3.88.47 \(\int \frac {50+50 x+(51+103 x+2 x^2) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx\)

Optimal. Leaf size=21 \[ x \left (x+(1+25 (\log (1+x)+\log (4 x \log (x))))^2\right ) \]

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Rubi [F]  time = 2.03, 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 {50+50 x+\left (51+103 x+2 x^2\right ) \log (x)+(1250+1250 x+(1300+2550 x) \log (x)) \log (1+x)+(625+625 x) \log (x) \log ^2(1+x)+(1250+1250 x+(1300+2550 x) \log (x)+(1250+1250 x) \log (x) \log (1+x)) \log (4 x \log (x))+(625+625 x) \log (x) \log ^2(4 x \log (x))}{(1+x) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(50 + 50*x + (51 + 103*x + 2*x^2)*Log[x] + (1250 + 1250*x + (1300 + 2550*x)*Log[x])*Log[1 + x] + (625 + 62
5*x)*Log[x]*Log[1 + x]^2 + (1250 + 1250*x + (1300 + 2550*x)*Log[x] + (1250 + 1250*x)*Log[x]*Log[1 + x])*Log[4*
x*Log[x]] + (625 + 625*x)*Log[x]*Log[4*x*Log[x]]^2)/((1 + x)*Log[x]),x]

[Out]

