {\sinh ^2(x)}{b}\right )d\sinh (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 \arctan \left (\frac {\sqrt {b} \sinh (x)}{\sqrt {a+b}}\right )}{b^{5/2} \sqrt {a+b}}-\frac {(a-b) \sinh (x)}{b^2}+\frac {\sinh ^3(x)}{3 b}\) |
Input:
Int[Cosh[x]^5/(a + b*Cosh[x]^2),x]
Output:
(a^2*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]) - ((a - b)*Sinh[x])/b^2 + Sinh[x]^3/(3*b)
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[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. \(145\) vs. \(2(46)=92\).
Time = 1.40 (sec) , antiderivative size = 146, normalized size of antiderivative = 2.61
method | result | size |
risch | \(\frac {{\mathrm e}^{3 x}}{24 b}-\frac {a \,{\mathrm e}^{x}}{2 b^{2}}+\frac {3 \,{\mathrm e}^{x}}{8 b}+\frac {{\mathrm e}^{-x} a}{2 b^{2}}-\frac {3 \,{\mathrm e}^{-x}}{8 b}-\frac {{\mathrm e}^{-3 x}}{24 b}-\frac {a^{2} \ln \left ({\mathrm e}^{2 x}-\frac {2 \left (a +b \right ) {\mathrm e}^{x}}{\sqrt {-a b -b^{2}}}-1\right )}{2 \sqrt {-a b -b^{2}}\, b^{2}}+\frac {a^{2} \ln \left ({\mathrm e}^{2 x}+\frac {2 \left (a +b \right ) {\mathrm e}^{x}}{\sqrt {-a b -b^{2}}}-1\right )}{2 \sqrt {-a b -b^{2}}\, b^{2}}\) | \(146\) |
default | \(-\frac {1}{3 b \left (1+\tanh \left (\frac {x}{2}\right )\right )^{3}}+\frac {1}{2 b \left (1+\tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {-a +b}{b^{2} \left (1+\tanh \left (\frac {x}{2}\right )\right )}+\frac {2 a^{2} \left (\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )+2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}+\frac {\arctan \left (\frac {2 \sqrt {a +b}\, \tanh \left (\frac {x}{2}\right )-2 \sqrt {a}}{2 \sqrt {b}}\right )}{2 \sqrt {a +b}\, \sqrt {b}}\right )}{b^{2}}-\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{2 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {-a +b}{b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}\) | \(165\) |
Input:
int(cosh(x)^5/(a+b*cosh(x)^2),x,method=_RETURNVERBOSE)
Output:
1/24/b*exp(3*x)-1/2*a/b^2*exp(x)+3/8/b*exp(x)+1/2/b^2*exp(-x)*a-3/8/b*exp( -x)-1/24/b*exp(-3*x)-1/2/(-a*b-b^2)^(1/2)*a^2/b^2*ln(exp(2*x)-2*(a+b)/(-a* b-b^2)^(1/2)*exp(x)-1)+1/2/(-a*b-b^2)^(1/2)*a^2/b^2*ln(exp(2*x)+2*(a+b)/(- a*b-b^2)^(1/2)*exp(x)-1)
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (46) = 92\).
