Optimal. Leaf size=66 \[ -\frac {d \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {b e \sqrt {1-c^2 x^2}}{c} \]
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Rubi [A] time = 0.07, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {14, 4732, 446, 80, 63, 208} \[ -\frac {d \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )-\frac {b e \sqrt {1-c^2 x^2}}{c} \]
Antiderivative was successfully verified.
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Rule 14
Rule 63
Rule 80
Rule 208
Rule 446
Rule 4732
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \cos ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+(b c) \int \frac {-d+e x^2}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+\frac {1}{2} (b c) \operatorname {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 \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )-\frac {1}{2} (b c d) \operatorname {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 \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+\frac {(b d) \operatorname {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 \left (a+b \cos ^{-1}(c x)\right )}{x}+e x \left (a+b \cos ^{-1}(c x)\right )+b c d \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.08, size = 80, normalized size = 1.21 \[ -\frac {a d}{x}+a e x+b c d \log \left (\sqrt {1-c^2 x^2}+1\right )-\frac {b e \sqrt {1-c^2 x^2}}{c}-b c d \log (x)-\frac {b d \cos ^{-1}(c x)}{x}+b e x \cos ^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 155, normalized size = 2.35 \[ \frac {b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - b c^{2} d x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 2 \, a c e x^{2} - 2 \, \sqrt {-c^{2} x^{2} + 1} b e x - 2 \, a c d - 2 \, {\left (b c d - b c e\right )} x \arctan \left (\frac {\sqrt {-c^{2} x^{2} + 1} c x}{c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c e x^{2} - b c d + {\left (b c d - b c e\right )} x\right )} \arccos \left (c x\right )}{2 \, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.62, size = 867, normalized size = 13.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 79, normalized size = 1.20 \[ c \left (\frac {a \left (c e x -\frac {c d}{x}\right )}{c^{2}}+\frac {b \left (\arccos \left (c x \right ) e c x -\frac {\arccos \left (c x \right ) c d}{x}-e \sqrt {-c^{2} x^{2}+1}+c^{2} d \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )}{c^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 81, normalized size = 1.23 \[ {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {\arccos \left (c x\right )}{x}\right )} b d + a e x + \frac {{\left (c x \arccos \left (c x\right ) - \sqrt {-c^{2} x^{2} + 1}\right )} b e}{c} - \frac {a d}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 71, normalized size = 1.08 \[ b\,c\,d\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {b\,d\,\mathrm {acos}\left (c\,x\right )}{x}-\frac {a\,\left (d-e\,x^2\right )}{x}-\frac {b\,e\,\left (\sqrt {1-c^2\,x^2}-c\,x\,\mathrm {acos}\left (c\,x\right )\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.85, size = 78, normalized size = 1.18 \[ - \frac {a d}{x} + a e x - b c d \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d \operatorname {acos}{\left (c x \right )}}{x} + b e \left (\begin {cases} \frac {\pi x}{2} & \text {for}\: c = 0 \\x \operatorname {acos}{\left (c x \right )} - \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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