Integrand size = 64, antiderivative size = 20 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=(4-x) \log ^2\left ((-4+x) \left (28+16 x^2\right )\right ) \] Output:
(4-x)*ln((16*x^2+28)*(-4+x))^2
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.27 (sec) , antiderivative size = 552, normalized size of antiderivative = 27.60 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=-4 \log ^2\left (i \sqrt {7}-2 x\right )-8 \log \left (i \sqrt {7}-2 x\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right )-8 \log \left (-\frac {i \sqrt {7}-2 x}{8-i \sqrt {7}}\right ) \log (-4+x)-4 \log ^2(-4+x)-8 \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )-8 \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (i \sqrt {7}+2 x\right )-4 \log ^2\left (i \sqrt {7}+2 x\right )-8 \log \left (i \sqrt {7}-2 x\right ) \log \left (-\frac {i \left (i \sqrt {7}+2 x\right )}{2 \sqrt {7}}\right )-8 \log (-4+x) \log \left (\frac {i \sqrt {7}+2 x}{8+i \sqrt {7}}\right )+8 \log \left (i \sqrt {7}-2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log (-4+x) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )+8 \log \left (i \sqrt {7}+2 x\right ) \log \left (-4 \left (28-7 x+16 x^2-4 x^3\right )\right )-x \log ^2\left (4 \left (-28+7 x-16 x^2+4 x^3\right )\right )-8 \operatorname {PolyLog}\left (2,-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right )-8 \operatorname {PolyLog}\left (2,-\frac {i \sqrt {7}-2 x}{8-i \sqrt {7}}\right )-8 \operatorname {PolyLog}\left (2,\frac {2 (4-x)}{8-i \sqrt {7}}\right )-8 \operatorname {PolyLog}\left (2,\frac {2 (4-x)}{8+i \sqrt {7}}\right )-8 \operatorname {PolyLog}\left (2,-\frac {i \left (i \sqrt {7}+2 x\right )}{2 \sqrt {7}}\right )-8 \operatorname {PolyLog}\left (2,\frac {i \sqrt {7}+2 x}{8+i \sqrt {7}}\right ) \] Input:
Integrate[((-14 + 64*x - 24*x^2)*Log[-112 + 28*x - 64*x^2 + 16*x^3] + (-7 - 4*x^2)*Log[-112 + 28*x - 64*x^2 + 16*x^3]^2)/(7 + 4*x^2),x]
Output:
-4*Log[I*Sqrt[7] - 2*x]^2 - 8*Log[I*Sqrt[7] - 2*x]*Log[(2*(4 - x))/(8 - I* Sqrt[7])] - 8*Log[-((I*Sqrt[7] - 2*x)/(8 - I*Sqrt[7]))]*Log[-4 + x] - 4*Lo g[-4 + x]^2 - 8*Log[((-1/2*I)*(I*Sqrt[7] - 2*x))/Sqrt[7]]*Log[I*Sqrt[7] + 2*x] - 8*Log[(2*(4 - x))/(8 + I*Sqrt[7])]*Log[I*Sqrt[7] + 2*x] - 4*Log[I*S qrt[7] + 2*x]^2 - 8*Log[I*Sqrt[7] - 2*x]*Log[((-1/2*I)*(I*Sqrt[7] + 2*x))/ Sqrt[7]] - 8*Log[-4 + x]*Log[(I*Sqrt[7] + 2*x)/(8 + I*Sqrt[7])] + 8*Log[I* Sqrt[7] - 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + 8*Log[-4 + x]*Log[-4* (28 - 7*x + 16*x^2 - 4*x^3)] + 8*Log[I*Sqrt[7] + 2*x]*Log[-4*(28 - 7*x + 1 6*x^2 - 4*x^3)] - x*Log[4*(-28 + 7*x - 16*x^2 + 4*x^3)]^2 - 8*PolyLog[2, ( (-1/2*I)*(I*Sqrt[7] - 2*x))/Sqrt[7]] - 8*PolyLog[2, -((I*Sqrt[7] - 2*x)/(8 - I*Sqrt[7]))] - 8*PolyLog[2, (2*(4 - x))/(8 - I*Sqrt[7])] - 8*PolyLog[2, (2*(4 - x))/(8 + I*Sqrt[7])] - 8*PolyLog[2, ((-1/2*I)*(I*Sqrt[7] + 2*x))/ Sqrt[7]] - 8*PolyLog[2, (I*Sqrt[7] + 2*x)/(8 + I*Sqrt[7])]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.00 (sec) , antiderivative size = 1101, normalized size of antiderivative = 55.05, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.031, Rules used = {7276, 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.