-2499*x + x^2 - 50*Log[1 + x] + 1300*(1 + x)*Log[1 + x] - 625*Log[1 + x]^2 + 625*(1 + x)*Log[1 + x]^2 + 2550*x
*Log[4*x*Log[x]] - 2550*LogIntegral[x] - 1250*Log[x]*LogIntegral[x] + 1250*Log[4*x*Log[x]]*LogIntegral[x] + 50
*Defer[Int][(1 + 25*Log[1 + x])/Log[x], x] - 1250*Defer[Int][Log[4*x*Log[x]]/(1 + x), x] + 1250*Defer[Int][Log
[1 + x]*Log[4*x*Log[x]], x] + 625*Defer[Int][Log[4*x*Log[x]]^2, x] - 1250*Defer[Int][LogIntegral[x]/(x*Log[x])
, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {50+50 x+51 \log (x)+103 x \log (x)+2 x^2 \log (x)+1250 \log (1+x)+1250 x \log (1+x)+1300 \log (x) \log (1+x)+2550 x \log (x) \log (1+x)+625 \log (x) \log ^2(1+x)+625 x \log (x) \log ^2(1+x)}{(1+x) \log (x)}+\frac {50 (25+25 x+26 \log (x)+51 x \log (x)+25 \log (x) \log (1+x)+25 x \log (x) \log (1+x)) \log (4 x \log (x))}{(1+x) \log (x)}+625 \log ^2(4 x \log (x))\right ) \, dx\\ &=50 \int \frac {(25+25 x+26 \log (x)+51 x \log (x)+25 \log (x) \log (1+x)+25 x \log (x) \log (1+x)) \log (4 x \log (x))}{(1+x) \log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+\int \frac {50+50 x+51 \log (x)+103 x \log (x)+2 x^2 \log (x)+1250 \log (1+x)+1250 x \log (1+x)+1300 \log (x) \log (1+x)+2550 x \log (x) \log (1+x)+625 \log (x) \log ^2(1+x)+625 x \log (x) \log ^2(1+x)}{(1+x) \log (x)} \, dx\\ &=50 \int \frac {(25 (1+x)+\log (x) (26+51 x+25 (1+x) \log (1+x))) \log (4 x \log (x))}{(1+x) \log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+\int \left (\frac {50 (1+25 \log (1+x))}{\log (x)}+\frac {51+103 x+2 x^2+50 (26+51 x) \log (1+x)+625 (1+x) \log ^2(1+x)}{1+x}\right ) \, dx\\ &=50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+50 \int \left (\frac {26 \log (4 x \log (x))}{1+x}+\frac {51 x \log (4 x \log (x))}{1+x}+\frac {25 \log (4 x \log (x))}{(1+x) \log (x)}+\frac {25 x \log (4 x \log (x))}{(1+x) \log (x)}+\frac {25 \log (1+x) \log (4 x \log (x))}{1+x}+\frac {25 x \log (1+x) \log (4 x \log (x))}{1+x}\right ) \, dx+625 \int \log ^2(4 x \log (x)) \, dx+\int \frac {51+103 x+2 x^2+50 (26+51 x) \log (1+x)+625 (1+x) \log ^2(1+x)}{1+x} \, dx\\ &=50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \frac {\log (4 x \log (x))}{(1+x) \log (x)} \, dx+1250 \int \frac {x \log (4 x \log (x))}{(1+x) \log (x)} \, dx+1250 \int \frac {\log (1+x) \log (4 x \log (x))}{1+x} \, dx+1250 \int \frac {x \log (1+x) \log (4 x \log (x))}{1+x} \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \int \frac {x \log (4 x \log (x))}{1+x} \, dx+\int \left (\frac {51+103 x+2 x^2}{1+x}+\frac {50 (26+51 x) \log (1+x)}{1+x}+625 \log ^2(1+x)\right ) \, dx\\ &=50 \int \frac {(26+51 x) \log (1+x)}{1+x} \, dx+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(1+x) \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \frac {\log (4 x \log (x))}{(1+x) \log (x)} \, dx+1250 \int \frac {\log (1+x) \log (4 x \log (x))}{1+x} \, dx+1250 \int \left (\frac {\log (4 x \log (x))}{\log (x)}-\frac {\log (4 x \log (x))}{(1+x) \log (x)}\right ) \, dx+1250 \int \left (\log (1+x) \log (4 x \log (x))-\frac {\log (1+x) \log (4 x \log (x))}{1+x}\right ) \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \int \left (\log (4 x \log (x))-\frac {\log (4 x \log (x))}{1+x}\right ) \, dx+\int \frac {51+103 x+2 x^2}{1+x} \, dx\\ &=50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+50 \operatorname {Subst}\left (\int \frac {(-25+51 x) \log (x)}{x} \, dx,x,1+x\right )+625 \int \log ^2(4 x \log (x)) \, dx+625 \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+x\right )+1250 \int \frac {\log (4 x \log (x))}{\log (x)} \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \int \log (4 x \log (x)) \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx+\int \left (101+2 x-\frac {50}{1+x}\right ) \, dx\\ &=101 x+x^2-50 \log (1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))+1250 \log (4 x \log (x)) \text {li}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \frac {(1+\log (x)) \text {li}(x)}{x \log (x)} \, dx-1250 \operatorname {Subst}(\int \log (x) \, dx,x,1+x)-1250 \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+x\right )+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \left (1+\frac {1}{\log (x)}\right ) \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx+2550 \operatorname {Subst}(\int \log (x) \, dx,x,1+x)\\ &=-3749 x+x^2-50 \log (1+x)+1300 (1+x) \log (1+x)-625 \log ^2(1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))+1250 \log (4 x \log (x)) \text {li}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \left (\frac {\text {li}(x)}{x}+\frac {\text {li}(x)}{x \log (x)}\right ) \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \frac {1}{\log (x)} \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx\\ &=-3749 x+x^2-50 \log (1+x)+1300 (1+x) \log (1+x)-625 \log ^2(1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))-2550 \text {li}(x)+1250 \log (4 x \log (x)) \text {li}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \frac {\text {li}(x)}{x} \, dx-1250 \int \frac {\text {li}(x)}{x \log (x)} \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx\\ &=-2499 x+x^2-50 \log (1+x)+1300 (1+x) \log (1+x)-625 \log ^2(1+x)+625 (1+x) \log ^2(1+x)+2550 x \log (4 x \log (x))-2550 \text {li}(x)-1250 \log (x) \text {li}(x)+1250 \log (4 x \log (x)) \text {li}(x)+50 \int \frac {1+25 \log (1+x)}{\log (x)} \, dx+625 \int \log ^2(4 x \log (x)) \, dx+1250 \int \log (1+x) \log (4 x \log (x)) \, dx-1250 \int \frac {\text {li}(x)}{x \log (x)} \, dx+1300 \int \frac {\log (4 x \log (x))}{1+x} \, dx-2550 \int \frac {\log (4 x \log (x))}{1+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.15, size = 45, normalized size = 2.14 \begin {gather*} x \left (1+x+50 \log (1+x)+625 \log ^2(1+x)+50 (1+25 \log (1+x)) \log (4 x \log (x))+625 \log ^2(4 x \log (x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(50 + 50*x + (51 + 103*x + 2*x^2)*Log[x] + (1250 + 1250*x + (1300 + 2550*x)*Log[x])*Log[1 + x] + (62
5 + 625*x)*Log[x]*Log[1 + x]^2 + (1250 + 1250*x + (1300 + 2550*x)*Log[x] + (1250 + 1250*x)*Log[x]*Log[1 + x])*
Log[4*x*Log[x]] + (625 + 625*x)*Log[x]*Log[4*x*Log[x]]^2)/((1 + x)*Log[x]),x]

