3.97.99 \(\int \frac {16-48 \log (\frac {-32-7 x}{4+x})+48 \log ^2(\frac {-32-7 x}{4+x})-16 \log ^3(\frac {-32-7 x}{4+x})}{128+60 x+7 x^2} \, dx\)

Optimal. Leaf size=16 \[ 261+\left (-1+\log \left (-8+\frac {x}{4+x}\right )\right )^4 \]

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Rubi [F]  time = 13.77, 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 {16-48 \log \left (\frac {-32-7 x}{4+x}\right )+48 \log ^2\left (\frac {-32-7 x}{4+x}\right )-16 \log ^3\left (\frac {-32-7 x}{4+x}\right )}{128+60 x+7 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16 - 48*Log[(-32 - 7*x)/(4 + x)] + 48*Log[(-32 - 7*x)/(4 + x)]^2 - 16*Log[(-32 - 7*x)/(4 + x)]^3)/(128 +
60*x + 7*x^2),x]

[Out]

-6*Log[-14*(4 + x)]^2 - 12*Log[(4 + x)^(-1)]^2*Log[8 + (7*x)/4] + 12*Log[8]*Log[32 + 7*x] - 12*Log[(-7*(4 + x)
)/4]*Log[32 + 7*x]^2 + 12*Log[4 + x]*Log[32 + 7*x]^2 + 4*Log[-56 - 14*x]*(1 - Log[-((32 + 7*x)/(4 + x))])^3 -
12*Log[-56 - 14*x]*Log[-((32 + 7*x)/(4 + x))]^2 - 12*Log[-4/(7*(4 + x))]*Log[-((32 + 7*x)/(4 + x))]^2 + 4*Log[
-56 - 14*x]*Log[-((32 + 7*x)/(4 + x))]^3 + 4*Log[-4/(7*(4 + x))]*Log[-((32 + 7*x)/(4 + x))]^3 + 12*Log[(4 + x)
^(-1)]^2*Log[64 + 14*x] + 12*Log[(-7*(4 + x))/4]*Log[64 + 14*x] - 24*Log[4 + x]*Log[32 + 7*x]*Log[64 + 14*x] -
4*(1 - Log[-((32 + 7*x)/(4 + x))])^3*Log[64 + 14*x] + 24*Log[(-7*(4 + x))/4]*(Log[(4 + x)^(-1)] + Log[32 + 7*
x] - Log[-((32 + 7*x)/(4 + x))])*Log[64 + 14*x] - 6*Log[64 + 14*x]^2 + 12*Log[4 + x]*Log[64 + 14*x]^2 + 12*Log
[-((32 + 7*x)/(4 + x))]*Log[64 + 14*x]^2 - 4*Log[64 + 14*x]^3 + 24*Log[(4 + x)^(-1)]*PolyLog[2, (-7*(4 + x))/4
] - 24*Log[32 + 7*x]*PolyLog[2, 8 + (7*x)/4] + 24*(Log[(4 + x)^(-1)] + Log[32 + 7*x] - Log[-((32 + 7*x)/(4 + x
))])*PolyLog[2, 8 + (7*x)/4] - 24*Log[-((32 + 7*x)/(4 + x))]*PolyLog[2, 1 + 4/(7*(4 + x))] + 12*Log[-((32 + 7*
x)/(4 + x))]^2*PolyLog[2, 1 + 4/(7*(4 + x))] + 24*PolyLog[3, (-7*(4 + x))/4] + 24*PolyLog[3, 8 + (7*x)/4] + 24
*PolyLog[3, 1 + 4/(7*(4 + x))] - 24*Log[-((32 + 7*x)/(4 + x))]*PolyLog[3, 1 + 4/(7*(4 + x))] + 24*PolyLog[4, 1
+ 4/(7*(4 + x))] + 12*Defer[Int][(Log[-((32 + 7*x)/(4 + x))]^2*Log[64 + 14*x])/(4 + x), x] - 84*Defer[Int][(L
og[-((32 + 7*x)/(4 + x))]^2*Log[64 + 14*x])/(32 + 7*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {16 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3}{128+60 x+7 x^2} \, dx\\ &=16 \int \frac {\left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3}{128+60 x+7 x^2} \, dx\\ &=16 \int \left (-\frac {7 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3}{2 (-56-14 x)}-\frac {7 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3}{2 (64+14 x)}\right ) \, dx\\ &=-\left (56 \int \frac {\left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3}{-56-14 x} \, dx\right )-56 \int \frac {\left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3}{64+14 x} \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)-12 \int \frac {(4+x) \left (-\frac {7}{4+x}+\frac {32+7 x}{(4+x)^2}\right ) \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^2}{32+7 x} \, dx+12 \int \frac {(4+x) \left (-\frac {7}{4+x}+\frac {32+7 x}{(4+x)^2}\right ) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^2 \log (64+14 x)}{32+7 x} \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)-12 \int \frac {4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^2}{(4+x) (32+7 x)} \, dx+12 \int \frac {4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^2 \log (64+14 x)}{(4+x) (32+7 x)} \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)-48 \int \frac {\log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^2}{(4+x) (32+7 x)} \, dx+48 \int \frac {\left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^2 \log (64+14 x)}{(4+x) (32+7 x)} \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)-48 \int \left (\frac {\log (-56-14 x)}{(4+x) (32+7 x)}+\frac {2 \log (-56-14 x) \log \left (-\frac {32+7 x}{4+x}\right )}{(-4-x) (32+7 x)}+\frac {\log (-56-14 x) \log ^2\left (-\frac {32+7 x}{4+x}\right )}{(4+x) (32+7 x)}\right ) \, dx+48 \int \left (\frac {\left (-1+\log \left (-\frac {32+7 x}{4+x}\right )\right )^2 \log (64+14 x)}{4 (4+x)}-\frac {7 \left (-1+\log \left (-\frac {32+7 x}{4+x}\right )\right )^2 \log (64+14 x)}{4 (32+7 x)}\right ) \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)+12 \int \frac {\left (-1+\log \left (-\frac {32+7 x}{4+x}\right )\right )^2 \log (64+14 x)}{4+x} \, dx-48 \int \frac {\log (-56-14 x)}{(4+x) (32+7 x)} \, dx-48 \int \frac {\log (-56-14 x) \log ^2\left (-\frac {32+7 x}{4+x}\right )}{(4+x) (32+7 x)} \, dx-84 \int \frac {\left (-1+\log \left (-\frac {32+7 x}{4+x}\right )\right )^2 \log (64+14 x)}{32+7 x} \, dx-96 \int \frac {\log (-56-14 x) \log \left (-\frac {32+7 x}{4+x}\right )}{(-4-x) (32+7 x)} \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-12 \log (-56-14 x) \log ^2\left (-\frac {32+7 x}{4+x}\right )+4 \log (-56-14 x) \log ^3\left (-\frac {32+7 x}{4+x}\right )-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)+\frac {24}{7} \operatorname {Subst}\left (\int -\frac {14 \log (x)}{\left (4-\frac {x}{2}\right ) x} \, dx,x,-56-14 x\right )+12 \int \left (\frac {\log (64+14 x)}{4+x}-\frac {2 \log \left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{4+x}+\frac {\log ^2\left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{4+x}\right ) \, dx+56 \int \frac {\log ^3\left (-\frac {32+7 x}{4+x}\right )}{-56-14 x} \, dx-84 \int \left (\frac {\log (64+14 x)}{32+7 x}-\frac {2 \log \left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{32+7 x}+\frac {\log ^2\left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{32+7 x}\right ) \, dx-168 \int \frac {\log ^2\left (-\frac {32+7 x}{4+x}\right )}{-56-14 x} \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-12 \log (-56-14 x) \log ^2\left (-\frac {32+7 x}{4+x}\right )-12 \log \left (-\frac {4}{7 (4+x)}\right ) \log ^2\left (-\frac {32+7 x}{4+x}\right )+4 \log (-56-14 x) \log ^3\left (-\frac {32+7 x}{4+x}\right )+4 \log \left (-\frac {4}{7 (4+x)}\right ) \log ^3\left (-\frac {32+7 x}{4+x}\right )-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)+12 \int \frac {\log (64+14 x)}{4+x} \, dx+12 \int \frac {\log ^2\left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{4+x} \, dx-24 \int \frac {\log \left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{4+x} \, dx+48 \int \frac {\log \left (-\frac {4}{7 (4+x)}\right ) \log ^2\left (-\frac {32+7 x}{4+x}\right )}{(4+x) (32+7 x)} \, dx-48 \operatorname {Subst}\left (\int \frac {\log (x)}{\left (4-\frac {x}{2}\right ) x} \, dx,x,-56-14 x\right )-84 \int \frac {\log (64+14 x)}{32+7 x} \, dx-84 \int \frac {\log ^2\left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{32+7 x} \, dx-96 \int \frac {\log \left (-\frac {4}{7 (4+x)}\right ) \log \left (-\frac {32+7 x}{4+x}\right )}{(4+x) (32+7 x)} \, dx+168 \int \frac {\log \left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{32+7 x} \, dx\\ &=4 \log (-56-14 x) \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3-12 \log (-56-14 x) \log ^2\left (-\frac {32+7 x}{4+x}\right )-12 \log \left (-\frac {4}{7 (4+x)}\right ) \log ^2\left (-\frac {32+7 x}{4+x}\right )+4 \log (-56-14 x) \log ^3\left (-\frac {32+7 x}{4+x}\right )+4 \log \left (-\frac {4}{7 (4+x)}\right ) \log ^3\left (-\frac {32+7 x}{4+x}\right )+12 \log \left (-\frac {7}{4} (4+x)\right ) \log (64+14 x)-4 \left (1-\log \left (-\frac {32+7 x}{4+x}\right )\right )^3 \log (64+14 x)+12 \log \left (-\frac {32+7 x}{4+x}\right ) \log ^2(64+14 x)-24 \log \left (-\frac {32+7 x}{4+x}\right ) \text {Li}_2\left (1+\frac {4}{7 (4+x)}\right )+12 \log ^2\left (-\frac {32+7 x}{4+x}\right ) \text {Li}_2\left (1+\frac {4}{7 (4+x)}\right )-6 \operatorname {Subst}\left (\int \frac {\log (x)}{4-\frac {x}{2}} \, dx,x,-56-14 x\right )-6 \operatorname {Subst}\left (\int \frac {2 \log (x)}{x} \, dx,x,64+14 x\right )+12 \int \frac {\log ^2\left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{4+x} \, dx+12 \int \frac {\log ^2(64+14 x)}{4+x} \, dx-12 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,-56-14 x\right )-24 \int \frac {\log \left (\frac {1}{4+x}\right ) \log (64+14 x)}{4+x} \, dx-24 \int \frac {\log (32+7 x) \log (64+14 x)}{4+x} \, dx-84 \int \frac {\log ^2\left (-\frac {32+7 x}{4+x}\right ) \log (64+14 x)}{32+7 x} \, dx-84 \int \frac {\log ^2(64+14 x)}{32+7 x} \, dx-96 \int \frac {\text {Li}_2\left (1+\frac {4}{7 (4+x)}\right )}{(4+x) (32+7 x)} \, dx+96 \int \frac {\log \left (-\frac {32+7 x}{4+x}\right ) \text {Li}_2\left (1+\frac {4}{7 (4+x)}\right )}{(4+x) (32+7 x)} \, dx-168 \int \frac {\log \left (-\frac {7}{4} (4+x)\right )}{64+14 x} \, dx-\left (24 \left (-\log \left (\frac {1}{4+x}\right )-\log (32+7 x)+\log \left (-\frac {32+7 x}{4+x}\right )\right )\right ) \int \frac {\log (64+14 x)}{4+x} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.30, size = 77, normalized size = 4.81 \begin {gather*} -16 \left (\frac {1}{4} \log \left (-\frac {32+7 x}{4+x}\right )-\frac {3}{8} \log ^2\left (-\frac {32+7 x}{4+x}\right )+\frac {1}{4} \log ^3\left (-\frac {32+7 x}{4+x}\right )-\frac {1}{16} \log ^4\left (-\frac {32+7 x}{4+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16 - 48*Log[(-32 - 7*x)/(4 + x)] + 48*Log[(-32 - 7*x)/(4 + x)]^2 - 16*Log[(-32 - 7*x)/(4 + x)]^3)/(
128 + 60*x + 7*x^2),x]