Time = 0.13 (sec) , antiderivative size = 1185, normalized size of antiderivative = 21.16 \[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate(cosh(x)^5/(a+b*cosh(x)^2),x, algorithm="fricas")
Output:
[1/24*((a*b^2 + b^3)*cosh(x)^6 + 6*(a*b^2 + b^3)*cosh(x)*sinh(x)^5 + (a*b^ 2 + b^3)*sinh(x)^6 - 3*(4*a^2*b + a*b^2 - 3*b^3)*cosh(x)^4 - 3*(4*a^2*b + a*b^2 - 3*b^3 - 5*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a*b^2 + b^3)* cosh(x)^3 - 3*(4*a^2*b + a*b^2 - 3*b^3)*cosh(x))*sinh(x)^3 - a*b^2 - b^3 + 3*(4*a^2*b + a*b^2 - 3*b^3)*cosh(x)^2 + 3*(5*(a*b^2 + b^3)*cosh(x)^4 + 4* a^2*b + a*b^2 - 3*b^3 - 6*(4*a^2*b + a*b^2 - 3*b^3)*cosh(x)^2)*sinh(x)^2 - 12*(a^2*cosh(x)^3 + 3*a^2*cosh(x)^2*sinh(x) + 3*a^2*cosh(x)*sinh(x)^2 + a ^2*sinh(x)^3)*sqrt(-a*b - b^2)*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*c osh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 - 1)*sinh(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)) + 6*((a*b^2 + b^3)*cosh(x)^5 - 2*(4*a^2 *b + a*b^2 - 3*b^3)*cosh(x)^3 + (4*a^2*b + a*b^2 - 3*b^3)*cosh(x))*sinh(x) )/((a*b^3 + b^4)*cosh(x)^3 + 3*(a*b^3 + b^4)*cosh(x)^2*sinh(x) + 3*(a*b^3 + b^4)*cosh(x)*sinh(x)^2 + (a*b^3 + b^4)*sinh(x)^3), 1/24*((a*b^2 + b^3)*c osh(x)^6 + 6*(a*b^2 + b^3)*cosh(x)*sinh(x)^5 + (a*b^2 + b^3)*sinh(x)^6 - 3 *(4*a^2*b + a*b^2 - 3*b^3)*cosh(x)^4 - 3*(4*a^2*b + a*b^2 - 3*b^3 - 5*(a*b ^2 + b^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a*b^2 + b^3)*cosh(x)^3 - 3*(4*a^...
Timed out. \[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(x)**5/(a+b*cosh(x)**2),x)
Output:
Timed out
\[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=\int { \frac {\cosh \left (x\right )^{5}}{b \cosh \left (x\right )^{2} + a} \,d x } \] Input:
integrate(cosh(x)^5/(a+b*cosh(x)^2),x, algorithm="maxima")
Output:
1/24*(b*e^(6*x) - 3*(4*a - 3*b)*e^(4*x) + 3*(4*a - 3*b)*e^(2*x) - b)*e^(-3 *x)/b^2 + 1/32*integrate(64*(a^2*e^(3*x) + a^2*e^x)/(b^3*e^(4*x) + b^3 + 2 *(2*a*b^2 + b^3)*e^(2*x)), x)
Exception generated. \[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cosh(x)^5/(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 = 2.54 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.34 \[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=\frac {{\mathrm {e}}^{3\,x}}{24\,b}-\frac {{\mathrm {e}}^{-3\,x}}{24\,b}+\frac {{\mathrm {e}}^{-x}\,\left (4\,a-3\,b\right )}{8\,b^2}+\frac {\sqrt {a^4}\,\left (2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^x\,\sqrt {b^5\,\left (a+b\right )}}{2\,b^2\,\left (a+b\right )\,\sqrt {a^4}}\right )-2\,\mathrm {atan}\left (\left (\frac {b^7\,\sqrt {b^6+a\,b^5}}{4}+\frac {a\,b^6\,\sqrt {b^6+a\,b^5}}{4}\right )\,\left ({\mathrm {e}}^x\,\left (\frac {2\,a^2}{b^8\,{\left (a+b\right )}^2\,\sqrt {a^4}}-\frac {4\,\left (2\,a^3\,b^3\,\sqrt {a^4}+2\,a^4\,b^2\,\sqrt {a^4}\right )}{a^5\,b^6\,\left (a+b\right )\,\sqrt {b^5\,\left (a+b\right )}\,\sqrt {b^6+a\,b^5}}\right )-\frac {2\,a^2\,{\mathrm {e}}^{3\,x}}{b^8\,{\left (a+b\right )}^2\,\sqrt {a^4}}\right )\right )\right )}{2\,\sqrt {b^6+a\,b^5}}-\frac {{\mathrm {e}}^x\,\left (4\,a-3\,b\right )}{8\,b^2} \] Input:
