Integrand size = 60, antiderivative size = 23 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {1}{4} \left (-3+x+\frac {1}{x+\frac {3}{i \pi +\log (25)}}\right ) \] Output:
1/4*x+1/4/(x+3/(2*ln(5)+I*Pi))-3/4
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {1}{4} \left (x-\frac {-\pi +i \log (25)}{-3 i+\pi x-i x \log (25)}\right ) \] Input:
Integrate[(9 + 6*x*(I*Pi + Log[25]) + (-1 + x^2)*(I*Pi + Log[25])^2)/(36 + 24*x*(I*Pi + Log[25]) + 4*x^2*(I*Pi + Log[25])^2),x]
Output:
(x - (-Pi + I*Log[25])/(-3*I + Pi*x - I*x*Log[25]))/4
Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2083, 1294, 27, 1107, 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.
\(\displaystyle \int \frac {\left (x^2-1\right ) (\log (25)+i \pi )^2+6 x (\log (25)+i \pi )+9}{4 x^2 (\log (25)+i \pi )^2+24 x (\log (25)+i \pi )+36} \, dx\) |
\(\Big \downarrow \) 2083 |
\(\displaystyle \int \frac {x^2 \left (-(\pi -i \log (25))^2\right )+6 x (\log (25)+i \pi )+\pi ^2+9-\log ^2(25)-2 i \pi \log (25)}{4 x^2 (\log (25)+i \pi )^2+24 x (\log (25)+i \pi )+36}dx\) |
\(\Big \downarrow \) 1294 |
\(\displaystyle 4 (\log (25)+i \pi )^2 \int \frac {-(\pi -i \log (25))^2 x^2+6 (i \pi +\log (25)) x-\log ^2(25)-2 i \pi \log (25)+\pi ^2+9}{16 (i \pi +\log (25))^2 ((i \pi +\log (25)) x+3)^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int \frac {-(\pi -i \log (25))^2 x^2+6 (i \pi +\log (25)) x-\log ^2(25)-2 i \pi \log (25)+\pi ^2+9}{((i \pi +\log (25)) x+3)^2}dx\) |
\(\Big \downarrow \) 1107 |
\(\displaystyle \frac {1}{4} \int \left (1+\frac {(\pi -i \log (25))^2}{((i \pi +\log (25)) x+3)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{4} \left (x+\frac {\log (25)+i \pi }{3+x (\log (25)+i \pi )}\right )\) |
Input:
Int[(9 + 6*x*(I*Pi + Log[25]) + (-1 + x^2)*(I*Pi + Log[25])^2)/(36 + 24*x* (I*Pi + Log[25]) + 4*x^2*(I*Pi + Log[25])^2),x]
Output:
(x + (I*Pi + Log[25])/(3 + x*(I*Pi + Log[25])))/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; F reeQ[{a, b, c, d, e, m}, x] && EqQ[2*c*d - b*e, 0] && IGtQ[p, 0] && !(EqQ[ m, 3] && NeQ[p, 1])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.)*((d_.) + (e_.)*(x_) + (f_.)*(x _)^2)^(q_.), x_Symbol] :> Simp[1/c^p Int[(b/2 + c*x)^(2*p)*(d + e*x + f*x ^2)^q, x], x] /; FreeQ[{a, b, c, d, e, f, q}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum [v, x]^q, x] /; FreeQ[{p, q}, x] && QuadraticQ[{u, v}, x] && !QuadraticMat chQ[{u, v}, x]
Time = 1.46 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78
method | result | size |
risch | \(\frac {x}{4}-\frac {i \ln \left (5\right )}{2 \left (-2 i x \ln \left (5\right )+\pi x -3 i\right )}+\frac {\pi }{-8 i x \ln \left (5\right )+4 \pi x -12 i}\) | \(41\) |
default | \(\frac {x}{4}-\frac {-\pi ^{2}+4 i \pi \ln \left (5\right )+4 \ln \left (5\right )^{2}}{4 \left (-2 i \ln \left (5\right )+\pi \right ) \left (-2 i x \ln \left (5\right )+\pi x -3 i\right )}\) | \(48\) |
parallelrisch | \(-\frac {-3 \pi \,x^{2}+i x \,\pi ^{2}-4 i x \ln \left (5\right )^{2}+6 i \ln \left (5\right ) x^{2}+4 \pi \ln \left (5\right ) x +9 i x}{12 \left (-2 i x \ln \left (5\right )+\pi x -3 i\right )}\) | \(57\) |
