\(\int \frac {1}{x^2 \log ^3(c (a+b x^2)^p)} \, dx\) [122]

Optimal result

Integrand size = 18, antiderivative size = 18 \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\text {Int}\left (\frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )},x\right ) \]

[Out]

Unintegrable(1/x^2/ln(c*(b*x^2+a)^p)^3,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 18, 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 = {} \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx \]

[In]

Int[1/(x^2*Log[c*(a + b*x^2)^p]^3),x]

[Out]

Defer[Int][1/(x^2*Log[c*(a + b*x^2)^p]^3), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx \]

[In]

Integrate[1/(x^2*Log[c*(a + b*x^2)^p]^3),x]

[Out]

Integrate[1/(x^2*Log[c*(a + b*x^2)^p]^3), x]

Maple [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}d x\]

[In]

int(1/x^2/ln(c*(b*x^2+a)^p)^3,x)

[Out]

int(1/x^2/ln(c*(b*x^2+a)^p)^3,x)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {1}{x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}} \,d x } \]

[In]

integrate(1/x^2/log(c*(b*x^2+a)^p)^3,x, algorithm="fricas")

[Out]

integral(1/(x^2*log((b*x^2 + a)^p*c)^3), x)

Sympy [N/A]

Not integrable

Time = 5.81 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}\, dx \]

[In]

integrate(1/x**2/ln(c*(b*x**2+a)**p)**3,x)

[Out]

Integral(1/(x**2*log(c*(a + b*x**2)**p)**3), x)

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 10.39 \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {1}{x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}} \,d x } \]

[In]

integrate(1/x^2/log(c*(b*x^2+a)^p)^3,x, algorithm="maxima")

[Out]

-1/8*(b^2*(2*p - log(c))*x^4 + 2*a*b*(p - 2*log(c))*x^2 - 3*a^2*log(c) - (b^2*p*x^4 + 4*a*b*p*x^2 + 3*a^2*p)*l
og(b*x^2 + a))/(b^2*p^4*x^5*log(b*x^2 + a)^2 + 2*b^2*p^3*x^5*log(b*x^2 + a)*log(c) + b^2*p^2*x^5*log(c)^2) + i
ntegrate(1/8*(b^2*x^4 + 12*a*b*x^2 + 15*a^2)/(b^2*p^3*x^6*log(b*x^2 + a) + b^2*p^2*x^6*log(c)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int { \frac {1}{x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}} \,d x } \]

[In]

integrate(1/x^2/log(c*(b*x^2+a)^p)^3,x, algorithm="giac")

[Out]

integrate(1/(x^2*log((b*x^2 + a)^p*c)^3), x)

Mupad [N/A]

Not integrable

Time = 1.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^2 \log ^3\left (c \left (a+b x^2\right )^p\right )} \, dx=\int \frac {1}{x^2\,{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3} \,d x \]

[In]

int(1/(x^2*log(c*(a + b*x^2)^p)^3),x)

[Out]

int(1/(x^2*log(c*(a + b*x^2)^p)^3), x)