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

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

[Out]

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

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 18.52, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e x + d} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 8.85, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {e x +d}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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