3.1.62 \(\int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx\) [62]

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 = 0.06, 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 {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx \end {gather*}

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 = 6.97, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \csc ^{-1}(c x)}{x^2 \sqrt {d+e x}} \, dx \end {gather*}

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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Maple [A]
time = 3.26, 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 \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))/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(x*e + d)*x^2), x) - a*x*e*log(x*e/(
x*e + 2*sqrt(x*e + d)*sqrt(d) + 2*d)) - 2*sqrt(x*e + d)*a*sqrt(d))/(d^(3/2)*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))/x^2/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

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

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

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

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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