\(\displaystyle \int \frac {\left (-4 x^2-7\right ) \log ^2\left (16 x^3-64 x^2+28 x-112\right )+\left (-24 x^2+64 x-14\right ) \log \left (16 x^3-64 x^2+28 x-112\right )}{4 x^2+7} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\log ^2\left (4 \left (4 x^3-16 x^2+7 x-28\right )\right )-\frac {2 \left (12 x^2-32 x+7\right ) \log \left (4 \left (4 x^3-16 x^2+7 x-28\right )\right )}{4 x^2+7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \left (8-i \sqrt {7}\right ) \log ^2\left (i \sqrt {7}-2 x\right )-\frac {1}{2} i \sqrt {7} \log ^2\left (i \sqrt {7}-2 x\right )-\left (8-i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-i \sqrt {7} \log \left (\frac {2 (4-x)}{8-i \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-\left (8-i \sqrt {7}\right ) \log \left (-\frac {i \left (2 x+i \sqrt {7}\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )-i \sqrt {7} \log \left (-\frac {i \left (2 x+i \sqrt {7}\right )}{2 \sqrt {7}}\right ) \log \left (i \sqrt {7}-2 x\right )+\left (8-i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right ) \log \left (i \sqrt {7}-2 x\right )+i \sqrt {7} \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right ) \log \left (i \sqrt {7}-2 x\right )-4 \log ^2(x-4)-\frac {1}{2} \left (8+i \sqrt {7}\right ) \log ^2\left (2 x+i \sqrt {7}\right )+\frac {1}{2} i \sqrt {7} \log ^2\left (2 x+i \sqrt {7}\right )-x \log ^2\left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-8 \log \left (-\frac {i \sqrt {7}-2 x}{8-i \sqrt {7}}\right ) \log (x-4)-\left (8+i \sqrt {7}\right ) \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )+i \sqrt {7} \log \left (-\frac {i \left (i \sqrt {7}-2 x\right )}{2 \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )-\left (8+i \sqrt {7}\right ) \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )+i \sqrt {7} \log \left (\frac {2 (4-x)}{8+i \sqrt {7}}\right ) \log \left (2 x+i \sqrt {7}\right )-8 \log (x-4) \log \left (\frac {2 x+i \sqrt {7}}{8+i \sqrt {7}}\right )+8 \log (x-4) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )+\left (8+i \sqrt {7}\right ) \log \left (2 x+i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-i \sqrt {7} \log \left (2 x+i \sqrt {7}\right ) \log \left (-4 \left (-4 x^3+16 x^2-7 x+28\right )\right )-8 \operatorname {PolyLog}\left (2,\frac {2 (4-x)}{8-i \sqrt {7}}\right )-8 \operatorname {PolyLog}\left (2,\frac {2 (4-x)}{8+i \sqrt {7}}\right )-\left (8+i \sqrt {7}\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {7}-2 