3.80.4 \(\int \frac {-126+e^x (-72-36 x)+(-63-36 e^x) \log (-35 x^2-20 e^x x^2)+(-7-4 e^x) \log ^2(-35 x^2-20 e^x x^2)}{(7 x^2+4 e^x x^2) \log ^2(-35 x^2-20 e^x x^2)} \, dx\)

Optimal. Leaf size=28 \[ \frac {1}{x}+\frac {9}{x \log \left (5 \left (x^2-4 \left (2+e^x\right ) x^2\right )\right )} \]

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Rubi [F]  time = 1.67, 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 {-126+e^x (-72-36 x)+\left (-63-36 e^x\right ) \log \left (-35 x^2-20 e^x x^2\right )+\left (-7-4 e^x\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )}{\left (7 x^2+4 e^x x^2\right ) \log ^2\left (-35 x^2-20 e^x x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-126 + E^x*(-72 - 36*x) + (-63 - 36*E^x)*Log[-35*x^2 - 20*E^x*x^2] + (-7 - 4*E^x)*Log[-35*x^2 - 20*E^x*x^
2]^2)/((7*x^2 + 4*E^x*x^2)*Log[-35*x^2 - 20*E^x*x^2]^2),x]

[Out]

x^(-1) - 18*Defer[Int][1/(x^2*Log[-5*(7 + 4*E^x)*x^2]^2), x] - 9*Defer[Int][1/(x*Log[-5*(7 + 4*E^x)*x^2]^2), x
] + 63*Defer[Int][1/((7 + 4*E^x)*x*Log[-5*(7 + 4*E^x)*x^2]^2), x] - 9*Defer[Int][1/(x^2*Log[-5*(7 + 4*E^x)*x^2
]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1-\frac {18 \left (7+2 e^x (2+x)\right )}{\left (7+4 e^x\right ) \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}-\frac {9}{\log \left (-5 \left (7+4 e^x\right ) x^2\right )}}{x^2} \, dx\\ &=\int \left (\frac {63}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}+\frac {-18-9 x-9 \log \left (-5 \left (7+4 e^x\right ) x^2\right )-\log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}{x^2 \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}\right ) \, dx\\ &=63 \int \frac {1}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx+\int \frac {-18-9 x-9 \log \left (-5 \left (7+4 e^x\right ) x^2\right )-\log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}{x^2 \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx\\ &=63 \int \frac {1}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx+\int \frac {-1-\frac {9 (2+x)}{\log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}-\frac {9}{\log \left (-5 \left (7+4 e^x\right ) x^2\right )}}{x^2} \, dx\\ &=63 \int \frac {1}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx+\int \left (-\frac {1}{x^2}-\frac {9 (2+x)}{x^2 \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}-\frac {9}{x^2 \log \left (-5 \left (7+4 e^x\right ) x^2\right )}\right ) \, dx\\ &=\frac {1}{x}-9 \int \frac {2+x}{x^2 \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx-9 \int \frac {1}{x^2 \log \left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx+63 \int \frac {1}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx\\ &=\frac {1}{x}-9 \int \left (\frac {2}{x^2 \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}+\frac {1}{x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )}\right ) \, dx-9 \int \frac {1}{x^2 \log \left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx+63 \int \frac {1}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx\\ &=\frac {1}{x}-9 \int \frac {1}{x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx-9 \int \frac {1}{x^2 \log \left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx-18 \int \frac {1}{x^2 \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx+63 \int \frac {1}{\left (7+4 e^x\right ) x \log ^2\left (-5 \left (7+4 e^x\right ) x^2\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.44, size = 24, normalized size = 0.86 \begin {gather*} \frac {1}{x}+\frac {9}{x \log \left (-5 \left (7+4 e^x\right ) x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-126 + E^x*(-72 - 36*x) + (-63 - 36*E^x)*Log[-35*x^2 - 20*E^x*x^2] + (-7 - 4*E^x)*Log[-35*x^2 - 20*
E^x*x^2]^2)/((7*x^2 + 4*E^x*x^2)*Log[-35*x^2 - 20*E^x*x^2]^2),x]

[Out]

x^(-1) + 9/(x*Log[-5*(7 + 4*E^x)*x^2])

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fricas [A]  time = 0.60, size = 36, normalized size = 1.29 \begin {gather*} \frac {\log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right ) + 9}{x \log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-7)*log(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*log(-20*exp(x)*x^2-35*x^2)+(-36*x-72)*ex
p(x)-126)/(4*exp(x)*x^2+7*x^2)/log(-20*exp(x)*x^2-35*x^2)^2,x, algorithm="fricas")

[Out]

(log(-20*x^2*e^x - 35*x^2) + 9)/(x*log(-20*x^2*e^x - 35*x^2))

