\(\int \frac {(f x)^m (a+b \sec ^{-1}(c x))}{d+e x^2} \, dx\) [164]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\text {Int}\left (\frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2},x\right ) \]

[Out]

Unintegrable((f*x)^m*(a+b*arcsec(c*x))/(e*x^2+d),x)

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx \]

[In]

Int[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2),x]

[Out]

Defer[Int][((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx \]

[In]

Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2),x]

[Out]

Integrate[((f*x)^m*(a + b*ArcSec[c*x]))/(d + e*x^2), x]

Maple [N/A] (verified)

Not integrable

Time = 2.82 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {\left (f x \right )^{m} \left (a +b \,\operatorname {arcsec}\left (c x \right )\right )}{e \,x^{2}+d}d x\]

[In]

int((f*x)^m*(a+b*arcsec(c*x))/(e*x^2+d),x)

[Out]

int((f*x)^m*(a+b*arcsec(c*x))/(e*x^2+d),x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arcsec(c*x))/(e*x^2+d),x, algorithm="fricas")

[Out]

integral((b*arcsec(c*x) + a)*(f*x)^m/(e*x^2 + d), x)

Sympy [N/A]

Not integrable

Time = 39.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {\left (f x\right )^{m} \left (a + b \operatorname {asec}{\left (c x \right )}\right )}{d + e x^{2}}\, dx \]

[In]

integrate((f*x)**m*(a+b*asec(c*x))/(e*x**2+d),x)

[Out]

Integral((f*x)**m*(a + b*asec(c*x))/(d + e*x**2), x)

Maxima [N/A]

Not integrable

Time = 0.55 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arcsec(c*x))/(e*x^2+d),x, algorithm="maxima")

[Out]

integrate((b*arcsec(c*x) + a)*(f*x)^m/(e*x^2 + d), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\int { \frac {{\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \left (f x\right )^{m}}{e x^{2} + d} \,d x } \]

[In]

integrate((f*x)^m*(a+b*arcsec(c*x))/(e*x^2+d),x, algorithm="giac")

[Out]

integrate((b*arcsec(c*x) + a)*(f*x)^m/(e*x^2 + d), x)

Mupad [N/A]

Not integrable

Time = 0.91 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {(f x)^m \left (a+b \sec ^{-1}(c x)\right )}{d+e x^2} \, dx=\int \frac {{\left (f\,x\right )}^m\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{e\,x^2+d} \,d x \]

[In]

int(((f*x)^m*(a + b*acos(1/(c*x))))/(d + e*x^2),x)

[Out]

int(((f*x)^m*(a + b*acos(1/(c*x))))/(d + e*x^2), x)