3.85.52 \(\int \frac {(-1024 x^4-1536 x^5) \log (x)+e^4 (-1024 x^3-1536 x^4) \log ^2(x)+e^8 (-384 x^2-576 x^3) \log ^3(x)+e^{12} (-64 x-96 x^2) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+(-2048 x^4-3072 x^5+(-768 x^5+e^4 (-1536 x^3-2304 x^4)) \log (x)+(e^8 (-384 x^2-576 x^3)+e^4 (-512 x^3-1536 x^4)) \log ^2(x)+(e^{12} (-32 x-48 x^2)+e^8 (-384 x^2-864 x^3)) \log ^3(x)+e^{12} (-96 x-192 x^2) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)) \log (x^2)}{(8 x^5+24 x^6+18 x^7) \log ^5(x) \log ^2(x^2)} \, dx\) [8452]

Optimal. Leaf size=30 \[ \frac {\left (\frac {e^4}{x}+\frac {4}{\log (x)}\right )^4}{(4+6 x) \log \left (x^2\right )} \]

[Out]

(4/ln(x)+exp(4)/x)^4/ln(x^2)/(4+6*x)

________________________________________________________________________________________

Rubi [F]
time = 21.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{\left (8 x^5+24 x^6+18 x^7\right ) \log ^5(x) \log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1024*x^4 - 1536*x^5)*Log[x] + E^4*(-1024*x^3 - 1536*x^4)*Log[x]^2 + E^8*(-384*x^2 - 576*x^3)*Log[x]^3 +
E^12*(-64*x - 96*x^2)*Log[x]^4 + E^16*(-4 - 6*x)*Log[x]^5 + (-2048*x^4 - 3072*x^5 + (-768*x^5 + E^4*(-1536*x^
3 - 2304*x^4))*Log[x] + (E^8*(-384*x^2 - 576*x^3) + E^4*(-512*x^3 - 1536*x^4))*Log[x]^2 + (E^12*(-32*x - 48*x^
2) + E^8*(-384*x^2 - 864*x^3))*Log[x]^3 + E^12*(-96*x - 192*x^2)*Log[x]^4 + E^16*(-8 - 15*x)*Log[x]^5)*Log[x^2
])/((8*x^5 + 24*x^6 + 18*x^7)*Log[x]^5*Log[x^2]^2),x]

[Out]

