Integrand size = 21, antiderivative size = 55 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=\frac {e \text {arctanh}\left (\frac {\sqrt {c} x^3}{\sqrt {a}}\right )}{3 \sqrt {a} \sqrt {c}}+\frac {d \log (x)}{a}-\frac {d \log \left (a-c x^6\right )}{6 a} \] Output:
1/3*e*arctanh(c^(1/2)*x^3/a^(1/2))/a^(1/2)/c^(1/2)+d*ln(x)/a-1/6*d*ln(-c*x ^6+a)/a
Leaf count is larger than twice the leaf count of optimal. \(156\) vs. \(2(55)=110\).
Time = 0.04 (sec) , antiderivative size = 156, normalized size of antiderivative = 2.84 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=\frac {6 \sqrt {c} d \log (x)-\sqrt {a} e \log \left (\sqrt [6]{a}-\sqrt [6]{c} x\right )+\sqrt {a} e \log \left (\sqrt [6]{a}+\sqrt [6]{c} x\right )+\sqrt {a} e \log \left (\sqrt [3]{a}-\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )-\sqrt {a} e \log \left (\sqrt [3]{a}+\sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )-\sqrt {c} d \log \left (a-c x^6\right )}{6 a \sqrt {c}} \] Input:
Integrate[(d + e*x^3)/(x*(a - c*x^6)),x]
Output:
(6*Sqrt[c]*d*Log[x] - Sqrt[a]*e*Log[a^(1/6) - c^(1/6)*x] + Sqrt[a]*e*Log[a ^(1/6) + c^(1/6)*x] + Sqrt[a]*e*Log[a^(1/3) - a^(1/6)*c^(1/6)*x + c^(1/3)* x^2] - Sqrt[a]*e*Log[a^(1/3) + a^(1/6)*c^(1/6)*x + c^(1/3)*x^2] - Sqrt[c]* d*Log[a - c*x^6])/(6*a*Sqrt[c])
Time = 0.22 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1803, 523, 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.
\(\displaystyle \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx\) |
\(\Big \downarrow \) 1803 |
\(\displaystyle \frac {1}{3} \int \frac {e x^3+d}{x^3 \left (a-c x^6\right )}dx^3\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \frac {1}{3} \int \left (\frac {d}{a x^3}+\frac {c d x^3+a e}{a \left (a-c x^6\right )}\right )dx^3\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} \left (\frac {e \text {arctanh}\left (\frac {\sqrt {c} x^3}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}-\frac {d \log \left (a-c x^6\right )}{2 a}+\frac {d \log \left (x^3\right )}{a}\right )\) |
Input:
Int[(d + e*x^3)/(x*(a - c*x^6)),x]
Output:
((e*ArcTanh[(Sqrt[c]*x^3)/Sqrt[a]])/(Sqrt[a]*Sqrt[c]) + (d*Log[x^3])/a - ( d*Log[a - c*x^6])/(2*a))/3
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(d + e*x )^q*(a + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && IntegerQ[Simplify[(m + 1)/n]]
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {\frac {d \ln \left (-c \,x^{6}+a \right )}{2}-\frac {a e \,\operatorname {arctanh}\left (\frac {c \,x^{3}}{\sqrt {a c}}\right )}{\sqrt {a c}}}{3 a}+\frac {d \ln \left (x \right )}{a}\) | \(47\) |
risch | \(\frac {d \ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{2} c \,\textit {\_Z}^{2}+2 a c d \textit {\_Z} -a \,e^{2}+c \,d^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (-7 a c \,\textit {\_R}^{2}-7 c d \textit {\_R} +6 e^{2}\right ) x^{3}-a e \textit {\_R} +6 d e \right )\right )}{6}\) | \(76\) |
