\(\int (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))) \, dx\) [8738]

Optimal result

Integrand size = 24, antiderivative size = 21 \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=x (3-x (-24+x+\log (i \pi -\log (2)))) \]

[Out]

x*(3-(x-24+ln(-ln(2)+I*Pi))*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6} \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=-x^3+x^2 (24-\log (-\log (2)-i \pi ))+3 x \]

[In]

Int[3 + 48*x - 3*x^2 - 2*x*Log[(-I)*Pi - Log[2]],x]

[Out]

3*x - x^3 + x^2*(24 - Log[(-I)*Pi - Log[2]])

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (3-3 x^2+x (48-2 \log (-i \pi -\log (2)))\right ) \, dx \\ & = 3 x-x^3+x^2 (24-\log (-i \pi -\log (2))) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=3 x+24 x^2-x^3-x^2 \log (-i \pi -\log (2)) \]

[In]

Integrate[3 + 48*x - 3*x^2 - 2*x*Log[(-I)*Pi - Log[2]],x]

[Out]

3*x + 24*x^2 - x^3 - x^2*Log[(-I)*Pi - Log[2]]

Maple [A] (verified)

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14

method	result	size
gosper	\(-x \left (x \ln \left (-\ln \left (2\right )-i \pi \right )+x^{2}-24 x -3\right )\)	\(24\)
norman	\(\left (-\ln \left (-\ln \left (2\right )-i \pi \right )+24\right ) x^{2}+3 x -x^{3}\)	\(28\)
default	\(-x^{2} \ln \left (-\ln \left (2\right )-i \pi \right )-x^{3}+24 x^{2}+3 x\)	\(30\)
risch	\(-x^{2} \ln \left (-\ln \left (2\right )-i \pi \right )-x^{3}+24 x^{2}+3 x\)	\(30\)
parallelrisch	\(-x^{2} \ln \left (-\ln \left (2\right )-i \pi \right )-x^{3}+24 x^{2}+3 x\)	\(30\)
parts	\(-x^{2} \ln \left (-\ln \left (2\right )-i \pi \right )-x^{3}+24 x^{2}+3 x\)	\(30\)

[In]

int(-2*x*ln(-ln(2)-I*Pi)-3*x^2+48*x+3,x,method=_RETURNVERBOSE)

[Out]

-x*(x*ln(-ln(2)-I*Pi)+x^2-24*x-3)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=-x^{3} - x^{2} \log \left (-i \, \pi - \log \left (2\right )\right ) + 24 \, x^{2} + 3 \, x \]

[In]

integrate(-2*x*log(-log(2)-I*pi)-3*x^2+48*x+3,x, algorithm="fricas")

[Out]

-x^3 - x^2*log(-I*pi - log(2)) + 24*x^2 + 3*x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=- x^{3} + x^{2} \cdot \left (24 - \log {\left (- \log {\left (2 \right )} - i \pi \right )}\right ) + 3 x \]

[In]

integrate(-2*x*ln(-ln(2)-I*pi)-3*x**2+48*x+3,x)

[Out]

-x**3 + x**2*(24 - log(-log(2) - I*pi)) + 3*x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=-x^{3} - x^{2} \log \left (-i \, \pi - \log \left (2\right )\right ) + 24 \, x^{2} + 3 \, x \]

[In]

integrate(-2*x*log(-log(2)-I*pi)-3*x^2+48*x+3,x, algorithm="maxima")

[Out]

-x^3 - x^2*log(-I*pi - log(2)) + 24*x^2 + 3*x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=-x^{3} - x^{2} \log \left (-i \, \pi - \log \left (2\right )\right ) + 24 \, x^{2} + 3 \, x \]

[In]

integrate(-2*x*log(-log(2)-I*pi)-3*x^2+48*x+3,x, algorithm="giac")

[Out]

-x^3 - x^2*log(-I*pi - log(2)) + 24*x^2 + 3*x

Mupad [B] (verification not implemented)

Time = 14.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int \left (3+48 x-3 x^2-2 x \log (-i \pi -\log (2))\right ) \, dx=-x^3+\left (24-\ln \left (-\ln \left (2\right )-\Pi \,1{}\mathrm {i}\right )\right )\,x^2+3\,x \]

[In]

int(48*x - 2*x*log(- Pi*1i - log(2)) - 3*x^2 + 3,x)

[Out]

3*x - x^3 - x^2*(log(- Pi*1i - log(2)) - 24)