(-6075*E^16*x*ExpIntegralEi[Log[x^2]/2])/(256*Sqrt[x^2]) + (81*E^16)/(64*Log[x^2]) + E^16/(4*x^4*Log[x^2]) - (
3*E^16)/(8*x^3*Log[x^2]) + (9*E^16)/(16*x^2*Log[x^2]) - (27*E^16)/(32*x*Log[x^2]) + (243*E^16*Defer[Int][1/((2
+ 3*x)*Log[x^2]^2), x])/32 - 128*Defer[Int][1/(x*Log[x]^4*Log[x^2]^2), x] + 384*Defer[Int][1/((2 + 3*x)*Log[x
]^4*Log[x^2]^2), x] - 128*E^4*Defer[Int][1/(x^2*Log[x]^3*Log[x^2]^2), x] + 192*E^4*Defer[Int][1/(x*Log[x]^3*Lo
g[x^2]^2), x] - 576*E^4*Defer[Int][1/((2 + 3*x)*Log[x]^3*Log[x^2]^2), x] - 48*E^8*Defer[Int][1/(x^3*Log[x]^2*L
og[x^2]^2), x] + 72*E^8*Defer[Int][1/(x^2*Log[x]^2*Log[x^2]^2), x] - 108*E^8*Defer[Int][1/(x*Log[x]^2*Log[x^2]
^2), x] + 324*E^8*Defer[Int][1/((2 + 3*x)*Log[x]^2*Log[x^2]^2), x] - 8*E^12*Defer[Int][1/(x^4*Log[x]*Log[x^2]^
2), x] + 12*E^12*Defer[Int][1/(x^3*Log[x]*Log[x^2]^2), x] - 18*E^12*Defer[Int][1/(x^2*Log[x]*Log[x^2]^2), x] +
27*E^12*Defer[Int][1/(x*Log[x]*Log[x^2]^2), x] - 81*E^12*Defer[Int][1/((2 + 3*x)*Log[x]*Log[x^2]^2), x] + (24
3*E^16*Defer[Int][1/((2 + 3*x)^2*Log[x^2]), x])/8 + (3645*E^16*Defer[Int][x/((2 + 3*x)^2*Log[x^2]), x])/64 + (
1215*E^16*Defer[Int][1/((2 + 3*x)*Log[x^2]), x])/16 + (18225*E^16*Defer[Int][x/((2 + 3*x)*Log[x^2]), x])/128 -
256*Defer[Int][1/(x*Log[x]^5*Log[x^2]), x] + 768*Defer[Int][1/((2 + 3*x)*Log[x]^5*Log[x^2]), x] - 192*E^4*Def
er[Int][1/(x^2*Log[x]^4*Log[x^2]), x] + 288*E^4*Defer[Int][1/(x*Log[x]^4*Log[x^2]), x] - 384*Defer[Int][1/((2
+ 3*x)^2*Log[x]^4*Log[x^2]), x] - 864*E^4*Defer[Int][1/((2 + 3*x)*Log[x]^4*Log[x^2]), x] - 48*E^8*Defer[Int][1
/(x^3*Log[x]^3*Log[x^2]), x] + 144*E^8*Defer[Int][1/(x^2*Log[x]^3*Log[x^2]), x] - 8*E^4*(8 + 9*E^4)*Defer[Int]
[1/(x^2*Log[x]^3*Log[x^2]), x] - 192*E^4*Defer[Int][1/(x*Log[x]^3*Log[x^2]), x] - 324*E^8*Defer[Int][1/(x*Log[
x]^3*Log[x^2]), x] + 24*E^4*(8 + 9*E^4)*Defer[Int][1/(x*Log[x]^3*Log[x^2]), x] + 1152*E^4*Defer[Int][1/((2 + 3
*x)^2*Log[x]^3*Log[x^2]), x] + 648*E^8*Defer[Int][1/((2 + 3*x)^2*Log[x]^3*Log[x^2]), x] - 72*E^4*(8 + 9*E^4)*D
efer[Int][1/((2 + 3*x)^2*Log[x]^3*Log[x^2]), x] + 576*E^4*Defer[Int][1/((2 + 3*x)*Log[x]^3*Log[x^2]), x] + 972
*E^8*Defer[Int][1/((2 + 3*x)*Log[x]^3*Log[x^2]), x] - 72*E^4*(8 + 9*E^4)*Defer[Int][1/((2 + 3*x)*Log[x]^3*Log[
x^2]), x] - 4*E^12*Defer[Int][1/(x^4*Log[x]^2*Log[x^2]), x] + 12*E^12*Defer[Int][1/(x^3*Log[x]^2*Log[x^2]), x]
- 6*E^8*(8 + E^4)*Defer[Int][1/(x^3*Log[x]^2*Log[x^2]), x] - 108*E^8*Defer[Int][1/(x^2*Log[x]^2*Log[x^2]), x]
- 27*E^12*Defer[Int][1/(x^2*Log[x]^2*Log[x^2]), x] + 18*E^8*(8 + E^4)*Defer[Int][1/(x^2*Log[x]^2*Log[x^2]), x
] + 324*E^8*Defer[Int][1/(x*Log[x]^2*Log[x^2]), x] + 54*E^12*Defer[Int][1/(x*Log[x]^2*Log[x^2]), x] - (81*E^8*
(8 + E^4)*Defer[Int][1/(x*Log[x]^2*Log[x^2]), x])/2 - 972*E^8*Defer[Int][1/((2 + 3*x)^2*Log[x]^2*Log[x^2]), x]
- 81*E^12*Defer[Int][1/((2 + 3*x)^2*Log[x]^2*Log[x^2]), x] + 81*E^8*(8 + E^4)*Defer[Int][1/((2 + 3*x)^2*Log[x
]^2*Log[x^2]), x] - 972*E^8*Defer[Int][1/((2 + 3*x)*Log[x]^2*Log[x^2]), x] - 162*E^12*Defer[Int][1/((2 + 3*x)*
Log[x]^2*Log[x^2]), x] + (243*E^8*(8 + E^4)*Defer[Int][1/((2 + 3*x)*Log[x]^2*Log[x^2]), x])/2 - 12*E^12*Defer[
Int][1/(x^4*Log[x]*Log[x^2]), x] + 12*E^12*Defer[Int][1/(x^3*Log[x]*Log[x^2]), x] - 9*E^12*Defer[Int][1/(x^2*L
og[x]*Log[x^2]), x] + 81*E^12*Defer[Int][1/((2 + 3*x)^2*Log[x]*Log[x^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{x^5 \left (8+24 x+18 x^2\right ) \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\int \frac {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{2 x^5 (2+3 x)^2 \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \frac {\left (-1024 x^4-1536 x^5\right ) \log (x)+e^4 \left (-1024 x^3-1536 x^4\right ) \log ^2(x)+e^8 \left (-384 x^2-576 x^3\right ) \log ^3(x)+e^{12} \left (-64 x-96 x^2\right ) \log ^4(x)+e^{16} (-4-6 x) \log ^5(x)+\left (-2048 x^4-3072 x^5+\left (-768 x^5+e^4 \left (-1536 x^3-2304 x^4\right )\right ) \log (x)+\left (e^8 \left (-384 x^2-576 x^3\right )+e^4 \left (-512 x^3-1536 x^4\right )\right ) \log ^2(x)+\left (e^{12} \left (-32 x-48 x^2\right )+e^8 \left (-384 x^2-864 x^3\right )\right ) \log ^3(x)+e^{12} \left (-96 x-192 x^2\right ) \log ^4(x)+e^{16} (-8-15 x) \log ^5(x)\right ) \log \left (x^2\right )}{x^5 (2+3 x)^2 \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (-16 x (2+3 x) \log \left (x^2\right )-4 x \log (x) \left (4+6 x+3 x \log \left (x^2\right )\right )-e^4 \log ^2(x) \left (4+6 x+(8+15 x) \log \left (x^2\right )\right )\right )}{x^5 (2+3 x)^2 \log ^5(x) \log ^2\left (x^2\right )} \, dx\\ &=\frac {1}{2} \int \left (-\frac {2 \left (4 x+e^4 \log (x)\right )^4}{x^5 (2+3 x) \log ^4(x) \log ^2\left (x^2\right )}-\frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^5 (2+3 x)^2 \log ^5(x) \log \left (x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^5 (2+3 x)^2 \log ^5(x) \log \left (x^2\right )} \, dx\right )-\int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^5 (2+3 x) \log ^4(x) \log ^2\left (x^2\right )} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{4 x^5 \log ^5(x) \log \left (x^2\right )}-\frac {3 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{4 x^4 \log ^5(x) \log \left (x^2\right )}+\frac {27 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{16 x^3 \log ^5(x) \log \left (x^2\right )}-\frac {27 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{8 x^2 \log ^5(x) \log \left (x^2\right )}+\frac {405 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{64 x \log ^5(x) \log \left (x^2\right )}-\frac {243 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{32 (2+3 x)^2 \log ^5(x) \log \left (x^2\right )}-\frac {1215 \left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{64 (2+3 x) \log ^5(x) \log \left (x^2\right )}\right ) \, dx\right )-\int \left (\frac {\left (4 x+e^4 \log (x)\right )^4}{2 x^5 \log ^4(x) \log ^2\left (x^2\right )}-\frac {3 \left (4 x+e^4 \log (x)\right )^4}{4 x^4 \log ^4(x) \log ^2\left (x^2\right )}+\frac {9 \left (4 x+e^4 \log (x)\right )^4}{8 x^3 \log ^4(x) \log ^2\left (x^2\right )}-\frac {27 \left (4 x+e^4 \log (x)\right )^4}{16 x^2 \log ^4(x) \log ^2\left (x^2\right )}+\frac {81 \left (4 x+e^4 \log (x)\right )^4}{32 x \log ^4(x) \log ^2\left (x^2\right )}-\frac {243 \left (4 x+e^4 \log (x)\right )^4}{32 (2+3 x) \log ^4(x) \log ^2\left (x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{8} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^5 \log ^5(x) \log \left (x^2\right )} \, dx\right )+\frac {3}{8} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^4 \log ^5(x) \log \left (x^2\right )} \, dx-\frac {1}{2} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^5 \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {3}{4} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^4 \log ^4(x) \log ^2\left (x^2\right )} \, dx-\frac {27}{32} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^3 \log ^5(x) \log \left (x^2\right )} \, dx-\frac {9}{8} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^3 \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {27}{16} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x^2 \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {27}{16} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x^2 \log ^5(x) \log \left (x^2\right )} \, dx-\frac {81}{32} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{x \log ^4(x) \log ^2\left (x^2\right )} \, dx-\frac {405}{128} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{x \log ^5(x) \log \left (x^2\right )} \, dx+\frac {243}{64} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{(2+3 x)^2 \log ^5(x) \log \left (x^2\right )} \, dx+\frac {243}{32} \int \frac {\left (4 x+e^4 \log (x)\right )^4}{(2+3 x) \log ^4(x) \log ^2\left (x^2\right )} \, dx+\frac {1215}{128} \int \frac {\left (4 x+e^4 \log (x)\right )^3 \left (32 x+48 x^2+12 x^2 \log (x)+8 e^4 \log ^2(x)+15 e^4 x \log ^2(x)\right )}{(2+3 x) \log ^5(x) \log \left (x^2\right )} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.25, size = 36, normalized size = 1.20 \begin {gather*} \frac {\left (4 x+e^4 \log (x)\right )^4}{2 x^4 (2+3 x) \log ^4(x) \log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1024*x^4 - 1536*x^5)*Log[x] + E^4*(-1024*x^3 - 1536*x^4)*Log[x]^2 + E^8*(-384*x^2 - 576*x^3)*Log[
x]^3 + E^12*(-64*x - 96*x^2)*Log[x]^4 + E^16*(-4 - 6*x)*Log[x]^5 + (-2048*x^4 - 3072*x^5 + (-768*x^5 + E^4*(-1
536*x^3 - 2304*x^4))*Log[x] + (E^8*(-384*x^2 - 576*x^3) + E^4*(-512*x^3 - 1536*x^4))*Log[x]^2 + (E^12*(-32*x -
48*x^2) + E^8*(-384*x^2 - 864*x^3))*Log[x]^3 + E^12*(-96*x - 192*x^2)*Log[x]^4 + E^16*(-8 - 15*x)*Log[x]^5)*L
og[x^2])/((8*x^5 + 24*x^6 + 18*x^7)*Log[x]^5*Log[x^2]^2),x]

