Optimal. Leaf size=219 \[ -\frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3-\frac {2}{125} b^2 c^4 d^2 x^5+\frac {16 b d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{15 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{45 c}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {ArcSin}(c x))^2+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2 \]
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Rubi [A]
time = 0.18, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4743, 4715,
4767, 8, 200} \begin {gather*} \frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))^2+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) (a+b \text {ArcSin}(c x))^2+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} (a+b \text {ArcSin}(c x))}{25 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} (a+b \text {ArcSin}(c x))}{45 c}+\frac {16 b d^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{15 c}+\frac {8}{15} d^2 x (a+b \text {ArcSin}(c x))^2-\frac {2}{125} b^2 c^4 d^2 x^5+\frac {76}{675} b^2 c^2 d^2 x^3-\frac {298}{225} b^2 d^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 200
Rule 4715
Rule 4743
Rule 4767
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} (4 d) \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{5} \left (2 b c d^2\right ) \int x \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{15} \left (8 d^2\right ) \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac {1}{25} \left (2 b^2 d^2\right ) \int \left (1-c^2 x^2\right )^2 \, dx-\frac {1}{15} \left (8 b c d^2\right ) \int x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{25} \left (2 b^2 d^2\right ) \int \left (1-2 c^2 x^2+c^4 x^4\right ) \, dx-\frac {1}{45} \left (8 b^2 d^2\right ) \int \left (1-c^2 x^2\right ) \, dx-\frac {1}{15} \left (16 b c d^2\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {58}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3-\frac {2}{125} b^2 c^4 d^2 x^5+\frac {16 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{15} \left (16 b^2 d^2\right ) \int 1 \, dx\\ &=-\frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3-\frac {2}{125} b^2 c^4 d^2 x^5+\frac {16 b d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{15 c}+\frac {8 b d^2 \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{45 c}+\frac {2 b d^2 \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{25 c}+\frac {8}{15} d^2 x \left (a+b \sin ^{-1}(c x)\right )^2+\frac {4}{15} d^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{5} d^2 x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 193, normalized size = 0.88 \begin {gather*} \frac {d^2 \left (225 a^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+30 a b \sqrt {1-c^2 x^2} \left (149-38 c^2 x^2+9 c^4 x^4\right )-2 b^2 c x \left (2235-190 c^2 x^2+27 c^4 x^4\right )+30 b \left (15 a c x \left (15-10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1-c^2 x^2} \left (149-38 c^2 x^2+9 c^4 x^4\right )\right ) \text {ArcSin}(c x)+225 b^2 c x \left (15-10 c^2 x^2+3 c^4 x^4\right ) \text {ArcSin}(c x)^2\right )}{3375 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 275, normalized size = 1.26
method | result | size |
derivativedivides | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arcsin \left (c x \right )}{3}+c x \arcsin \left (c x \right )+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}+\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}\right )}{c}\) | \(275\) |
default | \(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}-\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {\arcsin \left (c x \right )^{2} \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{15}+\frac {2 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right )^{2} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {2 \left (3 c^{4} x^{4}-10 c^{2} x^{2}+15\right ) c x}{375}-\frac {8 \arcsin \left (c x \right ) \left (c^{2} x^{2}-1\right ) \sqrt {-c^{2} x^{2}+1}}{45}+\frac {8 \left (c^{2} x^{2}-3\right ) c x}{135}-\frac {16 c x}{15}+\frac {16 \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{15}\right )+2 d^{2} a b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-\frac {2 c^{3} x^{3} \arcsin \left (c x \right )}{3}+c x \arcsin \left (c x \right )+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{225}+\frac {149 \sqrt {-c^{2} x^{2}+1}}{225}\right )}{c}\) | \(275\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 465 vs.
