Optimal. Leaf size=135 \[ -\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \sqrt {1-a^2 x^2}}{15 a^5}+\frac {2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^{3/2}}{45 a^5}-\frac {d^2 \left (1-a^2 x^2\right )^{5/2}}{25 a^5}+c^2 x \text {ArcCos}(a x)+\frac {2}{3} c d x^3 \text {ArcCos}(a x)+\frac {1}{5} d^2 x^5 \text {ArcCos}(a x) \]
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Rubi [A]
time = 0.10, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {200, 4756, 12,
1261, 712} \begin {gather*} \frac {2 d \left (1-a^2 x^2\right )^{3/2} \left (5 a^2 c+3 d\right )}{45 a^5}-\frac {d^2 \left (1-a^2 x^2\right )^{5/2}}{25 a^5}-\frac {\sqrt {1-a^2 x^2} \left (15 a^4 c^2+10 a^2 c d+3 d^2\right )}{15 a^5}+c^2 x \text {ArcCos}(a x)+\frac {2}{3} c d x^3 \text {ArcCos}(a x)+\frac {1}{5} d^2 x^5 \text {ArcCos}(a x) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 200
Rule 712
Rule 1261
Rule 4756
Rubi steps
\begin {align*} \int \left (c+d x^2\right )^2 \cos ^{-1}(a x) \, dx &=c^2 x \cos ^{-1}(a x)+\frac {2}{3} c d x^3 \cos ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cos ^{-1}(a x)+a \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{15 \sqrt {1-a^2 x^2}} \, dx\\ &=c^2 x \cos ^{-1}(a x)+\frac {2}{3} c d x^3 \cos ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cos ^{-1}(a x)+\frac {1}{15} a \int \frac {x \left (15 c^2+10 c d x^2+3 d^2 x^4\right )}{\sqrt {1-a^2 x^2}} \, dx\\ &=c^2 x \cos ^{-1}(a x)+\frac {2}{3} c d x^3 \cos ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cos ^{-1}(a x)+\frac {1}{30} a \text {Subst}\left (\int \frac {15 c^2+10 c d x+3 d^2 x^2}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=c^2 x \cos ^{-1}(a x)+\frac {2}{3} c d x^3 \cos ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cos ^{-1}(a x)+\frac {1}{30} a \text {Subst}\left (\int \left (\frac {15 a^4 c^2+10 a^2 c d+3 d^2}{a^4 \sqrt {1-a^2 x}}-\frac {2 d \left (5 a^2 c+3 d\right ) \sqrt {1-a^2 x}}{a^4}+\frac {3 d^2 \left (1-a^2 x\right )^{3/2}}{a^4}\right ) \, dx,x,x^2\right )\\ &=-\frac {\left (15 a^4 c^2+10 a^2 c d+3 d^2\right ) \sqrt {1-a^2 x^2}}{15 a^5}+\frac {2 d \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )^{3/2}}{45 a^5}-\frac {d^2 \left (1-a^2 x^2\right )^{5/2}}{25 a^5}+c^2 x \cos ^{-1}(a x)+\frac {2}{3} c d x^3 \cos ^{-1}(a x)+\frac {1}{5} d^2 x^5 \cos ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 99, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \left (24 d^2+4 a^2 d \left (25 c+3 d x^2\right )+a^4 \left (225 c^2+50 c d x^2+9 d^2 x^4\right )\right )}{225 a^5}+\left (c^2 x+\frac {2}{3} c d x^3+\frac {d^2 x^5}{5}\right ) \text {ArcCos}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 169, normalized size = 1.25
method | result | size |
derivativedivides | \(\frac {\arccos \left (a x \right ) c^{2} a x +\frac {2 a \arccos \left (a x \right ) c d \,x^{3}}{3}+\frac {a \arccos \left (a x \right ) d^{2} x^{5}}{5}+\frac {-15 c^{2} a^{4} \sqrt {-a^{2} x^{2}+1}+10 c \,a^{2} d \left (-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3}\right )+3 d^{2} \left (-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15}\right )}{15 a^{4}}}{a}\) | \(169\) |
default | \(\frac {\arccos \left (a x \right ) c^{2} a x +\frac {2 a \arccos \left (a x \right ) c d \,x^{3}}{3}+\frac {a \arccos \left (a x \right ) d^{2} x^{5}}{5}+\frac {-15 c^{2} a^{4} \sqrt {-a^{2} x^{2}+1}+10 c \,a^{2} d \left (-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3}\right )+3 d^{2} \left (-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15}\right )}{15 a^{4}}}{a}\) | \(169\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 160, normalized size = 1.19 \begin {gather*} -\frac {1}{225} \, {\left (\frac {9 \, \sqrt {-a^{2} x^{2} + 1} d^{2} x^{4}}{a^{2}} + \frac {50 \, \sqrt {-a^{2} x^{2} + 1} c d x^{2}}{a^{2}} + \frac {225 \, \sqrt {-a^{2} x^{2} + 1} c^{2}}{a^{2}} + \frac {12 \, \sqrt {-a^{2} x^{2} + 1} d^{2} x^{2}}{a^{4}} + \frac {100 \, \sqrt {-a^{2} x^{2} + 1} c d}{a^{4}} + \frac {24 \, \sqrt {-a^{2} x^{2} + 1} d^{2}}{a^{6}}\right )} a + \frac {1}{15} \, {\left (3 \, d^{2} x^{5} + 10 \, c d x^{3} + 15 \, c^{2} x\right )} \arccos \left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.32, size = 110, normalized size = 0.81 \begin {gather*} \frac {15 \, {\left (3 \, a^{5} d^{2} x^{5} + 10 \, a^{5} c d x^{3} + 15 \, a^{5} c^{2} x\right )} \arccos \left (a x\right ) - {\left (9 \, a^{4} d^{2} x^{4} + 225 \, a^{4} c^{2} + 100 \, a^{2} c d + 2 \, {\left (25 \, a^{4} c d + 6 \, a^{2} d^{2}\right )} x^{2} + 24 \, d^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{225 \, a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.34, size = 197, normalized size = 1.46 \begin {gather*} \begin {cases} c^{2} x \operatorname {acos}{\left (a x \right )} + \frac {2 c d x^{3} \operatorname {acos}{\left (a x \right )}}{3} + \frac {d^{2} x^{5} \operatorname {acos}{\left (a x \right )}}{5} - \frac {c^{2} \sqrt {- a^{2} x^{2} + 1}}{a} - \frac {2 c d x^{2} \sqrt {- a^{2} x^{2} + 1}}{9 a} - \frac {d^{2} x^{4} \sqrt {- a^{2} x^{2} + 1}}{25 a} - \frac {4 c d \sqrt {- a^{2} x^{2} + 1}}{9 a^{3}} - \frac {4 d^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{75 a^{3}} - \frac {8 d^{2} \sqrt {- a^{2} x^{2} + 1}}{75 a^{5}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{2} x + \frac {2 c d x^{3}}{3} + \frac {d^{2} x^{5}}{5}\right )}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 160, normalized size = 1.19 \begin {gather*} \frac {1}{5} \, d^{2} x^{5} \arccos \left (a x\right ) + \frac {2}{3} \, c d x^{3} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} d^{2} x^{4}}{25 \, a} + c^{2} x \arccos \left (a x\right ) - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c d x^{2}}{9 \, a} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2}}{a} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} d^{2} x^{2}}{75 \, a^{3}} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c d}{9 \, a^{3}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} d^{2}}{75 \, a^{5}} \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 \mathrm {acos}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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