[Out]

(4*x + E^4*Log[x])^4/(2*x^4*(2 + 3*x)*Log[x]^4*Log[x^2])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 75.35, size = 114, normalized size = 3.80 \[\frac {{\mathrm e}^{16} \ln \left (x \right )^{4}+16 \,{\mathrm e}^{12} x \ln \left (x \right )^{3}+96 \,{\mathrm e}^{8} x^{2} \ln \left (x \right )^{2}+256 \,{\mathrm e}^{4} x^{3} \ln \left (x \right )+256 x^{4}}{\ln \left (x \right )^{4} \left (4 \ln \left (x \right )-i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}\right ) x^{4} \left (3 x +2\right )}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-15*x-8)*exp(4)^4*ln(x)^5+(-192*x^2-96*x)*exp(4)^3*ln(x)^4+((-48*x^2-32*x)*exp(4)^3+(-864*x^3-384*x^2)*
exp(4)^2)*ln(x)^3+((-576*x^3-384*x^2)*exp(4)^2+(-1536*x^4-512*x^3)*exp(4))*ln(x)^2+((-2304*x^4-1536*x^3)*exp(4
)-768*x^5)*ln(x)-3072*x^5-2048*x^4)*ln(x^2)+(-6*x-4)*exp(4)^4*ln(x)^5+(-96*x^2-64*x)*exp(4)^3*ln(x)^4+(-576*x^
3-384*x^2)*exp(4)^2*ln(x)^3+(-1536*x^4-1024*x^3)*exp(4)*ln(x)^2+(-1536*x^5-1024*x^4)*ln(x))/(18*x^7+24*x^6+8*x
^5)/ln(x)^5/ln(x^2)^2,x)

