3.93.87 \(\int \frac {3\ 2^{4 x}-4 \log (2)}{1+2^{4 x} (-5+3 x)} \, dx\)

Optimal. Leaf size=11 \[ \log \left (-5+2^{-4 x}+3 x\right ) \]

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Rubi [F]  time = 0.52, 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 {3\ 2^{4 x}-4 \log (2)}{1+2^{4 x} (-5+3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3*2^(4*x) - 4*Log[2])/(1 + 2^(4*x)*(-5 + 3*x)),x]

[Out]

Log[5 - 3*x] - Log[16]*Defer[Int][(1 - 5*16^x + 3*16^x*x)^(-1), x] - 3*Defer[Int][1/((-5 + 3*x)*(1 - 5*16^x +
3*16^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {3}{-5+3 x}-\frac {3-20 \log (2)+12 x \log (2)}{(-5+3 x) \left (1-5\ 16^x+3\ 16^x x\right )}\right ) \, dx\\ &=\log (5-3 x)-\int \frac {3-20 \log (2)+12 x \log (2)}{(-5+3 x) \left (1-5\ 16^x+3\ 16^x x\right )} \, dx\\ &=\log (5-3 x)-\int \left (\frac {3}{(-5+3 x) \left (1-5\ 16^x+3\ 16^x x\right )}+\frac {\log (16)}{1-5\ 16^x+3\ 16^x x}\right ) \, dx\\ &=\log (5-3 x)-3 \int \frac {1}{(-5+3 x) \left (1-5\ 16^x+3\ 16^x x\right )} \, dx-\log (16) \int \frac {1}{1-5\ 16^x+3\ 16^x x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 3.12, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3\ 2^{4 x}-4 \log (2)}{1+2^{4 x} (-5+3 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(3*2^(4*x) - 4*Log[2])/(1 + 2^(4*x)*(-5 + 3*x)),x]

[Out]

Integrate[(3*2^(4*x) - 4*Log[2])/(1 + 2^(4*x)*(-5 + 3*x)), x]

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fricas [B]  time = 0.65, size = 34, normalized size = 3.09 \begin {gather*} -4 \, x \log \relax (2) + \log \left (3 \, x - 5\right ) + \log \left (\frac {2^{4 \, x} {\left (3 \, x - 5\right )} + 1}{3 \, x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4*x*log(2))-4*log(2))/((3*x-5)*exp(4*x*log(2))+1),x, algorithm="fricas")

[Out]

-4*x*log(2) + log(3*x - 5) + log((2^(4*x)*(3*x - 5) + 1)/(3*x - 5))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {3 \cdot 2^{4 \, x} - 4 \, \log \relax (2)}{2^{4 \, x} {\left (3 \, x - 5\right )} + 1}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4*x*log(2))-4*log(2))/((3*x-5)*exp(4*x*log(2))+1),x, algorithm="giac")

[Out]

integrate((3*2^(4*x) - 4*log(2))/(2^(4*x)*(3*x - 5) + 1), x)

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maple [A]  time = 0.16, size = 25, normalized size = 2.27

method	result	size
risch	\(\ln \left (3 x -5\right )-4 x \ln \relax (2)+\ln \left (16^{x}+\frac {1}{3 x -5}\right )\)	\(25\)
norman	\(-4 x \ln \relax (2)+\ln \left (3 \,{\mathrm e}^{4 x \ln \relax (2)} x -5 \,{\mathrm e}^{4 x \ln \relax (2)}+1\right )\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*exp(4*x*ln(2))-4*ln(2))/((3*x-5)*exp(4*x*ln(2))+1),x,method=_RETURNVERBOSE)

[Out]

ln(3*x-5)-4*x*ln(2)+ln(16^x+1/(3*x-5))

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maxima [B]  time = 0.48, size = 34, normalized size = 3.09 \begin {gather*} -4 \, x \log \relax (2) + \log \left (3 \, x - 5\right ) + \log \left (\frac {2^{4 \, x} {\left (3 \, x - 5\right )} + 1}{3 \, x - 5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4*x*log(2))-4*log(2))/((3*x-5)*exp(4*x*log(2))+1),x, algorithm="maxima")

[Out]

-4*x*log(2) + log(3*x - 5) + log((2^(4*x)*(3*x - 5) + 1)/(3*x - 5))

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mupad [B]  time = 0.18, size = 24, normalized size = 2.18 \begin {gather*} \ln \left (3\,2^{4\,x}\,x-5\,2^{4\,x}+1\right )-4\,x\,\ln \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(2) - 3*exp(4*x*log(2)))/(exp(4*x*log(2))*(3*x - 5) + 1),x)

[Out]

log(3*2^(4*x)*x - 5*2^(4*x) + 1) - 4*x*log(2)

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sympy [A]  time = 0.30, size = 29, normalized size = 2.64 \begin {gather*} - 4 x \log {\relax (2 )} + \log {\left (3 x - 5 \right )} + \log {\left (e^{4 x \log {\relax (2 )}} + \frac {1}{3 x - 5} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3*exp(4*x*ln(2))-4*ln(2))/((3*x-5)*exp(4*x*ln(2))+1),x)

[Out]

-4*x*log(2) + log(3*x - 5) + log(exp(4*x*log(2)) + 1/(3*x - 5))

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