3.99.48 \(\int -\frac {60}{-21+20 e^4-15 x} \, dx\)

Optimal. Leaf size=20 \[ 4 \log \left (-1+e^4+\frac {1}{4} \left (-\frac {1}{5}+x\right )-x\right ) \]

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Rubi [A]  time = 0.00, antiderivative size = 13, normalized size of antiderivative = 0.65, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 31} \begin {gather*} 4 \log \left (15 x-20 e^4+21\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-60/(-21 + 20*E^4 - 15*x),x]

[Out]

4*Log[21 - 20*E^4 + 15*x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (60 \int \frac {1}{-21+20 e^4-15 x} \, dx\right )\\ &=4 \log \left (21-20 e^4+15 x\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 13, normalized size = 0.65 \begin {gather*} 4 \log \left (21-20 e^4+15 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-60/(-21 + 20*E^4 - 15*x),x]

[Out]

4*Log[21 - 20*E^4 + 15*x]

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fricas [A]  time = 0.56, size = 12, normalized size = 0.60 \begin {gather*} 4 \, \log \left (15 \, x - 20 \, e^{4} + 21\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-60/(20*exp(4)-15*x-21),x, algorithm="fricas")

[Out]

4*log(15*x - 20*e^4 + 21)

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giac [A]  time = 0.18, size = 13, normalized size = 0.65 \begin {gather*} 4 \, \log \left ({\left | 15 \, x - 20 \, e^{4} + 21 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-60/(20*exp(4)-15*x-21),x, algorithm="giac")

[Out]

4*log(abs(15*x - 20*e^4 + 21))

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maple [A]  time = 0.12, size = 13, normalized size = 0.65

method	result	size
default	\(4 \ln \left (20 \,{\mathrm e}^{4}-15 x -21\right )\)	\(13\)
norman	\(4 \ln \left (20 \,{\mathrm e}^{4}-15 x -21\right )\)	\(13\)
risch	\(4 \ln \left (-20 \,{\mathrm e}^{4}+15 x +21\right )\)	\(13\)
meijerg	\(-\frac {60 \left (-\frac {4 \,{\mathrm e}^{4}}{3}+\frac {7}{5}\right ) \ln \left (1-\frac {15 x}{20 \,{\mathrm e}^{4}-21}\right )}{20 \,{\mathrm e}^{4}-21}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-60/(20*exp(4)-15*x-21),x,method=_RETURNVERBOSE)

[Out]

4*ln(20*exp(4)-15*x-21)

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maxima [A]  time = 0.35, size = 12, normalized size = 0.60 \begin {gather*} 4 \, \log \left (15 \, x - 20 \, e^{4} + 21\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-60/(20*exp(4)-15*x-21),x, algorithm="maxima")

[Out]

4*log(15*x - 20*e^4 + 21)

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mupad [B]  time = 0.09, size = 10, normalized size = 0.50 \begin {gather*} 4\,\ln \left (x-\frac {4\,{\mathrm {e}}^4}{3}+\frac {7}{5}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(60/(15*x - 20*exp(4) + 21),x)

[Out]

4*log(x - (4*exp(4))/3 + 7/5)

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sympy [A]  time = 0.07, size = 12, normalized size = 0.60 \begin {gather*} 4 \log {\left (15 x - 20 e^{4} + 21 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-60/(20*exp(4)-15*x-21),x)

[Out]

4*log(15*x - 20*exp(4) + 21)

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