[Out]

(exp(16)*ln(x)^4+16*exp(12)*x*ln(x)^3+96*exp(8)*x^2*ln(x)^2+256*exp(4)*x^3*ln(x)+256*x^4)/ln(x)^4/(4*ln(x)-I*P
i*csgn(I*x)^2*csgn(I*x^2)+2*I*Pi*csgn(I*x)*csgn(I*x^2)^2-I*Pi*csgn(I*x^2)^3)/x^4/(3*x+2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (30) = 60\).
time = 0.37, size = 61, normalized size = 2.03 \begin {gather*} \frac {256 \, x^{3} e^{4} \log \left (x\right ) + 96 \, x^{2} e^{8} \log \left (x\right )^{2} + 16 \, x e^{12} \log \left (x\right )^{3} + e^{16} \log \left (x\right )^{4} + 256 \, x^{4}}{4 \, {\left (3 \, x^{5} + 2 \, x^{4}\right )} \log \left (x\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x-8)*exp(4)^4*log(x)^5+(-192*x^2-96*x)*exp(4)^3*log(x)^4+((-48*x^2-32*x)*exp(4)^3+(-864*x^3-3
84*x^2)*exp(4)^2)*log(x)^3+((-576*x^3-384*x^2)*exp(4)^2+(-1536*x^4-512*x^3)*exp(4))*log(x)^2+((-2304*x^4-1536*
x^3)*exp(4)-768*x^5)*log(x)-3072*x^5-2048*x^4)*log(x^2)+(-4-6*x)*exp(4)^4*log(x)^5+(-96*x^2-64*x)*exp(4)^3*log
(x)^4+(-576*x^3-384*x^2)*exp(4)^2*log(x)^3+(-1536*x^4-1024*x^3)*exp(4)*log(x)^2+(-1536*x^5-1024*x^4)*log(x))/(
18*x^7+24*x^6+8*x^5)/log(x)^5/log(x^2)^2,x, algorithm="maxima")

[Out]

1/4*(256*x^3*e^4*log(x) + 96*x^2*e^8*log(x)^2 + 16*x*e^12*log(x)^3 + e^16*log(x)^4 + 256*x^4)/((3*x^5 + 2*x^4)
*log(x)^5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (30) = 60\).
time = 0.38, size = 61, normalized size = 2.03 \begin {gather*} \frac {256 \, x^{3} e^{4} \log \left (x\right ) + 96 \, x^{2} e^{8} \log \left (x\right )^{2} + 16 \, x e^{12} \log \left (x\right )^{3} + e^{16} \log \left (x\right )^{4} + 256 \, x^{4}}{4 \, {\left (3 \, x^{5} + 2 \, x^{4}\right )} \log \left (x\right )^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x-8)*exp(4)^4*log(x)^5+(-192*x^2-96*x)*exp(4)^3*log(x)^4+((-48*x^2-32*x)*exp(4)^3+(-864*x^3-3
84*x^2)*exp(4)^2)*log(x)^3+((-576*x^3-384*x^2)*exp(4)^2+(-1536*x^4-512*x^3)*exp(4))*log(x)^2+((-2304*x^4-1536*
x^3)*exp(4)-768*x^5)*log(x)-3072*x^5-2048*x^4)*log(x^2)+(-4-6*x)*exp(4)^4*log(x)^5+(-96*x^2-64*x)*exp(4)^3*log
(x)^4+(-576*x^3-384*x^2)*exp(4)^2*log(x)^3+(-1536*x^4-1024*x^3)*exp(4)*log(x)^2+(-1536*x^5-1024*x^4)*log(x))/(
18*x^7+24*x^6+8*x^5)/log(x)^5/log(x^2)^2,x, algorithm="fricas")