int(cosh(x)^5/(a + b*cosh(x)^2),x)
Output:
exp(3*x)/(24*b) - exp(-3*x)/(24*b) + (exp(-x)*(4*a - 3*b))/(8*b^2) + ((a^4 )^(1/2)*(2*atan((a^2*exp(x)*(b^5*(a + b))^(1/2))/(2*b^2*(a + b)*(a^4)^(1/2 ))) - 2*atan(((b^7*(a*b^5 + b^6)^(1/2))/4 + (a*b^6*(a*b^5 + b^6)^(1/2))/4) *(exp(x)*((2*a^2)/(b^8*(a + b)^2*(a^4)^(1/2)) - (4*(2*a^3*b^3*(a^4)^(1/2) + 2*a^4*b^2*(a^4)^(1/2)))/(a^5*b^6*(a + b)*(b^5*(a + b))^(1/2)*(a*b^5 + b^ 6)^(1/2))) - (2*a^2*exp(3*x))/(b^8*(a + b)^2*(a^4)^(1/2))))))/(2*(a*b^5 + b^6)^(1/2)) - (exp(x)*(4*a - 3*b))/(8*b^2)
Time = 0.26 (sec) , antiderivative size = 611, normalized size of antiderivative = 10.91 \[ \int \frac {\cosh ^5(x)}{a+b \cosh ^2(x)} \, dx=\frac {-24 e^{3 x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a^{2}+24 e^{3 x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a^{3}+24 e^{3 x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}\, \mathit {atan} \left (\frac {e^{x} b}{\sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}+2 a +b}}\right ) a^{2} b -12 e^{3 x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2}+12 e^{3 x} \sqrt {b}\, \sqrt {a}\, \sqrt {a +b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2}-12 e^{3 x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{3}-12 e^{3 x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (-\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2} b +12 e^{3 x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{3}+12 e^{3 x} \sqrt {b}\, \sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}\, \mathrm {log}\left (\sqrt {2 \sqrt {a}\, \sqrt {a +b}-2 a -b}+e^{x} \sqrt {b}\right ) a^{2} b +e^{6 x} a \,b^{3}+e^{6 x} b^{4}-12 e^{4 x} a^{2} b^{2}-3 e^{4 x} a \,b^{3}+9 e^{4 x} b^{4}+12 e^{2 x} a^{2} b^{2}+3 e^{2 x} a \,b^{3}-9 e^{2 x} b^{4}-a \,b^{3}-b^{4}}{24 e^{3 x} b^{4} \left (a +b \right )} \] Input:
int(cosh(x)^5/(a+b*cosh(x)^2),x)
Output:
( - 24*e**(3*x)*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 + 24*e**(3*x)*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**3 + 24*e**(3*x)*sqrt (b)*sqrt(2*sqrt(a)*sqrt(a + b) + 2*a + b)*atan((e**x*b)/(sqrt(b)*sqrt(2*sq rt(a)*sqrt(a + b) + 2*a + b)))*a**2*b - 12*e**(3*x)*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**2 + 12*e**(3*x)*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**2 - 12*e**(3*x)*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**3 - 12*e**(3*x)*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*b + 12*e**(3* x)*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**3 + 12*e**(3*x)*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*s qrt(b))*a**2*b + e**(6*x)*a*b**3 + e**(6*x)*b**4 - 12*e**(4*x)*a**2*b**2 - 3*e**(4*x)*a*b**3 + 9*e**(4*x)*b**4 + 12*e**(2*x)*a**2*b**2 + 3*e**(2*x)* a*b**3 - 9*e**(2*x)*b**4 - a*b**3 - b**4)/(24*e**(3*x)*b**4*(a + b))