\(\int \sqrt {d+e x^2} (a+b \sec ^{-1}(c x)) \, dx\) [117]

Optimal result

Integrand size = 20, antiderivative size = 20 \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Int}\left (\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ),x\right ) \]

[Out]

Unintegrable((a+b*arcsec(c*x))*(e*x^2+d)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, 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 = {} \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 16.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.41 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \,d x } \]

[In]

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

[Out]

integral(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a), x)

Sympy [N/A]

Not integrable

Time = 47.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asec}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

[In]

integrate((a+b*asec(c*x))*(e*x**2+d)**(1/2),x)

[Out]

Integral((a + b*asec(c*x))*sqrt(d + e*x**2), x)

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int { \sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(e*x^2 + d)*(b*arcsec(c*x) + a), x)

Mupad [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right ) \, dx=\int \sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right ) \,d x \]

[In]

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

[Out]

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