3.641 \(\int \frac{1}{(a+b \log (c (d+\frac{e}{f+g x})^p))^2} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{\left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2},x\right ) \]

[Out]

Unintegrable[(a + b*Log[c*(d + e/(f + g*x))^p])^(-2), x]

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Rubi [A]  time = 0.0057474, 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 = {} \[ \int \frac{1}{\left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*(d + e/(f + g*x))^p])^(-2),x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.858967, size = 0, normalized size = 0. \[ \int \frac{1}{\left (a+b \log \left (c \left (d+\frac{e}{f+g x}\right )^p\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^(-2),x]

[Out]

Integrate[(a + b*Log[c*(d + e/(f + g*x))^p])^(-2), x]

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Maple [A]  time = 0.135, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\ln \left ( c \left ( d+{\frac{e}{gx+f}} \right ) ^{p} \right ) \right ) ^{-2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*ln(c*(d+e/(g*x+f))^p))^2,x)

[Out]

int(1/(a+b*ln(c*(d+e/(g*x+f))^p))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*g^2*x^2 + d*f^2 + e*f + (2*d*f*g + e*g)*x)/(b^2*e*g*p*log((d*g*x + d*f + e)^p) - b^2*e*g*p*log((g*x + f)^p)
+ b^2*e*g*p*log(c) + a*b*e*g*p) - integrate((2*d*g*x + 2*d*f + e)/(b^2*e*p*log((d*g*x + d*f + e)^p) - b^2*e*p
*log((g*x + f)^p) + b^2*e*p*log(c) + a*b*e*p), 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 (c \left (\frac{d g x + d f + e}{g x + f}\right )^{p}\right )^{2} + 2 \, a b \log \left (c \left (\frac{d g x + d f + e}{g x + f}\right )^{p}\right ) + a^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(b^2*log(c*((d*g*x + d*f + e)/(g*x + f))^p)^2 + 2*a*b*log(c*((d*g*x + d*f + e)/(g*x + f))^p) + 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(c*(d+e/(g*x+f))**p))**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 (c{\left (d + \frac{e}{g x + f}\right )}^{p}\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*log(c*(d + e/(g*x + f))^p) + a)^(-2), x)