\(2 (193) = 386\).
time = 0.54, size = 465, normalized size = 2.12 \begin {gather*} \frac {1}{5} \, b^{2} c^{4} d^{2} x^{5} \arcsin \left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} - \frac {2}{3} \, b^{2} c^{2} d^{2} x^{3} \arcsin \left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{2} + \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c \arcsin \left (c x\right ) - \frac {9 \, c^{4} x^{5} + 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} - \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} - \frac {4}{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 )} a b c^{2} d^{2} - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac {c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} - 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.82, size = 247, normalized size = 1.13 \begin {gather*} \frac {27 \, {\left (25 \, a^{2} - 2 \, b^{2}\right )} c^{5} d^{2} x^{5} - 10 \, {\left (225 \, a^{2} - 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} - 298 \, b^{2}\right )} c d^{2} x + 225 \, {\left (3 \, b^{2} c^{5} d^{2} x^{5} - 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \arcsin \left (c x\right )^{2} + 450 \, {\left (3 \, a b c^{5} d^{2} x^{5} - 10 \, a b c^{3} d^{2} x^{3} + 15 \, a b c d^{2} x\right )} \arcsin \left (c x\right ) + 30 \, {\left (9 \, a b c^{4} d^{2} x^{4} - 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2} + {\left (9 \, b^{2} c^{4} d^{2} x^{4} - 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{3375 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.58, size = 389, normalized size = 1.78 \begin {gather*} \begin {cases} \frac {a^{2} c^{4} d^{2} x^{5}}{5} - \frac {2 a^{2} c^{2} d^{2} x^{3}}{3} + a^{2} d^{2} x + \frac {2 a b c^{4} d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {2 a b c^{3} d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25} - \frac {4 a b c^{2} d^{2} x^{3} \operatorname {asin}{\left (c x \right )}}{3} - \frac {76 a b c d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{225} + 2 a b d^{2} x \operatorname {asin}{\left (c x \right )} + \frac {298 a b d^{2} \sqrt {- c^{2} x^{2} + 1}}{225 c} + \frac {b^{2} c^{4} d^{2} x^{5} \operatorname {asin}^{2}{\left (c x \right )}}{5} - \frac {2 b^{2} c^{4} d^{2} x^{5}}{125} + \frac {2 b^{2} c^{3} d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{25} - \frac {2 b^{2} c^{2} d^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{3} + \frac {76 b^{2} c^{2} d^{2} x^{3}}{675} - \frac {76 b^{2} c d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{225} + b^{2} d^{2} x \operatorname {asin}^{2}{\left (c x \right )} - \frac {298 b^{2} d^{2} x}{225} + \frac {298 b^{2} d^{2} \sqrt {- c^{2} x^{2} + 1} \operatorname {asin}{\left (c x \right )}}{225 c} & \text {for}\: c \neq 0 \\a^{2} d^{2} x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 374, normalized size = 1.71 \begin {gather*} \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} - \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac {1}{5} \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac {2}{5} \, {\left (c^{2} x^{2} - 1\right )}^{2} a b d^{2} x \arcsin \left (c x\right ) - \frac {4}{15} \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x \arcsin \left (c x\right )^{2} - \frac {2}{125} \, {\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d^{2} x - \frac {8}{15} \, {\left (c^{2} x^{2} - 1\right )} a b d^{2} x \arcsin \left (c x\right ) + \frac {8}{15} \, b^{2} d^{2} x \arcsin \left (c x\right )^{2} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{25 \, c} + \frac {272}{3375} \, {\left (c^{2} x^{2} - 1\right )} b^{2} d^{2} x + \frac {16}{15} \, a b d^{2} x \arcsin \left (c x\right ) + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{25 \, c} + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b^{2} d^{2} \arcsin \left (c x\right )}{45 \, c} + a^{2} d^{2} x - \frac {4144}{3375} \, b^{2} d^{2} x + \frac {8 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} a b d^{2}}{45 \, c} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{15 \, c} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1} a b d^{2}}{15 \, c} \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 = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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