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

Optimal. Leaf size=24 \[ \text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.08, 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 = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 9.21, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 2.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{e x^{3} + d x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 7.83, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x^{2} \sqrt {e x +d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*b*d^(3/2)*x*integrate(arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/(sqrt(e*x + d)*x^2), x) - a*e*x*log(e*x/(
e*x + 2*sqrt(e*x + d)*sqrt(d) + 2*d)) - 2*sqrt(e*x + d)*a*sqrt(d))/(d^(3/2)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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