[Out]

-16*(Log[-((32 + 7*x)/(4 + x))]/4 - (3*Log[-((32 + 7*x)/(4 + x))]^2)/8 + Log[-((32 + 7*x)/(4 + x))]^3/4 - Log[
-((32 + 7*x)/(4 + x))]^4/16)

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fricas [B]  time = 0.71, size = 65, normalized size = 4.06 \begin {gather*} \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{4} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{3} + 6 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{2} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*log((-7*x-32)/(4+x))^3+48*log((-7*x-32)/(4+x))^2-48*log((-7*x-32)/(4+x))+16)/(7*x^2+60*x+128),x
, algorithm="fricas")

[Out]

log(-(7*x + 32)/(x + 4))^4 - 4*log(-(7*x + 32)/(x + 4))^3 + 6*log(-(7*x + 32)/(x + 4))^2 - 4*log(-(7*x + 32)/(
x + 4))

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giac [B]  time = 0.12, size = 65, normalized size = 4.06 \begin {gather*} \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{4} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{3} + 6 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right )^{2} - 4 \, \log \left (-\frac {7 \, x + 32}{x + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*log((-7*x-32)/(4+x))^3+48*log((-7*x-32)/(4+x))^2-48*log((-7*x-32)/(4+x))+16)/(7*x^2+60*x+128),x
, algorithm="giac")

[Out]

log(-(7*x + 32)/(x + 4))^4 - 4*log(-(7*x + 32)/(x + 4))^3 + 6*log(-(7*x + 32)/(x + 4))^2 - 4*log(-(7*x + 32)/(
x + 4))

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maple [B]  time = 0.15, size = 54, normalized size = 3.38

method	result	size
derivativedivides	\(\ln \left (-7-\frac {4}{4+x}\right )^{4}-4 \ln \left (-7-\frac {4}{4+x}\right )^{3}+6 \ln \left (-7-\frac {4}{4+x}\right )^{2}-4 \ln \left (-7-\frac {4}{4+x}\right )\)	\(54\)
default	\(\ln \left (-7-\frac {4}{4+x}\right )^{4}-4 \ln \left (-7-\frac {4}{4+x}\right )^{3}+6 \ln \left (-7-\frac {4}{4+x}\right )^{2}-4 \ln \left (-7-\frac {4}{4+x}\right )\)	\(54\)
risch	\(4 \ln \left (4+x \right )-4 \ln \left (7 x +32\right )+6 \ln \left (-7-\frac {4}{4+x}\right )^{2}-4 \ln \left (-7-\frac {4}{4+x}\right )^{3}+\ln \left (-7-\frac {4}{4+x}\right )^{4}\)	\(56\)
norman	\(\ln \left (\frac {-7 x -32}{4+x}\right )^{4}+6 \ln \left (\frac {-7 x -32}{4+x}\right )^{2}-4 \ln \left (\frac {-7 x -32}{4+x}\right )^{3}-4 \ln \left (7 x +32\right )+4 \ln \left (4+x \right )\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*ln((-7*x-32)/(4+x))^3+48*ln((-7*x-32)/(4+x))^2-48*ln((-7*x-32)/(4+x))+16)/(7*x^2+60*x+128),x,method=_
RETURNVERBOSE)

[Out]

ln(-7-4/(4+x))^4-4*ln(-7-4/(4+x))^3+6*ln(-7-4/(4+x))^2-4*ln(-7-4/(4+x))

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maxima [B]  time = 0.37, size = 411, normalized size = 25.69 \begin {gather*} -\log \left (7 \, x + 32\right )^{4} + 4 \, \log \left (7 \, x + 32\right )^{3} \log \left (x + 4\right ) - 6 \, \log \left (7 \, x + 32\right )^{2} \log \left (x + 4\right )^{2} + 4 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right )^{3} - \log \left (x + 4\right )^{4} + 4 \, {\left (\log \left (7 \, x + 32\right ) - \log \left (x + 4\right )\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right )^{3} - 4 \, \log \left (7 \, x + 32\right )^{3} + 12 \, \log \left (7 \, x + 32\right )^{2} \log \left (x + 4\right ) - 12 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right )^{2} + 4 \, \log \left (x + 4\right )^{3} - 6 \, {\left (\log \left (7 \, x + 32\right )^{2} - 2 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right ) + \log \left (x + 4\right )^{2}\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right )^{2} - 12 \, {\left (\log \left (7 \, x + 32\right ) - \log \left (x + 4\right )\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right )^{2} - 6 \, \log \left (7 \, x + 32\right )^{2} + 12 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right ) - 6 \, \log \left (x + 4\right )^{2} + 4 \, {\left (\log \left (7 \, x + 32\right )^{3} - 3 \, \log \left (7 \, x + 32\right )^{2} \log \left (x + 4\right ) + 3 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right )^{2} - \log \left (x + 4\right )^{3}\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right ) + 12 \, {\left (\log \left (7 \, x + 32\right )^{2} - 2 \, \log \left (7 \, x + 32\right ) \log \left (x + 4\right ) + \log \left (x + 4\right )^{2}\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right ) + 12 \, {\left (\log \left (7 \, x + 32\right ) - \log \left (x + 4\right )\right )} \log \left (-\frac {7 \, x}{x + 4} - \frac {32}{x + 4}\right ) - 4 \, \log \left (7 \, x + 32\right ) + 4 \, \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*log((-7*x-32)/(4+x))^3+48*log((-7*x-32)/(4+x))^2-48*log((-7*x-32)/(4+x))+16)/(7*x^2+60*x+128),x
, algorithm="maxima")