[Out]

x*(1 + x + 50*Log[1 + x] + 625*Log[1 + x]^2 + 50*(1 + 25*Log[1 + x])*Log[4*x*Log[x]] + 625*Log[4*x*Log[x]]^2)

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fricas [B]  time = 0.82, size = 49, normalized size = 2.33 \begin {gather*} 625 \, x \log \left (4 \, x \log \relax (x)\right )^{2} + 625 \, x \log \left (x + 1\right )^{2} + x^{2} + 50 \, {\left (25 \, x \log \left (x + 1\right ) + x\right )} \log \left (4 \, x \log \relax (x)\right ) + 50 \, x \log \left (x + 1\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*x+625)*log(x)*log(4*x*log(x))^2+((1250*x+1250)*log(x)*log(x+1)+(2550*x+1300)*log(x)+1250*x+125
0)*log(4*x*log(x))+(625*x+625)*log(x)*log(x+1)^2+((2550*x+1300)*log(x)+1250*x+1250)*log(x+1)+(2*x^2+103*x+51)*
log(x)+50*x+50)/log(x)/(x+1),x, algorithm="fricas")

[Out]

625*x*log(4*x*log(x))^2 + 625*x*log(x + 1)^2 + x^2 + 50*(25*x*log(x + 1) + x)*log(4*x*log(x)) + 50*x*log(x + 1
) + x

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giac [B]  time = 0.52, size = 70, normalized size = 3.33 \begin {gather*} 625 \, x \log \left (x + 1\right )^{2} + 625 \, x \log \relax (x)^{2} + 625 \, x \log \left (4 \, \log \relax (x)\right )^{2} + x^{2} + 50 \, {\left (25 \, x \log \relax (x) + x\right )} \log \left (x + 1\right ) + 50 \, x \log \relax (x) + 50 \, {\left (25 \, x \log \left (x + 1\right ) + 25 \, x \log \relax (x) + x\right )} \log \left (4 \, \log \relax (x)\right ) + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*x+625)*log(x)*log(4*x*log(x))^2+((1250*x+1250)*log(x)*log(x+1)+(2550*x+1300)*log(x)+1250*x+125
0)*log(4*x*log(x))+(625*x+625)*log(x)*log(x+1)^2+((2550*x+1300)*log(x)+1250*x+1250)*log(x+1)+(2*x^2+103*x+51)*
log(x)+50*x+50)/log(x)/(x+1),x, algorithm="giac")

[Out]

625*x*log(x + 1)^2 + 625*x*log(x)^2 + 625*x*log(4*log(x))^2 + x^2 + 50*(25*x*log(x) + x)*log(x + 1) + 50*x*log
(x) + 50*(25*x*log(x + 1) + 25*x*log(x) + x)*log(4*log(x)) + x

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maple [C]  time = 0.60, size = 729, normalized size = 34.71

method	result	size
risch	\(x +2500 x \ln \relax (2) \ln \relax (x )+x^{2}+50 \ln \left (x +1\right ) x +2500 x \ln \relax (2)^{2}+625 x \ln \relax (x )^{2}+100 x \ln \relax (2)+50 x \ln \relax (x )+625 x \ln \left (\ln \relax (x )\right )^{2}+625 x \ln \left (x +1\right )^{2}+\left (1250 \ln \left (x +1\right ) x +625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )+625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )-625 i \pi x \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )-625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}+2500 x \ln \relax (2)+1250 x \ln \relax (x )+50 x \right ) \ln \left (\ln \relax (x )\right )-625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \ln \left (x +1\right )-1250 i x \ln \relax (2) \pi \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )-625 i \pi x \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \ln \relax (x )-625 i \pi x \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right ) \ln \left (x +1\right )+1250 \ln \relax (x ) \ln \left (x +1\right ) x +625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \ln \left (x +1\right )-25 i \pi x \,\mathrm {csgn}\left (i x \ln \relax (x )\right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )+1250 i x \ln \relax (2) \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )+1250 i x \ln \relax (2) \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )+625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right ) \ln \relax (x )+625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right ) \ln \relax (x )+625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right ) \ln \left (x +1\right )+2500 \ln \relax (2) x \ln \left (x +1\right )-\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4} \mathrm {csgn}\left (i x \right )^{2}}{4}-625 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \ln \relax (x )+25 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )-25 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2}}{4}+\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5} \mathrm {csgn}\left (i \ln \relax (x )\right )}{2}+\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{5} \mathrm {csgn}\left (i x \right )}{2}-\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{6}}{4}+25 i \pi x \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )-625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{4} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \ln \relax (x )\right )-1250 i x \ln \relax (2) \pi \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3}-\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2}}{4}+\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i \ln \relax (x )\right )}{2}+\frac {625 x \,\pi ^{2} \mathrm {csgn}\left (i x \ln \relax (x )\right )^{3} \mathrm {csgn}\left (i \ln \relax (x )\right )^{2} \mathrm {csgn}\left (i x \right )}{2}\)	\(729\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((625*x+625)*ln(x)*ln(4*x*ln(x))^2+((1250*x+1250)*ln(x)*ln(x+1)+(2550*x+1300)*ln(x)+1250*x+1250)*ln(4*x*ln
(x))+(625*x+625)*ln(x)*ln(x+1)^2+((2550*x+1300)*ln(x)+1250*x+1250)*ln(x+1)+(2*x^2+103*x+51)*ln(x)+50*x+50)/ln(
x)/(x+1),x,method=_RETURNVERBOSE)