[Out]

1/4*(256*x^3*e^4*log(x) + 96*x^2*e^8*log(x)^2 + 16*x*e^12*log(x)^3 + e^16*log(x)^4 + 256*x^4)/((3*x^5 + 2*x^4)
*log(x)^5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (20) = 40\).
time = 0.17, size = 65, normalized size = 2.17 \begin {gather*} \frac {256 x^{4} + 256 x^{3} e^{4} \log {\left (x \right )} + 96 x^{2} e^{8} \log {\left (x \right )}^{2} + 16 x e^{12} \log {\left (x \right )}^{3} + e^{16} \log {\left (x \right )}^{4}}{\left (12 x^{5} + 8 x^{4}\right ) \log {\left (x \right )}^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x-8)*exp(4)**4*ln(x)**5+(-192*x**2-96*x)*exp(4)**3*ln(x)**4+((-48*x**2-32*x)*exp(4)**3+(-864*
x**3-384*x**2)*exp(4)**2)*ln(x)**3+((-576*x**3-384*x**2)*exp(4)**2+(-1536*x**4-512*x**3)*exp(4))*ln(x)**2+((-2
304*x**4-1536*x**3)*exp(4)-768*x**5)*ln(x)-3072*x**5-2048*x**4)*ln(x**2)+(-4-6*x)*exp(4)**4*ln(x)**5+(-96*x**2
-64*x)*exp(4)**3*ln(x)**4+(-576*x**3-384*x**2)*exp(4)**2*ln(x)**3+(-1536*x**4-1024*x**3)*exp(4)*ln(x)**2+(-153
6*x**5-1024*x**4)*ln(x))/(18*x**7+24*x**6+8*x**5)/ln(x)**5/ln(x**2)**2,x)

[Out]

(256*x**4 + 256*x**3*exp(4)*log(x) + 96*x**2*exp(8)*log(x)**2 + 16*x*exp(12)*log(x)**3 + exp(16)*log(x)**4)/((
12*x**5 + 8*x**4)*log(x)**5)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).
time = 0.46, size = 65, normalized size = 2.17 \begin {gather*} \frac {256 \, x^{3} e^{4} \log \left (x\right ) + 96 \, x^{2} e^{8} \log \left (x\right )^{2} + 16 \, x e^{12} \log \left (x\right )^{3} + e^{16} \log \left (x\right )^{4} + 256 \, x^{4}}{4 \, {\left (3 \, x^{5} \log \left (x\right )^{5} + 2 \, x^{4} \log \left (x\right )^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-15*x-8)*exp(4)^4*log(x)^5+(-192*x^2-96*x)*exp(4)^3*log(x)^4+((-48*x^2-32*x)*exp(4)^3+(-864*x^3-3
84*x^2)*exp(4)^2)*log(x)^3+((-576*x^3-384*x^2)*exp(4)^2+(-1536*x^4-512*x^3)*exp(4))*log(x)^2+((-2304*x^4-1536*
x^3)*exp(4)-768*x^5)*log(x)-3072*x^5-2048*x^4)*log(x^2)+(-4-6*x)*exp(4)^4*log(x)^5+(-96*x^2-64*x)*exp(4)^3*log
(x)^4+(-576*x^3-384*x^2)*exp(4)^2*log(x)^3+(-1536*x^4-1024*x^3)*exp(4)*log(x)^2+(-1536*x^5-1024*x^4)*log(x))/(
18*x^7+24*x^6+8*x^5)/log(x)^5/log(x^2)^2,x, algorithm="giac")

