Optimal. Leaf size=128 \[ b c^3 d^2 \sqrt {1-c^2 x^2}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {ArcSin}(c x))}{x}+c^4 d^2 x (a+b \text {ArcSin}(c x))+\frac {11}{6} b c^3 d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {276, 4777, 12,
1265, 911, 1171, 396, 214} \begin {gather*} c^4 d^2 x (a+b \text {ArcSin}(c x))+\frac {2 c^2 d^2 (a+b \text {ArcSin}(c x))}{x}-\frac {d^2 (a+b \text {ArcSin}(c x))}{3 x^3}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}+b c^3 d^2 \sqrt {1-c^2 x^2}+\frac {11}{6} b c^3 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 396
Rule 911
Rule 1171
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^4} \, dx &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^2 \left (-1+6 c^2 x^2+3 c^4 x^4\right )}{3 x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} \left (b c d^2\right ) \int \frac {-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} \left (b c d^2\right ) \text {Subst}\left (\int \frac {-1+6 c^2 x+3 c^4 x^2}{x^2 \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {8-12 x^2+3 x^4}{\left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )^2} \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c}\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} \left (b c d^2\right ) \text {Subst}\left (\int \frac {-17+6 x^2}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )\\ &=b c^3 d^2 \sqrt {1-c^2 x^2}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} \left (11 b c 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 )\\ &=b c^3 d^2 \sqrt {1-c^2 x^2}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {11}{6} b c^3 d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 136, normalized size = 1.06 \begin {gather*} \frac {d^2 \left (-2 a+12 a c^2 x^2+6 a c^4 x^4-b c x \sqrt {1-c^2 x^2}+6 b c^3 x^3 \sqrt {1-c^2 x^2}+2 b \left (-1+6 c^2 x^2+3 c^4 x^4\right ) \text {ArcSin}(c x)-11 b c^3 x^3 \log (x)+11 b c^3 x^3 \log \left (1+\sqrt {1-c^2 x^2}\right )\right )}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 115, normalized size = 0.90
method | result | size |
derivativedivides | \(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {11 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(115\) |
default | \(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {11 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 170, normalized size = 1.33 \begin {gather*} a c^{4} d^{2} x + {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b c^{3} d^{2} + 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 c^{2} d^{2} - \frac {1}{6} \, {\left ({\left (c^{2} \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\sqrt {-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac {2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d^{2} + \frac {2 \, a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.98, size = 162, normalized size = 1.27 \begin {gather*} \frac {12 \, a c^{4} d^{2} x^{4} + 11 \, b c^{3} d^{2} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - 11 \, b c^{3} d^{2} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 24 \, a c^{2} d^{2} x^{2} - 4 \, a d^{2} + 4 \, {\left (3 \, b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} \arcsin \left (c x\right ) + 2 \, {\left (6 \, b c^{3} d^{2} x^{3} - b c d^{2} x\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 4.38, size = 233, normalized size = 1.82 \begin {gather*} a c^{4} d^{2} x + \frac {2 a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 x^{3}} + b c^{4} 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 ) - 2 b c^{3} 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 {2 b c^{2} d^{2} \operatorname {asin}{\left (c x \right )}}{x} + \frac {b c d^{2} \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} + \frac {c}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} - \frac {1}{2 c x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac {i c^{2} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{2} - \frac {i c \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{3} - \frac {b d^{2} \operatorname {asin}{\left (c x \right )}}{3 x^{3}} \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. 1409 vs.
\(2 (116) = 232\).
time = 58.26, size = 1409, normalized size = 11.01 \begin {gather*} -\frac {b c^{11} d^{2} x^{8} \arcsin \left (c x\right )}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{8}} - \frac {a c^{11} d^{2} x^{8}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{8}} + \frac {b c^{10} d^{2} x^{7}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{7}} + \frac {5 \, b c^{9} d^{2} x^{6} \arcsin \left (c x\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{6}} + \frac {5 \, a c^{9} d^{2} x^{6}}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{6}} - \frac {11 \, b c^{8} d^{2} x^{5} \log \left ({\left | c \right |} {\left | x \right |}\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {11 \, b c^{8} d^{2} x^{5} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} - \frac {23 \, b c^{8} d^{2} x^{5}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {15 \, b c^{7} d^{2} x^{4} \arcsin \left (c x\right )}{4 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} + \frac {15 \, a c^{7} d^{2} x^{4}}{4 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} - \frac {11 \, b c^{6} d^{2} x^{3} \log \left ({\left | c \right |} {\left | x \right |}\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {11 \, b c^{6} d^{2} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {23 \, b c^{6} d^{2} x^{3}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {5 \, b c^{5} d^{2} x^{2} \arcsin \left (c x\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {5 \, a c^{5} d^{2} x^{2}}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {b c^{4} d^{2} x}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c^{3} d^{2} \arcsin \left (c x\right )}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )}} - \frac {a c^{3} d^{2}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )}} \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*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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