3.122 \(\int \frac {\sqrt {d+e x^2} (a+b \csc ^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left (\frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^3},x\right ) \]

[Out]

Unintegrable((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^3,x)

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Rubi [A]  time = 0.11, 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 = {} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^3,x]

[Out]

Defer[Int][(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^3, x]

Rubi steps

\begin {align*} \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx &=\int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx\\ \end {align*}

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Mathematica [A]  time = 4.82, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \csc ^{-1}(c x)\right )}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^3,x]

[Out]

Integrate[(Sqrt[d + e*x^2]*(a + b*ArcCsc[c*x]))/x^3, x]

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fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 1.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^3,x)

[Out]

int((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (\frac {e \operatorname {arsinh}\left (\frac {d}{\sqrt {d e} {\left | x \right |}}\right )}{\sqrt {d}} - \frac {\sqrt {e x^{2} + d} e}{d} + \frac {{\left (e x^{2} + d\right )}^{\frac {3}{2}}}{d x^{2}}\right )} a + b \int \frac {\sqrt {e x^{2} + d} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))*(e*x^2+d)^(1/2)/x^3,x, algorithm="maxima")

[Out]

-1/2*(e*arcsinh(d/(sqrt(d*e)*abs(x)))/sqrt(d) - sqrt(e*x^2 + d)*e/d + (e*x^2 + d)^(3/2)/(d*x^2))*a + b*integra
te(sqrt(e*x^2 + d)*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/x^3, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]

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^3,x)

[Out]

int(((d + e*x^2)^(1/2)*(a + b*asin(1/(c*x))))/x^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))*(e*x**2+d)**(1/2)/x**3,x)

[Out]

Integral((a + b*acsc(c*x))*sqrt(d + e*x**2)/x**3, x)

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