i x}{8 i-\sqrt {7}}\right )+i \sqrt {7} \operatorname {PolyLog}\left (2,-\frac {\sqrt {7}-2 i x}{8 i-\sqrt {7}}\right )-\left (8-i \sqrt {7}\right ) \operatorname {PolyLog}\left (2,\frac {2 i x+\sqrt {7}}{8 i+\sqrt {7}}\right )-i \sqrt {7} \operatorname {PolyLog}\left (2,\frac {2 i x+\sqrt {7}}{8 i+\sqrt {7}}\right )-\left (8+i \sqrt {7}\right ) \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {i x}{\sqrt {7}}\right )+i \sqrt {7} \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {i x}{\sqrt {7}}\right )-\left (8-i \sqrt {7}\right ) \operatorname {PolyLog}\left (2,\frac {i x}{\sqrt {7}}+\frac {1}{2}\right )-i \sqrt {7} \operatorname {PolyLog}\left (2,\frac {i x}{\sqrt {7}}+\frac {1}{2}\right )\) |
Input:
Int[((-14 + 64*x - 24*x^2)*Log[-112 + 28*x - 64*x^2 + 16*x^3] + (-7 - 4*x^ 2)*Log[-112 + 28*x - 64*x^2 + 16*x^3]^2)/(7 + 4*x^2),x]
Output:
(-1/2*I)*Sqrt[7]*Log[I*Sqrt[7] - 2*x]^2 - ((8 - I*Sqrt[7])*Log[I*Sqrt[7] - 2*x]^2)/2 - I*Sqrt[7]*Log[I*Sqrt[7] - 2*x]*Log[(2*(4 - x))/(8 - I*Sqrt[7] )] - (8 - I*Sqrt[7])*Log[I*Sqrt[7] - 2*x]*Log[(2*(4 - x))/(8 - I*Sqrt[7])] - 8*Log[-((I*Sqrt[7] - 2*x)/(8 - I*Sqrt[7]))]*Log[-4 + x] - 4*Log[-4 + x] ^2 + I*Sqrt[7]*Log[((-1/2*I)*(I*Sqrt[7] - 2*x))/Sqrt[7]]*Log[I*Sqrt[7] + 2 *x] - (8 + I*Sqrt[7])*Log[((-1/2*I)*(I*Sqrt[7] - 2*x))/Sqrt[7]]*Log[I*Sqrt [7] + 2*x] + I*Sqrt[7]*Log[(2*(4 - x))/(8 + I*Sqrt[7])]*Log[I*Sqrt[7] + 2* x] - (8 + I*Sqrt[7])*Log[(2*(4 - x))/(8 + I*Sqrt[7])]*Log[I*Sqrt[7] + 2*x] + (I/2)*Sqrt[7]*Log[I*Sqrt[7] + 2*x]^2 - ((8 + I*Sqrt[7])*Log[I*Sqrt[7] + 2*x]^2)/2 - I*Sqrt[7]*Log[I*Sqrt[7] - 2*x]*Log[((-1/2*I)*(I*Sqrt[7] + 2*x ))/Sqrt[7]] - (8 - I*Sqrt[7])*Log[I*Sqrt[7] - 2*x]*Log[((-1/2*I)*(I*Sqrt[7 ] + 2*x))/Sqrt[7]] - 8*Log[-4 + x]*Log[(I*Sqrt[7] + 2*x)/(8 + I*Sqrt[7])] + I*Sqrt[7]*Log[I*Sqrt[7] - 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + (8 - I*Sqrt[7])*Log[I*Sqrt[7] - 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + 8* Log[-4 + x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] - I*Sqrt[7]*Log[I*Sqrt[7] + 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] + (8 + I*Sqrt[7])*Log[I*Sqrt[7] + 2*x]*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)] - x*Log[-4*(28 - 7*x + 16*x^2 - 4*x^3)]^2 - 8*PolyLog[2, (2*(4 - x))/(8 - I*Sqrt[7])] - 8*PolyLog[2, (2* (4 - x))/(8 + I*Sqrt[7])] + I*Sqrt[7]*PolyLog[2, -((Sqrt[7] - (2*I)*x)/(8* I - Sqrt[7]))] - (8 + I*Sqrt[7])*PolyLog[2, -((Sqrt[7] - (2*I)*x)/(8*I ...
