\(\int \frac {1}{\log ^2(c (a+b x^2)^p)} \, dx\) [114]

Optimal result

Integrand size = 14, antiderivative size = 14 \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\text {Int}\left (\frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx \]

[In]

Int[Log[c*(a + b*x^2)^p]^(-2),x]

[Out]

Defer[Int][Log[c*(a + b*x^2)^p]^(-2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^(-2),x]

[Out]

Integrate[Log[c*(a + b*x^2)^p]^(-2), x]

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00

\[\int \frac {1}{{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {1}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

integral(log((b*x^2 + a)^p*c)^(-2), x)

Sympy [N/A]

Not integrable

Time = 1.89 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}\, dx \]

[In]

integrate(1/ln(c*(b*x**2+a)**p)**2,x)

[Out]

Integral(log(c*(a + b*x**2)**p)**(-2), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 73, normalized size of antiderivative = 5.21 \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {1}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

-1/2*(b*x^2 + a)/(b*p^2*x*log(b*x^2 + a) + b*p*x*log(c)) + integrate(1/2*(b*x^2 - a)/(b*p^2*x^2*log(b*x^2 + a)
+ b*p*x^2*log(c)), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {1}{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}} \,d x } \]

[In]

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

[Out]

integrate(log((b*x^2 + a)^p*c)^(-2), x)

Mupad [N/A]

Not integrable

Time = 1.30 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\log ^2\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2} \,d x \]

[In]

int(1/log(c*(a + b*x^2)^p)^2,x)

[Out]

int(1/log(c*(a + b*x^2)^p)^2, x)