[Out]

1/4*(256*x^3*e^4*log(x) + 96*x^2*e^8*log(x)^2 + 16*x*e^12*log(x)^3 + e^16*log(x)^4 + 256*x^4)/(3*x^5*log(x)^5
+ 2*x^4*log(x)^5)

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Mupad [B]
time = 6.45, size = 892, normalized size = 29.73 \begin {gather*} \frac {\frac {4\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+8\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^5+12288\,x^5\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )+16384\,x^4+24576\,x^5-8192\,x^3\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-12288\,x^4\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-128\,x\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-192\,x\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+6\,x\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+15\,x\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^5-4096\,x^3\,{\mathrm {e}}^4\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-12288\,x^4\,{\mathrm {e}}^4\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+1536\,x^2\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+1536\,x^2\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3+2304\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+3456\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-192\,x^2\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-384\,x^2\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}{4\,x^4\,{\left (3\,x+2\right )}^2\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}+\frac {\ln \left (x\right )\,\left (8\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+12288\,x^5-4096\,x^3\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-12288\,x^4\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-192\,x\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3+15\,x\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+1536\,x^2\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+3456\,x^3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-384\,x^2\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3\right )}{2\,x^4\,{\left (3\,x+2\right )}^2\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}}{\ln \left (x^2\right )}-\frac {8\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4-x\,\left (192\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-15\,{\mathrm {e}}^{16}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4\right )+x^3\,\left (3456\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-4096\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )+x^2\,\left (1536\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2-384\,{\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3\right )+12288\,x^5-12288\,x^4\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}{16\,x^4\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+48\,x^5\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4+36\,x^6\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}+\frac {128}{{\ln \left (x\right )}^4\,\left (3\,x+2\right )\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}+\frac {8\,\left ({\mathrm {e}}^{12}\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3-128\,x^3+64\,x^2\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )-12\,x\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2\right )}{x^3\,\ln \left (x\right )\,\left (3\,x+2\right )\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^4}+\frac {16\,\left (3\,{\mathrm {e}}^8\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2+32\,x^2-16\,x\,{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )}{x^2\,{\ln \left (x\right )}^2\,\left (3\,x+2\right )\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^3}-\frac {128\,\left (2\,x-{\mathrm {e}}^4\,\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )\right )}{x\,{\ln \left (x\right )}^3\,\left (3\,x+2\right )\,{\left (\ln \left (x^2\right )-2\,\ln \left (x\right )\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(1024*x^4 + 1536*x^5) + log(x^2)*(log(x)^2*(exp(8)*(384*x^2 + 576*x^3) + exp(4)*(512*x^3 + 1536*x
^4)) + log(x)*(exp(4)*(1536*x^3 + 2304*x^4) + 768*x^5) + log(x)^3*(exp(12)*(32*x + 48*x^2) + exp(8)*(384*x^2 +
864*x^3)) + 2048*x^4 + 3072*x^5 + exp(16)*log(x)^5*(15*x + 8) + exp(12)*log(x)^4*(96*x + 192*x^2)) + exp(16)*
log(x)^5*(6*x + 4) + exp(12)*log(x)^4*(64*x + 96*x^2) + exp(8)*log(x)^3*(384*x^2 + 576*x^3) + exp(4)*log(x)^2*
(1024*x^3 + 1536*x^4))/(log(x^2)^2*log(x)^5*(8*x^5 + 24*x^6 + 18*x^7)),x)

[Out]

