3.2.18 \(\int \frac {\sqrt {d+e x^2} (a+b \sec ^{-1}(c x))}{x^2} \, dx\) [118]

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.06, 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 {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 0.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 0.73, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arcsec}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

(arcsinh(x*e^(1/2)/sqrt(d))*e^(1/2) - sqrt(x^2*e + d)/x)*a + b*integrate(sqrt(x^2*e + d)*arctan(sqrt(c*x + 1)*
sqrt(c*x - 1))/x^2, x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________