Integrand size = 19, antiderivative size = 66 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=-\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))+b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]
[Out]
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 4816, 457, 81, 65, 214} \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))+b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {b e \sqrt {1-c^2 x^2}}{c} \]
[In]
[Out]
Rule 14
Rule 65
Rule 81
Rule 214
Rule 457
Rule 4816
Rubi steps \begin{align*} \text {integral}& = -\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))+(b c) \int \frac {-d+e x^2}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {-d+e x}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))-\frac {1}{2} (b c d) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right ) \\ & = -\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))+\frac {(b d) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{c} \\ & = -\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {d (a+b \arccos (c x))}{x}+e x (a+b \arccos (c x))+b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.21 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=-\frac {a d}{x}+a e x-\frac {b e \sqrt {1-c^2 x^2}}{c}-\frac {b d \arccos (c x)}{x}+b e x \arccos (c x)-b c d \log (x)+b c d \log \left (1+\sqrt {1-c^2 x^2}\right ) \]
[In]
[Out]
Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arccos \left (c x \right ) e c x -\frac {\arccos \left (c x \right ) d c}{x}-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(79\) |
default | \(c \left (\frac {a \left (c e x -\frac {d c}{x}\right )}{c^{2}}+\frac {b \left (\arccos \left (c x \right ) e c x -\frac {\arccos \left (c x \right ) d c}{x}-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right )\) | \(79\) |
parts | \(a \left (e x -\frac {d}{x}\right )+b c \left (\frac {\arccos \left (c x \right ) e x}{c}-\frac {\arccos \left (c x \right ) d}{c x}+\frac {-e \sqrt {-c^{2} x^{2}+1}+d \,c^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{c^{2}}\right )\) | \(79\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (62) = 124\).
Time = 0.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 2.35 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=\frac {b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 2 \, a c e x^{2} - 2 \, \sqrt {-c^{2} x^{2} + 1} b e x - 2 \, a c d - 2 \, {\left (b c d - b c e\right )} x \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \arccos \left (c x\right )}{2 \, c x} \]
[In]
[Out]
Time = 2.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.18 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=- \frac {a d}{x} + a e x - b c d \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d \operatorname {acos}{\left (c x \right )}}{x} + b e \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: c = 0 \\x \operatorname {acos}{\left (c x \right )} - \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.23 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx={\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arccos \left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b e}{c} - \frac {a d}{x} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 859 vs. \(2 (62) = 124\).
Time = 0.57 (sec) , antiderivative size = 859, normalized size of antiderivative = 13.02 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=\text {Too large to display} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right ) (a+b \arccos (c x))}{x^2} \, dx=b\,c\,d\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {b\,d\,\mathrm {acos}\left (c\,x\right )}{x}-\frac {a\,\left (d-e\,x^2\right )}{x}-\frac {b\,e\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c} \]
[In]
[Out]