3.2.34 \(\int \frac {a+b \sec ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx\) [134]

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

[Out]

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

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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 {a+b \sec ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \sec ^{-1}(c x)}{x \sqrt {d+e x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.94, size = 0, normalized size = 0.00 \[\int \frac {a +b \,\mathrm {arcsec}\left (c x \right )}{x \sqrt {e \,x^{2}+d}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

integral(sqrt(x^2*e + d)*(b*arcsec(c*x) + a)/(x^3*e + d*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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