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.85 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.25
method | result | size |
risch | \(\left (-x +4\right ) \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}\) | \(25\) |
norman | \(4 \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}-x \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}\) | \(43\) |
parallelrisch | \(4 \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}-x \ln \left (16 x^{3}-64 x^{2}+28 x -112\right )^{2}\) | \(43\) |
orering | \(\text {Expression too large to display}\) | \(1107\) |
Input:
int(((-4*x^2-7)*ln(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*ln(16*x^3-6 4*x^2+28*x-112))/(4*x^2+7),x,method=_RETURNVERBOSE)
Output:
(-x+4)*ln(16*x^3-64*x^2+28*x-112)^2
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=-{\left (x - 4\right )} \log \left (16 \, x^{3} - 64 \, x^{2} + 28 \, x - 112\right )^{2} \] Input:
integrate(((-4*x^2-7)*log(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*log( 16*x^3-64*x^2+28*x-112))/(4*x^2+7),x, algorithm="fricas")
Output:
-(x - 4)*log(16*x^3 - 64*x^2 + 28*x - 112)^2
Time = 0.09 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=\left (4 - x\right ) \log {\left (16 x^{3} - 64 x^{2} + 28 x - 112 \right )}^{2} \] Input:
integrate(((-4*x**2-7)*ln(16*x**3-64*x**2+28*x-112)**2+(-24*x**2+64*x-14)* ln(16*x**3-64*x**2+28*x-112))/(4*x**2+7),x)
Output:
(4 - x)*log(16*x**3 - 64*x**2 + 28*x - 112)**2
Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (20) = 40\).
Time = 0.45 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.85 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=-4 \, x \log \left (2\right )^{2} - {\left (x - 4\right )} \log \left (4 \, x^{2} + 7\right )^{2} - {\left (x - 4\right )} \log \left (x - 4\right )^{2} - 2 \, {\left (2 \, x \log \left (2\right ) + {\left (x - 4\right )} \log \left (x - 4\right ) - 8 \, \log \left (2\right )\right )} \log \left (4 \, x^{2} + 7\right ) - 4 \, {\left (x \log \left (2\right ) - 4 \, \log \left (2\right )\right )} \log \left (x - 4\right ) \] Input:
integrate(((-4*x^2-7)*log(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*log( 16*x^3-64*x^2+28*x-112))/(4*x^2+7),x, algorithm="maxima")
Output:
-4*x*log(2)^2 - (x - 4)*log(4*x^2 + 7)^2 - (x - 4)*log(x - 4)^2 - 2*(2*x*l og(2) + (x - 4)*log(x - 4) - 8*log(2))*log(4*x^2 + 7) - 4*(x*log(2) - 4*lo g(2))*log(x - 4)
Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=-{\left (x - 4\right )} \log \left (16 \, x^{3} - 64 \, x^{2} + 28 \, x - 112\right )^{2} \] Input:
integrate(((-4*x^2-7)*log(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*log( 16*x^3-64*x^2+28*x-112))/(4*x^2+7),x, algorithm="giac")
Output:
-(x - 4)*log(16*x^3 - 64*x^2 + 28*x - 112)^2
Time = 3.57 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=-{\ln \left (16\,x^3-64\,x^2+28\,x-112\right )}^2\,\left (x-4\right ) \] Input:
int(-(log(28*x - 64*x^2 + 16*x^3 - 112)^2*(4*x^2 + 7) + log(28*x - 64*x^2 + 16*x^3 - 112)*(24*x^2 - 64*x + 14))/(4*x^2 + 7),x)
Output:
-log(28*x - 64*x^2 + 16*x^3 - 112)^2*(x - 4)
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {\left (-14+64 x-24 x^2\right ) \log \left (-112+28 x-64 x^2+16 x^3\right )+\left (-7-4 x^2\right ) \log ^2\left (-112+28 x-64 x^2+16 x^3\right )}{7+4 x^2} \, dx=\mathrm {log}\left (16 x^{3}-64 x^{2}+28 x -112\right )^{2} \left (-x +4\right ) \] Input:
int(((-4*x^2-7)*log(16*x^3-64*x^2+28*x-112)^2+(-24*x^2+64*x-14)*log(16*x^3 -64*x^2+28*x-112))/(4*x^2+7),x)
Output:
log(16*x**3 - 64*x**2 + 28*x - 112)**2*( - x + 4)