[Out]

x+2500*x*ln(2)*ln(x)+x^2+50*ln(x+1)*x+2500*x*ln(2)^2+625*x*ln(x)^2+100*x*ln(2)+50*x*ln(x)+625*x*ln(ln(x))^2+62
5*x*ln(x+1)^2+625*I*Pi*x*csgn(I*x*ln(x))^2*csgn(I*ln(x))*ln(x+1)-25*I*Pi*x*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln
(x))+1250*I*x*ln(2)*Pi*csgn(I*x*ln(x))^2*csgn(I*x)-625/4*x*Pi^2*csgn(I*x*ln(x))^4*csgn(I*x)^2-625*I*Pi*x*csgn(
I*x*ln(x))^3*ln(x+1)-625*I*Pi*x*csgn(I*x*ln(x))^3*ln(x)+25*I*Pi*x*csgn(I*x*ln(x))^2*csgn(I*x)+25*I*Pi*x*csgn(I
*x*ln(x))^2*csgn(I*ln(x))+(1250*ln(x+1)*x+625*I*Pi*x*csgn(I*x*ln(x))^2*csgn(I*x)+625*I*Pi*x*csgn(I*x*ln(x))^2*
csgn(I*ln(x))-625*I*Pi*x*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))-625*I*Pi*x*csgn(I*x*ln(x))^3+2500*x*ln(2)+125
0*x*ln(x)+50*x)*ln(ln(x))-1250*I*x*ln(2)*Pi*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))-625*I*Pi*x*csgn(I*x*ln(x))
*csgn(I*x)*csgn(I*ln(x))*ln(x)-625*I*Pi*x*csgn(I*x*ln(x))*csgn(I*x)*csgn(I*ln(x))*ln(x+1)-625*x*Pi^2*csgn(I*x*
ln(x))^4*csgn(I*x)*csgn(I*ln(x))+1250*I*x*ln(2)*Pi*csgn(I*x*ln(x))^2*csgn(I*ln(x))+625*I*Pi*x*csgn(I*x*ln(x))^
2*csgn(I*x)*ln(x)+625*I*Pi*x*csgn(I*x*ln(x))^2*csgn(I*ln(x))*ln(x)+625*I*Pi*x*csgn(I*x*ln(x))^2*csgn(I*x)*ln(x
+1)+1250*ln(x)*ln(x+1)*x-25*I*Pi*x*csgn(I*x*ln(x))^3-625/4*x*Pi^2*csgn(I*x*ln(x))^4*csgn(I*ln(x))^2+2500*ln(2)
*x*ln(x+1)-1250*I*x*ln(2)*Pi*csgn(I*x*ln(x))^3+625/2*x*Pi^2*csgn(I*x*ln(x))^5*csgn(I*ln(x))+625/2*x*Pi^2*csgn(
I*x*ln(x))^5*csgn(I*x)-625/4*x*Pi^2*csgn(I*x*ln(x))^2*csgn(I*x)^2*csgn(I*ln(x))^2-625/4*x*Pi^2*csgn(I*x*ln(x))
^6+625/2*x*Pi^2*csgn(I*x*ln(x))^3*csgn(I*x)^2*csgn(I*ln(x))+625/2*x*Pi^2*csgn(I*x*ln(x))^3*csgn(I*ln(x))^2*csg
n(I*x)

