Integrand size = 23, antiderivative size = 69 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=-b c d \sqrt {1-c^2 x^2}-\frac {d (a+b \arcsin (c x))}{x}-c^2 d x (a+b \arcsin (c x))-b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {14, 4777, 12, 457, 81, 65, 214} \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=c^2 (-d) x (a+b \arcsin (c x))-\frac {d (a+b \arcsin (c x))}{x}-b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-b c d \sqrt {1-c^2 x^2} \]
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Rule 12
Rule 14
Rule 65
Rule 81
Rule 214
Rule 457
Rule 4777
Rubi steps \begin{align*} \text {integral}& = -\frac {d (a+b \arcsin (c x))}{x}-c^2 d x (a+b \arcsin (c x))-(b c) \int \frac {d \left (-1-c^2 x^2\right )}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {d (a+b \arcsin (c x))}{x}-c^2 d x (a+b \arcsin (c x))-(b c d) \int \frac {-1-c^2 x^2}{x \sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {d (a+b \arcsin (c x))}{x}-c^2 d x (a+b \arcsin (c x))-\frac {1}{2} (b c d) \text {Subst}\left (\int \frac {-1-c^2 x}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right ) \\ & = -b c d \sqrt {1-c^2 x^2}-\frac {d (a+b \arcsin (c x))}{x}-c^2 d x (a+b \arcsin (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 ) \\ & = -b c d \sqrt {1-c^2 x^2}-\frac {d (a+b \arcsin (c x))}{x}-c^2 d x (a+b \arcsin (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} \\ & = -b c d \sqrt {1-c^2 x^2}-\frac {d (a+b \arcsin (c x))}{x}-c^2 d x (a+b \arcsin (c x))-b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=-\frac {a d}{x}-a c^2 d x-b c d \sqrt {1-c^2 x^2}-\frac {b d \arcsin (c x)}{x}-b c^2 d x \arcsin (c x)-b c d \text {arctanh}\left (\sqrt {1-c^2 x^2}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.93
method | result | size |
parts | \(-d a \left (c^{2} x +\frac {1}{x}\right )-d b c \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\) | \(64\) |
derivativedivides | \(c \left (-d a \left (c x +\frac {1}{c x}\right )-d b \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(67\) |
default | \(c \left (-d a \left (c x +\frac {1}{c x}\right )-d b \left (c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(67\) |
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.42 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=-\frac {2 \, a c^{2} d x^{2} + b c d x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c d x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 2 \, \sqrt {-c^{2} x^{2} + 1} b c d x + 2 \, a d + 2 \, {\left (b c^{2} d x^{2} + b d\right )} \arcsin \left (c x\right )}{2 \, x} \]
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Time = 1.80 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=- a c^{2} d x - \frac {a d}{x} - b c^{2} d \left (\begin {cases} 0 & \text {for}\: c = 0 \\x \operatorname {asin}{\left (c x \right )} + \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) + b c d \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d \operatorname {asin}{\left (c x \right )}}{x} \]
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Time = 0.29 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=-a c^{2} d x - {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b c d - {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b d - \frac {a d}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 856 vs. \(2 (65) = 130\).
Time = 1.10 (sec) , antiderivative size = 856, normalized size of antiderivative = 12.41 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=-\frac {b c^{5} d x^{4} \arcsin \left (c x\right )}{2 \, {\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} - \frac {a c^{5} d x^{4}}{2 \, {\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} + \frac {b c^{4} d x^{3} \log \left ({\left | c \right |} {\left | x \right |}\right )}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {b c^{4} d x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {b c^{4} d x^{3}}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac {3 \, b c^{3} d x^{2} \arcsin \left (c x\right )}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {3 \, a c^{3} d x^{2}}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {b c^{2} d x \log \left ({\left | c \right |} {\left | x \right |}\right )}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c^{2} d x}{{\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c d \arcsin \left (c x\right )}{2 \, {\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )}} - \frac {a c d}{2 \, {\left (\frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {c x}{\sqrt {-c^{2} x^{2} + 1} + 1}\right )}} \]
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Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.03 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))}{x^2} \, dx=-\frac {a\,d\,\left (c^2\,x^2+1\right )}{x}-b\,c\,d\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )-\frac {b\,d\,\mathrm {asin}\left (c\,x\right )}{x}-b\,c\,d\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right ) \]
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