3.1.55 \(\int \frac {\sqrt {d+e x} (a+b \csc ^{-1}(c x))}{x^2} \, dx\) [55]

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

[Out]

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

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Rubi [A]
time = 0.05, 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 {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 6.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d+e x} \left (a+b \csc ^{-1}(c x)\right )}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 2.53, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccsc}\left (c x \right )\right ) \sqrt {e x +d}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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