Optimal. Leaf size=123 \[ -\frac {5}{3} b c d^2 \sqrt {1-c^2 x^2}-\frac {1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac {d^2 (a+b \text {ArcSin}(c x))}{x}-2 c^2 d^2 x (a+b \text {ArcSin}(c x))+\frac {1}{3} c^4 d^2 x^3 (a+b \text {ArcSin}(c x))-b c d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {276, 4777, 12,
1265, 911, 1167, 214} \begin {gather*} \frac {1}{3} c^4 d^2 x^3 (a+b \text {ArcSin}(c x))-2 c^2 d^2 x (a+b \text {ArcSin}(c x))-\frac {d^2 (a+b \text {ArcSin}(c x))}{x}-\frac {1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac {5}{3} b c d^2 \sqrt {1-c^2 x^2}-b c d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 214
Rule 276
Rule 911
Rule 1167
Rule 1265
Rule 4777
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^2 \left (-3-6 c^2 x^2+c^4 x^4\right )}{3 x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} \left (b c d^2\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} \left (b c d^2\right ) \text {Subst}\left (\int \frac {-3-6 c^2 x+c^4 x^2}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {-8+4 x^2+x^4}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c}\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \left (-5 c^2-c^2 x^2-\frac {3}{\frac {1}{c^2}-\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c}\\ &=-\frac {5}{3} b c d^2 \sqrt {1-c^2 x^2}-\frac {1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \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 {5}{3} b c d^2 \sqrt {1-c^2 x^2}-\frac {1}{9} b c d^2 \left (1-c^2 x^2\right )^{3/2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}-2 c^2 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} c^4 d^2 x^3 \left (a+b \sin ^{-1}(c x)\right )-b c d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 126, normalized size = 1.02 \begin {gather*} \frac {d^2 \left (-9 a-18 a c^2 x^2+3 a c^4 x^4-16 b c x \sqrt {1-c^2 x^2}+b c^3 x^3 \sqrt {1-c^2 x^2}+3 b \left (-3-6 c^2 x^2+c^4 x^4\right ) \text {ArcSin}(c x)+9 b c x \log (x)-9 b c x \log \left (1+\sqrt {1-c^2 x^2}\right )\right )}{9 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 117, normalized size = 0.95
method | result | size |
derivativedivides | \(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-2 c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{9}-\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(117\) |
default | \(c \left (d^{2} a \left (\frac {c^{3} x^{3}}{3}-2 c x -\frac {1}{c x}\right )+d^{2} b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-2 c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{9}-\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right )\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 160, normalized size = 1.30 \begin {gather*} \frac {1}{3} \, a c^{4} d^{2} x^{3} + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{4} d^{2} - 2 \, a c^{2} d^{2} x - 2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b c d^{2} - {\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^{2} - \frac {a d^{2}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 152, normalized size = 1.24 \begin {gather*} \frac {6 \, a c^{4} d^{2} x^{4} - 36 \, a c^{2} d^{2} x^{2} - 9 \, b c d^{2} x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) + 9 \, b c d^{2} x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) - 18 \, a d^{2} + 6 \, {\left (b c^{4} d^{2} x^{4} - 6 \, b c^{2} d^{2} x^{2} - 3 \, b d^{2}\right )} \arcsin \left (c x\right ) + 2 \, {\left (b c^{3} d^{2} x^{3} - 16 \, b c d^{2} x\right )} \sqrt {-c^{2} x^{2} + 1}}{18 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.61, size = 182, normalized size = 1.48 \begin {gather*} \frac {a c^{4} d^{2} x^{3}}{3} - 2 a c^{2} d^{2} x - \frac {a d^{2}}{x} - \frac {b c^{5} d^{2} \left (\begin {cases} - \frac {x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right )}{3} + \frac {b c^{4} d^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} - 2 b c^{2} d^{2} \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^{2} \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^{2} \operatorname {asin}{\left (c x \right )}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2717 vs.
\(2 (111) = 222\).
time = 6.29, size = 2717, normalized size = 22.09 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \left \{\begin {array}{cl} b\,c^4\,d^2\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )-\frac {a\,d^2\,\left (-c^4\,x^4+6\,c^2\,x^2+3\right )}{3\,x}-2\,b\,c\,d^2\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )-b\,c\,d^2\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-c^2\,x^2}}\right )-\frac {b\,d^2\,\mathrm {asin}\left (c\,x\right )}{x} & \text {\ if\ \ }0<c\\ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^2} \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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