[Out]

-log(7*x + 32)^4 + 4*log(7*x + 32)^3*log(x + 4) - 6*log(7*x + 32)^2*log(x + 4)^2 + 4*log(7*x + 32)*log(x + 4)^
3 - log(x + 4)^4 + 4*(log(7*x + 32) - log(x + 4))*log(-7*x/(x + 4) - 32/(x + 4))^3 - 4*log(7*x + 32)^3 + 12*lo
g(7*x + 32)^2*log(x + 4) - 12*log(7*x + 32)*log(x + 4)^2 + 4*log(x + 4)^3 - 6*(log(7*x + 32)^2 - 2*log(7*x + 3
2)*log(x + 4) + log(x + 4)^2)*log(-7*x/(x + 4) - 32/(x + 4))^2 - 12*(log(7*x + 32) - log(x + 4))*log(-7*x/(x +
4) - 32/(x + 4))^2 - 6*log(7*x + 32)^2 + 12*log(7*x + 32)*log(x + 4) - 6*log(x + 4)^2 + 4*(log(7*x + 32)^3 -
3*log(7*x + 32)^2*log(x + 4) + 3*log(7*x + 32)*log(x + 4)^2 - log(x + 4)^3)*log(-7*x/(x + 4) - 32/(x + 4)) + 1
2*(log(7*x + 32)^2 - 2*log(7*x + 32)*log(x + 4) + log(x + 4)^2)*log(-7*x/(x + 4) - 32/(x + 4)) + 12*(log(7*x +
32) - log(x + 4))*log(-7*x/(x + 4) - 32/(x + 4)) - 4*log(7*x + 32) + 4*log(x + 4)

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mupad [B]  time = 6.69, size = 61, normalized size = 3.81 \begin {gather*} {\ln \left (-\frac {7\,x+32}{x+4}\right )}^4-4\,{\ln \left (-\frac {7\,x+32}{x+4}\right )}^3+6\,{\ln \left (-\frac {7\,x+32}{x+4}\right )}^2+\mathrm {atan}\left (\frac {x\,7{}\mathrm {i}}{2}+15{}\mathrm {i}\right )\,8{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(48*log(-(7*x + 32)/(x + 4)) - 48*log(-(7*x + 32)/(x + 4))^2 + 16*log(-(7*x + 32)/(x + 4))^3 - 16)/(60*x
+ 7*x^2 + 128),x)

[Out]

atan((x*7i)/2 + 15i)*8i + 6*log(-(7*x + 32)/(x + 4))^2 - 4*log(-(7*x + 32)/(x + 4))^3 + log(-(7*x + 32)/(x + 4
))^4

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sympy [B]  time = 0.19, size = 58, normalized size = 3.62 \begin {gather*} \log {\left (\frac {- 7 x - 32}{x + 4} \right )}^{4} - 4 \log {\left (\frac {- 7 x - 32}{x + 4} \right )}^{3} + 6 \log {\left (\frac {- 7 x - 32}{x + 4} \right )}^{2} + 4 \log {\left (x + 4 \right )} - 4 \log {\left (x + \frac {32}{7} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*ln((-7*x-32)/(4+x))**3+48*ln((-7*x-32)/(4+x))**2-48*ln((-7*x-32)/(4+x))+16)/(7*x**2+60*x+128),x
)

[Out]

log((-7*x - 32)/(x + 4))**4 - 4*log((-7*x - 32)/(x + 4))**3 + 6*log((-7*x - 32)/(x + 4))**2 + 4*log(x + 4) - 4
*log(x + 32/7)

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