3.7.41 \(\int \frac {1}{(a+b \log (c (d+\frac {e}{f+g x})^p))^2} \, dx\) [641]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx \end {gather*}

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*}

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Mathematica [A]
time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2} \, dx \end {gather*}

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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )\right )^{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}\right )^{2}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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