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giac [A]  time = 0.33, size = 36, normalized size = 1.29 \begin {gather*} \frac {\log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right ) + 9}{x \log \left (-20 \, x^{2} e^{x} - 35 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-7)*log(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*log(-20*exp(x)*x^2-35*x^2)+(-36*x-72)*ex
p(x)-126)/(4*exp(x)*x^2+7*x^2)/log(-20*exp(x)*x^2-35*x^2)^2,x, algorithm="giac")

[Out]

(log(-20*x^2*e^x - 35*x^2) + 9)/(x*log(-20*x^2*e^x - 35*x^2))

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maple [A]  time = 0.15, size = 37, normalized size = 1.32

method	result	size
norman	\(\frac {9+\ln \left (-20 \,{\mathrm e}^{x} x^{2}-35 x^{2}\right )}{x \ln \left (-20 \,{\mathrm e}^{x} x^{2}-35 x^{2}\right )}\)	\(37\)
risch	\(\frac {1}{x}-\frac {18 i}{x \left (\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right ) \mathrm {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right ) \mathrm {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right ) \mathrm {csgn}\left (i x^{2}\right )+\pi \mathrm {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{3}-2 \pi \mathrm {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{2}+\pi \mathrm {csgn}\left (i x^{2} \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )^{2} \mathrm {csgn}\left (i x^{2}\right )-\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+2 \pi -2 i \ln \relax (5)-4 i \ln \relax (x )-2 i \ln \left ({\mathrm e}^{x}+\frac {7}{4}\right )\right )}\)	\(186\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*exp(x)-7)*ln(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*ln(-20*exp(x)*x^2-35*x^2)+(-36*x-72)*exp(x)-126
)/(4*exp(x)*x^2+7*x^2)/ln(-20*exp(x)*x^2-35*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

(9+ln(-20*exp(x)*x^2-35*x^2))/x/ln(-20*exp(x)*x^2-35*x^2)

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maxima [C]  time = 0.50, size = 44, normalized size = 1.57 \begin {gather*} \frac {i \, \pi + \log \relax (5) + 2 \, \log \relax (x) + \log \left (4 \, e^{x} + 7\right ) + 9}{{\left (i \, \pi + \log \relax (5)\right )} x + 2 \, x \log \relax (x) + x \log \left (4 \, e^{x} + 7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-7)*log(-20*exp(x)*x^2-35*x^2)^2+(-36*exp(x)-63)*log(-20*exp(x)*x^2-35*x^2)+(-36*x-72)*ex
p(x)-126)/(4*exp(x)*x^2+7*x^2)/log(-20*exp(x)*x^2-35*x^2)^2,x, algorithm="maxima")

[Out]

(I*pi + log(5) + 2*log(x) + log(4*e^x + 7) + 9)/((I*pi + log(5))*x + 2*x*log(x) + x*log(4*e^x + 7))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {\left (4\,{\mathrm {e}}^x+7\right )\,{\ln \left (-20\,x^2\,{\mathrm {e}}^x-35\,x^2\right )}^2+\left (36\,{\mathrm {e}}^x+63\right )\,\ln \left (-20\,x^2\,{\mathrm {e}}^x-35\,x^2\right )+{\mathrm {e}}^x\,\left (36\,x+72\right )+126}{{\ln \left (-20\,x^2\,{\mathrm {e}}^x-35\,x^2\right )}^2\,\left (4\,x^2\,{\mathrm {e}}^x+7\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(- 20*x^2*exp(x) - 35*x^2)*(36*exp(x) + 63) + log(- 20*x^2*exp(x) - 35*x^2)^2*(4*exp(x) + 7) + exp(x)
*(36*x + 72) + 126)/(log(- 20*x^2*exp(x) - 35*x^2)^2*(4*x^2*exp(x) + 7*x^2)),x)

[Out]

int(-(log(- 20*x^2*exp(x) - 35*x^2)*(36*exp(x) + 63) + log(- 20*x^2*exp(x) - 35*x^2)^2*(4*exp(x) + 7) + exp(x)
*(36*x + 72) + 126)/(log(- 20*x^2*exp(x) - 35*x^2)^2*(4*x^2*exp(x) + 7*x^2)), x)

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sympy [A]  time = 0.20, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{x} + \frac {9}{x \log {\left (- 20 x^{2} e^{x} - 35 x^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(x)-7)*ln(-20*exp(x)*x**2-35*x**2)**2+(-36*exp(x)-63)*ln(-20*exp(x)*x**2-35*x**2)+(-36*x-72)
*exp(x)-126)/(4*exp(x)*x**2+7*x**2)/ln(-20*exp(x)*x**2-35*x**2)**2,x)

[Out]

1/x + 9/(x*log(-20*x**2*exp(x) - 35*x**2))

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