norman | \(\frac {\left (\frac {\pi ^{2}}{4}+\ln \left (5\right )^{2}\right ) x^{3}+\left (\frac {9}{4}+\frac {\pi ^{2}}{4}-\ln \left (5\right )^{2}-i \pi \ln \left (5\right )\right ) x +\left (-\frac {i \pi ^{3}}{12}-\frac {i \pi \ln \left (5\right )^{2}}{3}-\frac {\pi ^{2} \ln \left (5\right )}{6}-\frac {2 \ln \left (5\right )^{3}}{3}+3 \ln \left (5\right )\right ) x^{2}}{\pi ^{2} x^{2}+4 x^{2} \ln \left (5\right )^{2}+12 x \ln \left (5\right )+9}\) | \(99\) |
gosper | \(-\frac {\left (-2 i x \ln \left (5\right )+\pi x -3 i\right ) x \left (4 \pi \ln \left (5\right )+i \pi ^{2}-4 i \ln \left (5\right )^{2}+6 i x \ln \left (5\right )-3 \pi x +9 i\right ) \left (-4 i \pi \ln \left (5\right ) x^{2}+\pi ^{2} x^{2}-4 x^{2} \ln \left (5\right )^{2}+4 i \pi \ln \left (5\right )-6 i \pi x -\pi ^{2}+4 \ln \left (5\right )^{2}-12 x \ln \left (5\right )-9\right )}{12 \left (-2 i x \ln \left (5\right )+\pi x +2 i \ln \left (5\right )-\pi -3 i\right ) \left (-2 i x \ln \left (5\right )+\pi x -2 i \ln \left (5\right )+\pi -3 i\right ) \left (-4 i \pi \ln \left (5\right ) x^{2}+\pi ^{2} x^{2}-4 x^{2} \ln \left (5\right )^{2}-6 i \pi x -12 x \ln \left (5\right )-9\right )}\) | \(182\) |
Input:
int(((x^2-1)*(2*ln(5)+I*Pi)^2+6*x*(2*ln(5)+I*Pi)+9)/(4*x^2*(2*ln(5)+I*Pi)^ 2+24*x*(2*ln(5)+I*Pi)+36),x,method=_RETURNVERBOSE)
Output:
1/4*x-1/2*I/(-2*I*x*ln(5)+Pi*x-3*I)*ln(5)+1/4/(-2*I*x*ln(5)+Pi*x-3*I)*Pi
Time = 0.06 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.57 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {i \, \pi + i \, \pi x^{2} + 2 \, {\left (x^{2} + 1\right )} \log \left (5\right ) + 3 \, x}{4 i \, \pi x + 8 \, x \log \left (5\right ) + 12} \] Input:
integrate(((x^2-1)*(2*log(5)+I*pi)^2+6*x*(2*log(5)+I*pi)+9)/(4*x^2*(2*log( 5)+I*pi)^2+24*x*(2*log(5)+I*pi)+36),x, algorithm="fricas")
Output:
(I*pi + I*pi*x^2 + 2*(x^2 + 1)*log(5) + 3*x)/(4*I*pi*x + 8*x*log(5) + 12)
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {x}{4} + \frac {2 \log {\left (5 \right )} + i \pi }{x \left (8 \log {\left (5 \right )} + 4 i \pi \right ) + 12} \] Input:
integrate(((x**2-1)*(2*ln(5)+I*pi)**2+6*x*(2*ln(5)+I*pi)+9)/(4*x**2*(2*ln( 5)+I*pi)**2+24*x*(2*ln(5)+I*pi)+36),x)
Output:
x/4 + (2*log(5) + I*pi)/(x*(8*log(5) + 4*I*pi) + 12)
Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {1}{4} \, x + \frac {i \, \pi + 2 \, \log \left (5\right )}{4 \, {\left ({\left (i \, \pi + 2 \, \log \left (5\right )\right )} x + 3\right )}} \] Input:
integrate(((x^2-1)*(2*log(5)+I*pi)^2+6*x*(2*log(5)+I*pi)+9)/(4*x^2*(2*log( 5)+I*pi)^2+24*x*(2*log(5)+I*pi)+36),x, algorithm="maxima")
Output:
1/4*x + 1/4*(I*pi + 2*log(5))/((I*pi + 2*log(5))*x + 3)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 62 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.70 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {\pi ^{2} x - 4 i \, \pi x \log \left (5\right ) - 4 \, x \log \left (5\right )^{2}}{4 \, {\left (\pi ^{2} - 4 i \, \pi \log \left (5\right ) - 4 \, \log \left (5\right )^{2}\right )}} + \frac {-i \, \pi - 2 \, \log \left (5\right )}{4 \, {\left (-i \, \pi x - 2 \, x \log \left (5\right ) - 3\right )}} \] Input:
integrate(((x^2-1)*(2*log(5)+I*pi)^2+6*x*(2*log(5)+I*pi)+9)/(4*x^2*(2*log( 5)+I*pi)^2+24*x*(2*log(5)+I*pi)+36),x, algorithm="giac")
Output:
1/4*(pi^2*x - 4*I*pi*x*log(5) - 4*x*log(5)^2)/(pi^2 - 4*I*pi*log(5) - 4*lo g(5)^2) + 1/4*(-I*pi - 2*log(5))/(-I*pi*x - 2*x*log(5) - 3)
Time = 0.65 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.87 \[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\frac {-\Pi +x\,3{}\mathrm {i}+\ln \left (5\right )\,2{}\mathrm {i}-\Pi \,x^2+x^2\,\ln \left (5\right )\,2{}\mathrm {i}}{4\,\left (-\Pi \,x+x\,\ln \left (5\right )\,2{}\mathrm {i}+3{}\mathrm {i}\right )} \] Input:
int((6*x*(Pi*1i + 2*log(5)) + (x^2 - 1)*(Pi*1i + 2*log(5))^2 + 9)/(4*x^2*( Pi*1i + 2*log(5))^2 + 24*x*(Pi*1i + 2*log(5)) + 36),x)
Output:
(x*3i - Pi + log(5)*2i - Pi*x^2 + x^2*log(5)*2i)/(4*(x*log(5)*2i - Pi*x + 3i))
\[ \int \frac {9+6 x (i \pi +\log (25))+\left (-1+x^2\right ) (i \pi +\log (25))^2}{36+24 x (i \pi +\log (25))+4 x^2 (i \pi +\log (25))^2} \, dx=\left (\int \frac {x^{2}}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) \mathrm {log}\left (5\right )^{2}+\left (\int \frac {x^{2}}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) \mathrm {log}\left (5\right ) i \pi -\frac {\left (\int \frac {x^{2}}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) \pi ^{2}}{4}+3 \left (\int \frac {x}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) \mathrm {log}\left (5\right )+\frac {3 \left (\int \frac {x}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) i \pi }{2}-\left (\int \frac {1}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) \mathrm {log}\left (5\right )^{2}-\left (\int \frac {1}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) \mathrm {log}\left (5\right ) i \pi +\frac {\left (\int \frac {1}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right ) \pi ^{2}}{4}+\frac {9 \left (\int \frac {1}{4 \mathrm {log}\left (5\right )^{2} x^{2}+4 \,\mathrm {log}\left (5\right ) i \pi \,x^{2}+12 \,\mathrm {log}\left (5\right ) x +6 i \pi x -\pi ^{2} x^{2}+9}d x \right )}{4} \] Input:
int(((x^2-1)*(2*log(5)+I*Pi)^2+6*x*(2*log(5)+I*Pi)+9)/(4*x^2*(2*log(5)+I*P i)^2+24*x*(2*log(5)+I*Pi)+36),x)
Output:
(4*int(x**2/(4*log(5)**2*x**2 + 4*log(5)*i*pi*x**2 + 12*log(5)*x + 6*i*pi* x - pi**2*x**2 + 9),x)*log(5)**2 + 4*int(x**2/(4*log(5)**2*x**2 + 4*log(5) *i*pi*x**2 + 12*log(5)*x + 6*i*pi*x - pi**2*x**2 + 9),x)*log(5)*i*pi - int (x**2/(4*log(5)**2*x**2 + 4*log(5)*i*pi*x**2 + 12*log(5)*x + 6*i*pi*x - pi **2*x**2 + 9),x)*pi**2 + 12*int(x/(4*log(5)**2*x**2 + 4*log(5)*i*pi*x**2 + 12*log(5)*x + 6*i*pi*x - pi**2*x**2 + 9),x)*log(5) + 6*int(x/(4*log(5)**2 *x**2 + 4*log(5)*i*pi*x**2 + 12*log(5)*x + 6*i*pi*x - pi**2*x**2 + 9),x)*i *pi - 4*int(1/(4*log(5)**2*x**2 + 4*log(5)*i*pi*x**2 + 12*log(5)*x + 6*i*p i*x - pi**2*x**2 + 9),x)*log(5)**2 - 4*int(1/(4*log(5)**2*x**2 + 4*log(5)* i*pi*x**2 + 12*log(5)*x + 6*i*pi*x - pi**2*x**2 + 9),x)*log(5)*i*pi + int( 1/(4*log(5)**2*x**2 + 4*log(5)*i*pi*x**2 + 12*log(5)*x + 6*i*pi*x - pi**2* x**2 + 9),x)*pi**2 + 9*int(1/(4*log(5)**2*x**2 + 4*log(5)*i*pi*x**2 + 12*l og(5)*x + 6*i*pi*x - pi**2*x**2 + 9),x))/4