\(\int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx\) [178]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\text {Int}\left (\frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}},x\right ) \]

[Out]

Unintegrable((a+b*arccsc(c*x))/x^5/(-c^4*x^4+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx \]

[In]

Int[(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]),x]

[Out]

Defer[Int][(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 8.12 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx \]

[In]

Integrate[(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]),x]

[Out]

Integrate[(a + b*ArcCsc[c*x])/(x^5*Sqrt[1 - c^4*x^4]), x]

Maple [N/A] (verified)

Not integrable

Time = 3.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

\[\int \frac {a +b \,\operatorname {arccsc}\left (c x \right )}{x^{5} \sqrt {-c^{4} x^{4}+1}}d x\]

[In]

int((a+b*arccsc(c*x))/x^5/(-c^4*x^4+1)^(1/2),x)

[Out]

int((a+b*arccsc(c*x))/x^5/(-c^4*x^4+1)^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {-c^{4} x^{4} + 1} x^{5}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))/x^5/(-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^9 - x^5), x)

Sympy [N/A]

Not integrable

Time = 83.53 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.38 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\int \frac {a + b \operatorname {acsc}{\left (c x \right )}}{x^{5} \sqrt {- \left (c x - 1\right ) \left (c x + 1\right ) \left (c^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate((a+b*acsc(c*x))/x**5/(-c**4*x**4+1)**(1/2),x)

[Out]

Integral((a + b*acsc(c*x))/(x**5*sqrt(-(c*x - 1)*(c*x + 1)*(c**2*x**2 + 1))), x)

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.31 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {-c^{4} x^{4} + 1} x^{5}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))/x^5/(-c^4*x^4+1)^(1/2),x, algorithm="maxima")

[Out]

-1/8*(c^4*log(sqrt(-c^4*x^4 + 1) + 1) - c^4*log(sqrt(-c^4*x^4 + 1) - 1) + 2*sqrt(-c^4*x^4 + 1)/x^4)*a + b*inte
grate(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^5), x)

Giac [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{\sqrt {-c^{4} x^{4} + 1} x^{5}} \,d x } \]

[In]

integrate((a+b*arccsc(c*x))/x^5/(-c^4*x^4+1)^(1/2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5 \sqrt {1-c^4 x^4}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^5\,\sqrt {1-c^4\,x^4}} \,d x \]

[In]

int((a + b*asin(1/(c*x)))/(x^5*(1 - c^4*x^4)^(1/2)),x)

[Out]

int((a + b*asin(1/(c*x)))/(x^5*(1 - c^4*x^4)^(1/2)), x)