Integrand size = 11, antiderivative size = 101 \[ \int \frac {1}{a-b \cosh ^4(x)} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \tanh (x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a^{3/4} \sqrt {\sqrt {a}+\sqrt {b}}} \]
1/2*arctanh(a^(1/4)*tanh(x)/(a^(1/2)-b^(1/2))^(1/2))/a^(3/4)/(a^(1/2)-b^(1 /2))^(1/2)+1/2*arctanh(a^(1/4)*tanh(x)/(a^(1/2)+b^(1/2))^(1/2))/a^(3/4)/(a ^(1/2)+b^(1/2))^(1/2)
Time = 0.35 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.08 \[ \int \frac {1}{a-b \cosh ^4(x)} \, dx=-\frac {\arctan \left (\frac {\sqrt {a} \tanh (x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {-a+\sqrt {a} \sqrt {b}}}+\frac {\text {arctanh}\left (\frac {\sqrt {a} \tanh (x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{2 \sqrt {a} \sqrt {a+\sqrt {a} \sqrt {b}}} \]
-1/2*ArcTan[(Sqrt[a]*Tanh[x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]]/(Sqrt[a]*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + Sqrt[a]*Sqrt[b]] ]/(2*Sqrt[a]*Sqrt[a + Sqrt[a]*Sqrt[b]])
Time = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.54, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 3688, 1480, 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.
| \(\displaystyle \int \frac {1}{a-b \cosh ^4(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a-b \sin \left (\frac {\pi }{2}+i x\right )^4}dx\) |
| \(\Big \downarrow \) 3688 |
| \(\displaystyle \int \frac {1-\coth ^2(x)}{(a-b) \coth ^4(x)-2 a \coth ^2(x)+a}d\coth (x)\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle -\frac {1}{2} \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \int \frac {1}{(a-b) \coth ^2(x)-\sqrt {a} \left (\sqrt {a}-\sqrt {b}\right )}d\coth (x)-\frac {1}{2} \left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \int \frac {1}{(a-b) \coth ^2(x)-\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )}d\coth (x)\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\left (\frac {\sqrt {b}}{\sqrt {a}}+1\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \coth (x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \left (\sqrt {a}+\sqrt {b}\right )}+\frac {\left (1-\frac {\sqrt {b}}{\sqrt {a}}\right ) \text {arctanh}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \coth (x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a}+\sqrt {b}}}\) |
((1 + Sqrt[b]/Sqrt[a])*ArcTanh[(Sqrt[Sqrt[a] - Sqrt[b]]*Coth[x])/a^(1/4)]) /(2*a^(1/4)*Sqrt[Sqrt[a] - Sqrt[b]]*(Sqrt[a] + Sqrt[b])) + ((1 - Sqrt[b]/S qrt[a])*ArcTanh[(Sqrt[Sqrt[a] + Sqrt[b]]*Coth[x])/a^(1/4)])/(2*a^(1/4)*(Sq rt[a] - Sqrt[b])*Sqrt[Sqrt[a] + Sqrt[b]])
3.1.61.3.1 Defintions of rubi rules used
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[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + 2*a*ff^2*x^2 + ( a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] / ; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.34 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.95
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (256 a^{4}-256 a^{3} b \right ) \textit {\_Z}^{4}-32 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 x}+\left (\frac {128 a^{4}}{b}-128 a^{3}\right ) \textit {\_R}^{3}+\left (-\frac {32 a^{3}}{b}+32 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {8 a^{2}}{b}-8 a \right ) \textit {\_R} +\frac {2 a}{b}+1\right )\) | \(96\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a -b \right ) \textit {\_Z}^{8}+\left (-4 a -4 b \right ) \textit {\_Z}^{6}+\left (6 a -6 b \right ) \textit {\_Z}^{4}+\left (-4 a -4 b \right ) \textit {\_Z}^{2}+a -b \right )}{\sum }\frac {\left (-\textit {\_R}^{6}+3 \textit {\_R}^{4}-3 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{7} a -\textit {\_R}^{7} b -3 \textit {\_R}^{5} a -3 \textit {\_R}^{5} b +3 \textit {\_R}^{3} a -3 \textit {\_R}^{3} b -\textit {\_R} a -\textit {\_R} b}\right )}{4}\) | \(127\) |
sum(_R*ln(exp(2*x)+(128*a^4/b-128*a^3)*_R^3+(-32/b*a^3+32*a^2)*_R^2+(-8*a^ 2/b-8*a)*_R+2*a/b+1),_R=RootOf(1+(256*a^4-256*a^3*b)*_Z^4-32*a^2*_Z^2))
Leaf count of result is larger than twice the leaf count of optimal. 779 vs. \(2 (65) = 130\).
