3.2.77 \(\int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}} \, dx\) [177]

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 = 0.06, 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 \sqrt {1-c^4 x^4}} \, dx \end {gather*}

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 = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \csc ^{-1}(c x)}{x \sqrt {1-c^4 x^4}} \, dx \end {gather*}

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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Maple [A]
time = 180.00, 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 \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/(-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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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/(-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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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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

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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