((4*exp(16)*(log(x^2) - 2*log(x))^4 + 8*exp(16)*(log(x^2) - 2*log(x))^5 + 12288*x^5*(log(x^2) - 2*log(x)) + 16
384*x^4 + 24576*x^5 - 8192*x^3*exp(4)*(log(x^2) - 2*log(x)) - 12288*x^4*exp(4)*(log(x^2) - 2*log(x)) - 128*x*e
xp(12)*(log(x^2) - 2*log(x))^3 - 192*x*exp(12)*(log(x^2) - 2*log(x))^4 + 6*x*exp(16)*(log(x^2) - 2*log(x))^4 +
15*x*exp(16)*(log(x^2) - 2*log(x))^5 - 4096*x^3*exp(4)*(log(x^2) - 2*log(x))^2 - 12288*x^4*exp(4)*(log(x^2) -
2*log(x))^2 + 1536*x^2*exp(8)*(log(x^2) - 2*log(x))^2 + 1536*x^2*exp(8)*(log(x^2) - 2*log(x))^3 + 2304*x^3*ex
p(8)*(log(x^2) - 2*log(x))^2 + 3456*x^3*exp(8)*(log(x^2) - 2*log(x))^3 - 192*x^2*exp(12)*(log(x^2) - 2*log(x))
^3 - 384*x^2*exp(12)*(log(x^2) - 2*log(x))^4)/(4*x^4*(3*x + 2)^2*(log(x^2) - 2*log(x))^4) + (log(x)*(8*exp(16)
*(log(x^2) - 2*log(x))^4 + 12288*x^5 - 4096*x^3*exp(4)*(log(x^2) - 2*log(x)) - 12288*x^4*exp(4)*(log(x^2) - 2*
log(x)) - 192*x*exp(12)*(log(x^2) - 2*log(x))^3 + 15*x*exp(16)*(log(x^2) - 2*log(x))^4 + 1536*x^2*exp(8)*(log(
x^2) - 2*log(x))^2 + 3456*x^3*exp(8)*(log(x^2) - 2*log(x))^2 - 384*x^2*exp(12)*(log(x^2) - 2*log(x))^3))/(2*x^
4*(3*x + 2)^2*(log(x^2) - 2*log(x))^4))/log(x^2) - (8*exp(16)*(log(x^2) - 2*log(x))^4 - x*(192*exp(12)*(log(x^
2) - 2*log(x))^3 - 15*exp(16)*(log(x^2) - 2*log(x))^4) + x^3*(3456*exp(8)*(log(x^2) - 2*log(x))^2 - 4096*exp(4
)*(log(x^2) - 2*log(x))) + x^2*(1536*exp(8)*(log(x^2) - 2*log(x))^2 - 384*exp(12)*(log(x^2) - 2*log(x))^3) + 1
2288*x^5 - 12288*x^4*exp(4)*(log(x^2) - 2*log(x)))/(16*x^4*(log(x^2) - 2*log(x))^4 + 48*x^5*(log(x^2) - 2*log(
x))^4 + 36*x^6*(log(x^2) - 2*log(x))^4) + 128/(log(x)^4*(3*x + 2)*(log(x^2) - 2*log(x))) + (8*(exp(12)*(log(x^
2) - 2*log(x))^3 - 128*x^3 + 64*x^2*exp(4)*(log(x^2) - 2*log(x)) - 12*x*exp(8)*(log(x^2) - 2*log(x))^2))/(x^3*
log(x)*(3*x + 2)*(log(x^2) - 2*log(x))^4) + (16*(3*exp(8)*(log(x^2) - 2*log(x))^2 + 32*x^2 - 16*x*exp(4)*(log(
x^2) - 2*log(x))))/(x^2*log(x)^2*(3*x + 2)*(log(x^2) - 2*log(x))^3) - (128*(2*x - exp(4)*(log(x^2) - 2*log(x))
))/(x*log(x)^3*(3*x + 2)*(log(x^2) - 2*log(x))^2)

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