Time = 0.30 (sec) , antiderivative size = 779, normalized size of antiderivative = 7.71 \[ \int \frac {1}{a-b \cosh ^4(x)} \, dx=-\frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} \log \left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, {\left (a b - {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} - 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) + \frac {1}{4} \, \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} \log \left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} - 2 \, {\left (a b - {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + 1}{a^{2} - a b}} - 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) - \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} \log \left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} + 2 \, {\left (a b + {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} + 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) + \frac {1}{4} \, \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} \log \left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2} - 2 \, {\left (a b + {\left (a^{4} - a^{3} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}}\right )} \sqrt {-\frac {{\left (a^{2} - a b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} - 1}{a^{2} - a b}} + 2 \, {\left (a^{3} - a^{2} b\right )} \sqrt {\frac {b}{a^{5} - 2 \, a^{4} b + a^{3} b^{2}}} + b\right ) \]
-1/4*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b)) *log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 + 2*(a*b - (a^4 - a^3 *b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a ^4*b + a^3*b^2)) + 1)/(a^2 - a*b)) - 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) + 1/4*sqrt(((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2) ) + 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 - 2*(a*b - (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)))*sqrt(((a^2 - a*b )*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + 1)/(a^2 - a*b)) - 2*(a^3 - a^2*b)*sq rt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) - 1/4*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(b*cosh(x)^2 + 2*b*cosh(x)*sinh (x) + b*sinh(x)^2 + 2*(a*b + (a^4 - a^3*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2 )))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b)) + 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b) + 1/4*sqrt(-((a^ 2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) - 1)/(a^2 - a*b))*log(b*cosh(x) ^2 + 2*b*cosh(x)*sinh(x) + b*sinh(x)^2 - 2*(a*b + (a^4 - a^3*b)*sqrt(b/(a^ 5 - 2*a^4*b + a^3*b^2)))*sqrt(-((a^2 - a*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^ 2)) - 1)/(a^2 - a*b)) + 2*(a^3 - a^2*b)*sqrt(b/(a^5 - 2*a^4*b + a^3*b^2)) + b)
Timed out. \[ \int \frac {1}{a-b \cosh ^4(x)} \, dx=\text {Timed out} \]
\[ \int \frac {1}{a-b \cosh ^4(x)} \, dx=\int { -\frac {1}{b \cosh \left (x\right )^{4} - a} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 1697 vs. \(2 (65) = 130\).
Time = 2.46 (sec) , antiderivative size = 1697, normalized size of antiderivative = 16.80 \[ \int \frac {1}{a-b \cosh ^4(x)} \, dx=\text {Too large to display} \]
1/4*sqrt((a^2 - sqrt(a*b)*a)/(a^4 - a^3*b))*log(abs(60*a^4*b*e^(2*x) - 68* a^3*b^2*e^(2*x) - 16*a^2*b^3*e^(2*x) + 24*sqrt(a*b)*a^4*e^(2*x) + 48*sqrt( a^2 + sqrt(a*b)*a)*a^3*b*e^(2*x) + 16*sqrt(a*b)*a^3*b*e^(2*x) - 61*sqrt(a^ 2 + sqrt(a*b)*a)*a^2*b^2*e^(2*x) - 64*sqrt(a*b)*a^2*b^2*e^(2*x) - 4*sqrt(a ^2 + sqrt(a*b)*a)*a*b^3*e^(2*x) + 6*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 + 24*sqr t(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^3*e^(2*x) - 5*sqrt(a^2 + sqrt(a*b)*a)*sqr t(a*b)*a^2*b*e^(2*x) - 36*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a*b^2*e^(2*x) + 6*sqrt(a^2 + sqrt(a*b)*a)*a^3*b + 12*sqrt(a*b)*a^3*b - 5*sqrt(a^2 + sqrt (a*b)*a)*a^2*b^2 - 16*sqrt(a*b)*a^2*b^2 - 4*sqrt(a^2 + sqrt(a*b)*a)*a*b^3 + 9*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^2*b - 12*sqrt(a^2 + sqrt(a*b)*a)*s qrt(a*b)*a*b^2)) - 1/4*sqrt((a^2 - sqrt(a*b)*a)/(a^4 - a^3*b))*log(abs(60* a^4*b*e^(2*x) - 68*a^3*b^2*e^(2*x) - 16*a^2*b^3*e^(2*x) + 24*sqrt(a*b)*a^4 *e^(2*x) - 48*sqrt(a^2 + sqrt(a*b)*a)*a^3*b*e^(2*x) + 16*sqrt(a*b)*a^3*b*e ^(2*x) + 61*sqrt(a^2 + sqrt(a*b)*a)*a^2*b^2*e^(2*x) - 64*sqrt(a*b)*a^2*b^2 *e^(2*x) + 4*sqrt(a^2 + sqrt(a*b)*a)*a*b^3*e^(2*x) + 6*a^4*b - 2*a^3*b^2 - 8*a^2*b^3 - 24*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^3*e^(2*x) + 5*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^2*b*e^(2*x) + 36*sqrt(a^2 + sqrt(a*b)*a)*sqrt( a*b)*a*b^2*e^(2*x) - 6*sqrt(a^2 + sqrt(a*b)*a)*a^3*b + 12*sqrt(a*b)*a^3*b + 5*sqrt(a^2 + sqrt(a*b)*a)*a^2*b^2 - 16*sqrt(a*b)*a^2*b^2 + 4*sqrt(a^2 + sqrt(a*b)*a)*a*b^3 - 9*sqrt(a^2 + sqrt(a*b)*a)*sqrt(a*b)*a^2*b + 12*sqr...
