Optimal. Leaf size=32 \[ -\frac{a+b \csc ^{-1}(c x)}{x}-b c \sqrt{1-\frac{1}{c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0197825, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {5221, 261} \[ -\frac{a+b \csc ^{-1}(c x)}{x}-b c \sqrt{1-\frac{1}{c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5221
Rule 261
Rubi steps
\begin{align*} \int \frac{a+b \csc ^{-1}(c x)}{x^2} \, dx &=-\frac{a+b \csc ^{-1}(c x)}{x}-\frac{b \int \frac{1}{\sqrt{1-\frac{1}{c^2 x^2}} x^3} \, dx}{c}\\ &=-b c \sqrt{1-\frac{1}{c^2 x^2}}-\frac{a+b \csc ^{-1}(c x)}{x}\\ \end{align*}
Mathematica [A] time = 0.0287449, size = 41, normalized size = 1.28 \[ -\frac{a}{x}-b c \sqrt{\frac{c^2 x^2-1}{c^2 x^2}}-\frac{b \csc ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.171, size = 63, normalized size = 2. \begin{align*} c \left ( -{\frac{a}{cx}}+b \left ( -{\frac{{\rm arccsc} \left (cx\right )}{cx}}-{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}-1}{{c}^{2}{x}^{2}}}}}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.982462, size = 45, normalized size = 1.41 \begin{align*} -{\left (c \sqrt{-\frac{1}{c^{2} x^{2}} + 1} + \frac{\operatorname{arccsc}\left (c x\right )}{x}\right )} b - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.20002, size = 62, normalized size = 1.94 \begin{align*} -\frac{b \operatorname{arccsc}\left (c x\right ) + \sqrt{c^{2} x^{2} - 1} b + a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 2.6338, size = 37, normalized size = 1.16 \begin{align*} \begin{cases} - \frac{a}{x} - b c \sqrt{1 - \frac{1}{c^{2} x^{2}}} - \frac{b \operatorname{acsc}{\left (c x \right )}}{x} & \text{for}\: c \neq 0 \\- \frac{a + \tilde{\infty } b}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{arccsc}\left (c x\right ) + a}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]