Input:
int((e*x^3+d)/x/(-c*x^6+a),x,method=_RETURNVERBOSE)
Output:
-1/3/a*(1/2*d*ln(-c*x^6+a)-a*e/(a*c)^(1/2)*arctanh(c*x^3/(a*c)^(1/2)))+d*l n(x)/a
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.15 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=\left [-\frac {c d \log \left (c x^{6} - a\right ) - 6 \, c d \log \left (x\right ) - \sqrt {a c} e \log \left (\frac {c x^{6} + 2 \, \sqrt {a c} x^{3} + a}{c x^{6} - a}\right )}{6 \, a c}, -\frac {c d \log \left (c x^{6} - a\right ) - 6 \, c d \log \left (x\right ) + 2 \, \sqrt {-a c} e \arctan \left (\frac {\sqrt {-a c} x^{3}}{a}\right )}{6 \, a c}\right ] \] Input:
integrate((e*x^3+d)/x/(-c*x^6+a),x, algorithm="fricas")
Output:
[-1/6*(c*d*log(c*x^6 - a) - 6*c*d*log(x) - sqrt(a*c)*e*log((c*x^6 + 2*sqrt (a*c)*x^3 + a)/(c*x^6 - a)))/(a*c), -1/6*(c*d*log(c*x^6 - a) - 6*c*d*log(x ) + 2*sqrt(-a*c)*e*arctan(sqrt(-a*c)*x^3/a))/(a*c)]
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (48) = 96\).
Time = 1.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.13 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=- \left (\frac {d}{6 a} - \frac {e \sqrt {a^{3} c}}{6 a^{2} c}\right ) \log {\left (x^{3} + \frac {- 6 a \left (\frac {d}{6 a} - \frac {e \sqrt {a^{3} c}}{6 a^{2} c}\right ) + d}{e} \right )} - \left (\frac {d}{6 a} + \frac {e \sqrt {a^{3} c}}{6 a^{2} c}\right ) \log {\left (x^{3} + \frac {- 6 a \left (\frac {d}{6 a} + \frac {e \sqrt {a^{3} c}}{6 a^{2} c}\right ) + d}{e} \right )} + \frac {d \log {\left (x \right )}}{a} \] Input:
integrate((e*x**3+d)/x/(-c*x**6+a),x)
Output:
-(d/(6*a) - e*sqrt(a**3*c)/(6*a**2*c))*log(x**3 + (-6*a*(d/(6*a) - e*sqrt( a**3*c)/(6*a**2*c)) + d)/e) - (d/(6*a) + e*sqrt(a**3*c)/(6*a**2*c))*log(x* *3 + (-6*a*(d/(6*a) + e*sqrt(a**3*c)/(6*a**2*c)) + d)/e) + d*log(x)/a
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.15 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=-\frac {e \log \left (\frac {c x^{3} - \sqrt {a c}}{c x^{3} + \sqrt {a c}}\right )}{6 \, \sqrt {a c}} - \frac {d \log \left (c x^{6} - a\right )}{6 \, a} + \frac {d \log \left (x^{3}\right )}{3 \, a} \] Input:
integrate((e*x^3+d)/x/(-c*x^6+a),x, algorithm="maxima")
Output:
-1/6*e*log((c*x^3 - sqrt(a*c))/(c*x^3 + sqrt(a*c)))/sqrt(a*c) - 1/6*d*log( c*x^6 - a)/a + 1/3*d*log(x^3)/a
Time = 0.11 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=-\frac {e \arctan \left (\frac {c x^{3}}{\sqrt {-a c}}\right )}{3 \, \sqrt {-a c}} - \frac {d \log \left (c x^{6} - a\right )}{6 \, a} + \frac {d \log \left ({\left | x \right |}\right )}{a} \] Input:
integrate((e*x^3+d)/x/(-c*x^6+a),x, algorithm="giac")
Output:
-1/3*e*arctan(c*x^3/sqrt(-a*c))/sqrt(-a*c) - 1/6*d*log(c*x^6 - a)/a + d*lo g(abs(x))/a