Time = 10.27 (sec) , antiderivative size = 1487, normalized size of antiderivative = 14.72 \[ \int \frac {1}{a-b \cosh ^4(x)} \, dx=\text {Too large to display} \]
log(((((1/(a^2 - (a^3*b)^(1/2)))^(1/2)*((4194304*(b^4*exp(2*x) + 253*a*b^3 + 1184*a^3*b - 512*a^4 + b^4 - 930*a^2*b^2 - 1392*a^2*b^2*exp(2*x) + 627* a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))/(b^6*(a - b)^2) - (8388608*a*(1/(a^2 - (a^3*b)^(1/2)))^(1/2)*(512*a^3*exp(2*x) + 6*b^3*exp(2*x) + 181*a*b^2 - 432*a^2*b + 256*a^3 + 5*b^3 + 622*a*b^2*exp(2*x) - 1152*a^2*b*exp(2*x)))/( b^6*(a - b))))/4 - (2097152*(176*a*b - 1536*a^2*exp(2*x) + 134*b^2*exp(2*x ) - 256*a^2 + 75*b^2 + 1408*a*b*exp(2*x)))/(b^6*(a - b)))*(1/(a^2 - (a^3*b )^(1/2)))^(1/2))/4 + (524288*(1024*a^3*exp(2*x) + 35*b^3*exp(2*x) + 185*a* b^2 - 464*a^2*b + 256*a^3 + 24*b^3 + 988*a*b^2*exp(2*x) - 2048*a^2*b*exp(2 *x)))/(a*b^6*(a - b)^2))*(-(a^2 + (a^3*b)^(1/2))/(16*(a^3*b - a^4)))^(1/2) - log(((((1/(a^2 - (a^3*b)^(1/2)))^(1/2)*((4194304*(b^4*exp(2*x) + 253*a* b^3 + 1184*a^3*b - 512*a^4 + b^4 - 930*a^2*b^2 - 1392*a^2*b^2*exp(2*x) + 6 27*a*b^3*exp(2*x) + 768*a^3*b*exp(2*x)))/(b^6*(a - b)^2) + (8388608*a*(1/( a^2 - (a^3*b)^(1/2)))^(1/2)*(512*a^3*exp(2*x) + 6*b^3*exp(2*x) + 181*a*b^2 - 432*a^2*b + 256*a^3 + 5*b^3 + 622*a*b^2*exp(2*x) - 1152*a^2*b*exp(2*x)) )/(b^6*(a - b))))/4 + (2097152*(176*a*b - 1536*a^2*exp(2*x) + 134*b^2*exp( 2*x) - 256*a^2 + 75*b^2 + 1408*a*b*exp(2*x)))/(b^6*(a - b)))*(1/(a^2 - (a^ 3*b)^(1/2)))^(1/2))/4 + (524288*(1024*a^3*exp(2*x) + 35*b^3*exp(2*x) + 185 *a*b^2 - 464*a^2*b + 256*a^3 + 24*b^3 + 988*a*b^2*exp(2*x) - 2048*a^2*b*ex p(2*x)))/(a*b^6*(a - b)^2))*(-(a^2 + (a^3*b)^(1/2))/(16*(a^3*b - a^4)))...