3.177 \(\int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}} \, dx\)

Optimal. Leaf size=29 \[ \text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}},x\right ) \]

[Out]

Unintegrable((a+b*arccsc(c*x))/x/(-c^4*x^4+1)^(1/2),x)

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Rubi [A]  time = 0.09, 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 = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*ArcCsc[c*x])/(x*Sqrt[1 - c^4*x^4]),x]

[Out]

Defer[Int][(a + b*ArcCsc[c*x])/(x*Sqrt[1 - c^4*x^4]), x]

Rubi steps

\begin {align*} \int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}} \, dx &=\int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 0, normalized size = 0.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[1 - c^4*x^4]),x]

[Out]

Integrate[(a + b*ArcCsc[c*x])/(x*Sqrt[1 - c^4*x^4]), x]

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fricas [A]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{4} x^{4} + 1} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{c^{4} x^{5} - x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x/(-c^4*x^4+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^4*x^4 + 1)*(b*arccsc(c*x) + a)/(c^4*x^5 - x), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {-c^{4} x^{4} + 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x/(-c^4*x^4+1)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arccsc(c*x) + a)/(sqrt(-c^4*x^4 + 1)*x), x)

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maple [A]  time = 6.20, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsc}\left (c x \right )}{x \sqrt {-c^{4} x^{4}+1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arccsc(c*x))/x/(-c^4*x^4+1)^(1/2),x)

[Out]

int((a+b*arccsc(c*x))/x/(-c^4*x^4+1)^(1/2),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, a {\left (\log \left (\sqrt {-c^{4} x^{4} + 1} + 1\right ) - \log \left (\sqrt {-c^{4} x^{4} + 1} - 1\right )\right )} + b \int \frac {\arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right )}{\sqrt {c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {-c x + 1} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arccsc(c*x))/x/(-c^4*x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/4*a*(log(sqrt(-c^4*x^4 + 1) + 1) - log(sqrt(-c^4*x^4 + 1) - 1)) + b*integrate(arctan2(1, sqrt(c*x + 1)*sqrt
(c*x - 1))/(sqrt(c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(-c*x + 1)*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x\,\sqrt {1-c^4\,x^4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))/(x*(1 - c^4*x^4)^(1/2)),x)

[Out]

int((a + b*asin(1/(c*x)))/(x*(1 - c^4*x^4)^(1/2)), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x \sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))/x/(-c**4*x**4+1)**(1/2),x)

[Out]

Integral((a + b*acsc(c*x))/(x*sqrt(-(c*x - 1)*(c*x + 1)*(c**2*x**2 + 1))), x)

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