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maxima [B]  time = 0.47, size = 98, normalized size = 4.67 \begin {gather*} 625 \, x \log \left (x + 1\right )^{2} + 50 \, x {\left (50 \, \log \relax (2) + 1\right )} \log \relax (x) + 625 \, x \log \relax (x)^{2} + 625 \, x \log \left (\log \relax (x)\right )^{2} + {\left (2500 \, \log \relax (2)^{2} + 100 \, \log \relax (2) + 1\right )} x + x^{2} + 50 \, {\left (x {\left (50 \, \log \relax (2) + 1\right )} + 25 \, x \log \relax (x) + 25 \, x \log \left (\log \relax (x)\right )\right )} \log \left (x + 1\right ) + 50 \, {\left (x {\left (50 \, \log \relax (2) + 1\right )} + 25 \, x \log \relax (x)\right )} \log \left (\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*x+625)*log(x)*log(4*x*log(x))^2+((1250*x+1250)*log(x)*log(x+1)+(2550*x+1300)*log(x)+1250*x+125
0)*log(4*x*log(x))+(625*x+625)*log(x)*log(x+1)^2+((2550*x+1300)*log(x)+1250*x+1250)*log(x+1)+(2*x^2+103*x+51)*
log(x)+50*x+50)/log(x)/(x+1),x, algorithm="maxima")

[Out]

625*x*log(x + 1)^2 + 50*x*(50*log(2) + 1)*log(x) + 625*x*log(x)^2 + 625*x*log(log(x))^2 + (2500*log(2)^2 + 100
*log(2) + 1)*x + x^2 + 50*(x*(50*log(2) + 1) + 25*x*log(x) + 25*x*log(log(x)))*log(x + 1) + 50*(x*(50*log(2) +
1) + 25*x*log(x))*log(log(x))

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mupad [B]  time = 5.99, size = 127, normalized size = 6.05 \begin {gather*} x+\ln \left (4\,x\,\ln \relax (x)\right )\,\left (\frac {\ln \left (x+1\right )\,\left (1250\,x^2+1250\,x\right )}{x+1}-\frac {1250\,x^4+2500\,x^3+1250\,x^2}{x\,{\left (x+1\right )}^2}+\frac {1300\,x^4+2600\,x^3+1300\,x^2}{x\,{\left (x+1\right )}^2}\right )+50\,x\,\ln \left (x+1\right )+625\,x\,{\ln \left (x+1\right )}^2+x^2+\frac {{\ln \left (4\,x\,\ln \relax (x)\right )}^2\,\left (625\,x^3+625\,x^2\right )}{x\,\left (x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((50*x + log(x + 1)*(1250*x + log(x)*(2550*x + 1300) + 1250) + log(x)*(103*x + 2*x^2 + 51) + log(4*x*log(x)
)*(1250*x + log(x)*(2550*x + 1300) + log(x + 1)*log(x)*(1250*x + 1250) + 1250) + log(4*x*log(x))^2*log(x)*(625
*x + 625) + log(x + 1)^2*log(x)*(625*x + 625) + 50)/(log(x)*(x + 1)),x)

[Out]

x + log(4*x*log(x))*((log(x + 1)*(1250*x + 1250*x^2))/(x + 1) - (1250*x^2 + 2500*x^3 + 1250*x^4)/(x*(x + 1)^2)
+ (1300*x^2 + 2600*x^3 + 1300*x^4)/(x*(x + 1)^2)) + 50*x*log(x + 1) + 625*x*log(x + 1)^2 + x^2 + (log(4*x*log
(x))^2*(625*x^2 + 625*x^3))/(x*(x + 1))

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sympy [B]  time = 11.03, size = 56, normalized size = 2.67 \begin {gather*} x^{2} + 625 x \log {\left (4 x \log {\relax (x )} \right )}^{2} + 625 x \log {\left (x + 1 \right )}^{2} + 50 x \log {\left (x + 1 \right )} + x + \left (1250 x \log {\left (x + 1 \right )} + 50 x\right ) \log {\left (4 x \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((625*x+625)*ln(x)*ln(4*x*ln(x))**2+((1250*x+1250)*ln(x)*ln(x+1)+(2550*x+1300)*ln(x)+1250*x+1250)*ln
(4*x*ln(x))+(625*x+625)*ln(x)*ln(x+1)**2+((2550*x+1300)*ln(x)+1250*x+1250)*ln(x+1)+(2*x**2+103*x+51)*ln(x)+50*
x+50)/ln(x)/(x+1),x)

[Out]

x**2 + 625*x*log(4*x*log(x))**2 + 625*x*log(x + 1)**2 + 50*x*log(x + 1) + x + (1250*x*log(x + 1) + 50*x)*log(4
*x*log(x))

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