3.289 \(\int \frac{1}{(A+B \log (\frac{e (a+b x)^2}{(c+d x)^2}))^2} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{1}{\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2},x\right ) \]

[Out]

Unintegrable[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^(-2), x]

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Rubi [A]  time = 0.0139057, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^(-2),x]

[Out]

Defer[Int][(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^(-2), x]

Rubi steps

\begin{align*} \int \frac{1}{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx &=\int \frac{1}{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.213572, size = 0, normalized size = 0. \[ \int \frac{1}{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^(-2),x]

[Out]

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^(-2), x]

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Maple [A]  time = 1.013, size = 0, normalized size = 0. \begin{align*} \int \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{-2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,x)

[Out]

int(1/(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b d x^{2} + a c +{\left (b c + a d\right )} x}{2 \,{\left (2 \,{\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - 2 \,{\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) +{\left (b c - a d\right )} A B +{\left (b c \log \left (e\right ) - a d \log \left (e\right )\right )} B^{2}\right )}} + \int \frac{2 \, b d x + b c + a d}{2 \,{\left (2 \,{\left (b c - a d\right )} B^{2} \log \left (b x + a\right ) - 2 \,{\left (b c - a d\right )} B^{2} \log \left (d x + c\right ) +{\left (b c - a d\right )} A B +{\left (b c \log \left (e\right ) - a d \log \left (e\right )\right )} B^{2}\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="maxima")

[Out]

-1/2*(b*d*x^2 + a*c + (b*c + a*d)*x)/(2*(b*c - a*d)*B^2*log(b*x + a) - 2*(b*c - a*d)*B^2*log(d*x + c) + (b*c -
a*d)*A*B + (b*c*log(e) - a*d*log(e))*B^2) + integrate(1/2*(2*b*d*x + b*c + a*d)/(2*(b*c - a*d)*B^2*log(b*x +
a) - 2*(b*c - a*d)*B^2*log(d*x + c) + (b*c - a*d)*A*B + (b*c*log(e) - a*d*log(e))*B^2), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{B^{2} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \, A B \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) + A^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="fricas")

[Out]

integral(1/(B^2*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2))^2 + 2*A*B*log((b^2*e*x^2 + 2*a*
b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)) + A^2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(A+B*ln(e*(b*x+a)**2/(d*x+c)**2))**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (B \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^2*e/(d*x + c)^2) + A)^(-2), x)