Time = 21.28 (sec) , antiderivative size = 226, normalized size of antiderivative = 4.11 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=\frac {d\,\ln \left (x\right )}{a}-\frac {d\,\ln \left (a\,e\,\sqrt {a^3\,c}+7\,a^2\,c\,d-a^2\,c\,e\,x^3-7\,c\,d\,x^3\,\sqrt {a^3\,c}\right )}{6\,a}-\frac {d\,\ln \left (7\,a^2\,c\,d-a\,e\,\sqrt {a^3\,c}-a^2\,c\,e\,x^3+7\,c\,d\,x^3\,\sqrt {a^3\,c}\right )}{6\,a}-\frac {e\,\ln \left (a\,e\,\sqrt {a^3\,c}+7\,a^2\,c\,d-a^2\,c\,e\,x^3-7\,c\,d\,x^3\,\sqrt {a^3\,c}\right )\,\sqrt {a^3\,c}}{6\,a^2\,c}+\frac {e\,\ln \left (7\,a^2\,c\,d-a\,e\,\sqrt {a^3\,c}-a^2\,c\,e\,x^3+7\,c\,d\,x^3\,\sqrt {a^3\,c}\right )\,\sqrt {a^3\,c}}{6\,a^2\,c} \] Input:
int((d + e*x^3)/(x*(a - c*x^6)),x)
Output:
(d*log(x))/a - (d*log(a*e*(a^3*c)^(1/2) + 7*a^2*c*d - a^2*c*e*x^3 - 7*c*d* x^3*(a^3*c)^(1/2)))/(6*a) - (d*log(7*a^2*c*d - a*e*(a^3*c)^(1/2) - a^2*c*e *x^3 + 7*c*d*x^3*(a^3*c)^(1/2)))/(6*a) - (e*log(a*e*(a^3*c)^(1/2) + 7*a^2* c*d - a^2*c*e*x^3 - 7*c*d*x^3*(a^3*c)^(1/2))*(a^3*c)^(1/2))/(6*a^2*c) + (e *log(7*a^2*c*d - a*e*(a^3*c)^(1/2) - a^2*c*e*x^3 + 7*c*d*x^3*(a^3*c)^(1/2) )*(a^3*c)^(1/2))/(6*a^2*c)
Time = 0.19 (sec) , antiderivative size = 234, normalized size of antiderivative = 4.25 \[ \int \frac {d+e x^3}{x \left (a-c x^6\right )} \, dx=\frac {c^{\frac {1}{6}} a^{\frac {7}{6}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e +c^{\frac {1}{6}} a^{\frac {7}{6}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e -c^{\frac {1}{6}} a^{\frac {7}{6}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) e -c^{\frac {1}{6}} a^{\frac {7}{6}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) e -c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (-c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d -c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}} x +a^{\frac {1}{3}}+c^{\frac {1}{3}} x^{2}\right ) d -c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (c^{\frac {1}{6}} a^{\frac {1}{6}}-c^{\frac {1}{3}} x \right ) d +6 c^{\frac {2}{3}} a^{\frac {2}{3}} \mathrm {log}\left (x \right ) d}{6 c^{\frac {2}{3}} a^{\frac {5}{3}}} \] Input:
int((e*x^3+d)/x/(-c*x^6+a),x)
Output:
(c**(1/6)*a**(1/6)*log( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)* a*e + c**(1/6)*a**(1/6)*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)*a*e - c**(1 /6)*a**(1/6)*log(c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*a*e - c** (1/6)*a**(1/6)*log(c**(1/6)*a**(1/6) - c**(1/3)*x)*a*e - c**(2/3)*a**(2/3) *log( - c**(1/6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*d - c**(2/3)*a**(2 /3)*log( - c**(1/6)*a**(1/6) - c**(1/3)*x)*d - c**(2/3)*a**(2/3)*log(c**(1 /6)*a**(1/6)*x + a**(1/3) + c**(1/3)*x**2)*d - c**(2/3)*a**(2/3)*log(c**(1 /6)*a**(1/6) - c**(1/3)*x)*d + 6*c**(2/3)*a**(2/3)*log(x)*d)/(6*c**(2/3)*a **(2/3)*a)