\(\int \frac {1}{a+b \csc ^{-1}(c x)} \, dx\) [34]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\text {Int}\left (\frac {1}{a+b \csc ^{-1}(c x)},x\right ) \]

[Out]

Unintegrable(1/(a+b*arccsc(c*x)),x)

Rubi [N/A]

Not integrable

Time = 0.00 (sec) , antiderivative size = 10, 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 = {} \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\int \frac {1}{a+b \csc ^{-1}(c x)} \, dx \]

[In]

Int[(a + b*ArcCsc[c*x])^(-1),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.56 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\int \frac {1}{a+b \csc ^{-1}(c x)} \, dx \]

[In]

Integrate[(a + b*ArcCsc[c*x])^(-1),x]

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.59 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

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

[In]

int(1/(a+b*arccsc(c*x)),x)

[Out]

int(1/(a+b*arccsc(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\int { \frac {1}{b \operatorname {arccsc}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integral(1/(b*arccsc(c*x) + a), x)

Sympy [N/A]

Not integrable

Time = 0.40 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\int \frac {1}{a + b \operatorname {acsc}{\left (c x \right )}}\, dx \]

[In]

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

[Out]

Integral(1/(a + b*acsc(c*x)), x)

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\int { \frac {1}{b \operatorname {arccsc}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(1/(b*arccsc(c*x) + a), x)

Giac [N/A]

Not integrable

Time = 11.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\int { \frac {1}{b \operatorname {arccsc}\left (c x\right ) + a} \,d x } \]

[In]

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

[Out]

integrate(1/(b*arccsc(c*x) + a), x)

Mupad [N/A]

Not integrable

Time = 0.79 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.60 \[ \int \frac {1}{a+b \csc ^{-1}(c x)} \, dx=\int \frac {1}{a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )} \,d x \]

[In]

int(1/(a + b*asin(1/(c*x))),x)

[Out]

int(1/(a + b*asin(1/(c*x))), x)