3.124 \(\int \sqrt {d+e x^2} (a+b \csc ^{-1}(c x)) \, dx\)

Optimal. Leaf size=23 \[ \text {Int}\left (\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ),x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 6.67, size = 0, normalized size = 0.00 \[ \int \sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 4.38, size = 0, normalized size = 0.00 \[ \int \left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, {\left (\sqrt {e x^{2} + d} x + \frac {d \operatorname {arsinh}\left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {e}}\right )} a + b \int \sqrt {e x^{2} + d} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(e*x^2 + d)*x + d*arcsinh(e*x/sqrt(d*e))/sqrt(e))*a + b*integrate(sqrt(e*x^2 + d)*arctan2(1, sqrt(c*x
+ 1)*sqrt(c*x - 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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