Optimal. Leaf size=292 \[ \frac {4 d^3 \left (1-a^2 x^2\right )^{7/2} \left (9 a^2 c+7 d\right )}{441 a^9}-\frac {d^4 \left (1-a^2 x^2\right )^{9/2}}{81 a^9}-\frac {2 d^2 \left (1-a^2 x^2\right )^{5/2} \left (63 a^4 c^2+90 a^2 c d+35 d^2\right )}{525 a^9}+\frac {4 d \left (1-a^2 x^2\right )^{3/2} \left (105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right )}{945 a^9}-\frac {\sqrt {1-a^2 x^2} \left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right )}{315 a^9}+c^4 x \cos ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cos ^{-1}(a x) \]
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Rubi [A] time = 0.33, antiderivative size = 292, 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 = {194, 4666, 12, 1799, 1850} \[ -\frac {2 d^2 \left (1-a^2 x^2\right )^{5/2} \left (63 a^4 c^2+90 a^2 c d+35 d^2\right )}{525 a^9}+\frac {4 d \left (1-a^2 x^2\right )^{3/2} \left (189 a^4 c^2 d+105 a^6 c^3+135 a^2 c d^2+35 d^3\right )}{945 a^9}-\frac {\sqrt {1-a^2 x^2} \left (378 a^4 c^2 d^2+420 a^6 c^3 d+315 a^8 c^4+180 a^2 c d^3+35 d^4\right )}{315 a^9}+\frac {4 d^3 \left (1-a^2 x^2\right )^{7/2} \left (9 a^2 c+7 d\right )}{441 a^9}-\frac {d^4 \left (1-a^2 x^2\right )^{9/2}}{81 a^9}+\frac {6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cos ^{-1}(a x)+c^4 x \cos ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cos ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 12
Rule 194
Rule 1799
Rule 1850
Rule 4666
Rubi steps
\begin {align*} \int \left (c+d x^2\right )^4 \cos ^{-1}(a x) \, dx &=c^4 x \cos ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cos ^{-1}(a x)+a \int \frac {x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right )}{315 \sqrt {1-a^2 x^2}} \, dx\\ &=c^4 x \cos ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cos ^{-1}(a x)+\frac {1}{315} a \int \frac {x \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right )}{\sqrt {1-a^2 x^2}} \, dx\\ &=c^4 x \cos ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cos ^{-1}(a x)+\frac {1}{630} a \operatorname {Subst}\left (\int \frac {315 c^4+420 c^3 d x+378 c^2 d^2 x^2+180 c d^3 x^3+35 d^4 x^4}{\sqrt {1-a^2 x}} \, dx,x,x^2\right )\\ &=c^4 x \cos ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cos ^{-1}(a x)+\frac {1}{630} a \operatorname {Subst}\left (\int \left (\frac {315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4}{a^8 \sqrt {1-a^2 x}}-\frac {4 d \left (105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right ) \sqrt {1-a^2 x}}{a^8}+\frac {6 d^2 \left (63 a^4 c^2+90 a^2 c d+35 d^2\right ) \left (1-a^2 x\right )^{3/2}}{a^8}-\frac {20 d^3 \left (9 a^2 c+7 d\right ) \left (1-a^2 x\right )^{5/2}}{a^8}+\frac {35 d^4 \left (1-a^2 x\right )^{7/2}}{a^8}\right ) \, dx,x,x^2\right )\\ &=-\frac {\left (315 a^8 c^4+420 a^6 c^3 d+378 a^4 c^2 d^2+180 a^2 c d^3+35 d^4\right ) \sqrt {1-a^2 x^2}}{315 a^9}+\frac {4 d \left (105 a^6 c^3+189 a^4 c^2 d+135 a^2 c d^2+35 d^3\right ) \left (1-a^2 x^2\right )^{3/2}}{945 a^9}-\frac {2 d^2 \left (63 a^4 c^2+90 a^2 c d+35 d^2\right ) \left (1-a^2 x^2\right )^{5/2}}{525 a^9}+\frac {4 d^3 \left (9 a^2 c+7 d\right ) \left (1-a^2 x^2\right )^{7/2}}{441 a^9}-\frac {d^4 \left (1-a^2 x^2\right )^{9/2}}{81 a^9}+c^4 x \cos ^{-1}(a x)+\frac {4}{3} c^3 d x^3 \cos ^{-1}(a x)+\frac {6}{5} c^2 d^2 x^5 \cos ^{-1}(a x)+\frac {4}{7} c d^3 x^7 \cos ^{-1}(a x)+\frac {1}{9} d^4 x^9 \cos ^{-1}(a x)\\ \end {align*}
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Mathematica [A] time = 0.20, size = 212, normalized size = 0.73 \[ \frac {1}{315} x \cos ^{-1}(a x) \left (315 c^4+420 c^3 d x^2+378 c^2 d^2 x^4+180 c d^3 x^6+35 d^4 x^8\right )-\frac {\sqrt {1-a^2 x^2} \left (a^8 \left (99225 c^4+44100 c^3 d x^2+23814 c^2 d^2 x^4+8100 c d^3 x^6+1225 d^4 x^8\right )+8 a^6 d \left (11025 c^3+3969 c^2 d x^2+1215 c d^2 x^4+175 d^3 x^6\right )+48 a^4 d^2 \left (1323 c^2+270 c d x^2+35 d^2 x^4\right )+320 a^2 d^3 \left (81 c+7 d x^2\right )+4480 d^4\right )}{99225 a^9} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 239, normalized size = 0.82 \[ \frac {315 \, {\left (35 \, a^{9} d^{4} x^{9} + 180 \, a^{9} c d^{3} x^{7} + 378 \, a^{9} c^{2} d^{2} x^{5} + 420 \, a^{9} c^{3} d x^{3} + 315 \, a^{9} c^{4} x\right )} \arccos \left (a x\right ) - {\left (1225 \, a^{8} d^{4} x^{8} + 99225 \, a^{8} c^{4} + 88200 \, a^{6} c^{3} d + 63504 \, a^{4} c^{2} d^{2} + 100 \, {\left (81 \, a^{8} c d^{3} + 14 \, a^{6} d^{4}\right )} x^{6} + 25920 \, a^{2} c d^{3} + 6 \, {\left (3969 \, a^{8} c^{2} d^{2} + 1620 \, a^{6} c d^{3} + 280 \, a^{4} d^{4}\right )} x^{4} + 4480 \, d^{4} + 4 \, {\left (11025 \, a^{8} c^{3} d + 7938 \, a^{6} c^{2} d^{2} + 3240 \, a^{4} c d^{3} + 560 \, a^{2} d^{4}\right )} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{99225 \, a^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 408, normalized size = 1.40 \[ \frac {1}{9} \, d^{4} x^{9} \arccos \left (a x\right ) + \frac {4}{7} \, c d^{3} x^{7} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} d^{4} x^{8}}{81 \, a} + \frac {6}{5} \, c^{2} d^{2} x^{5} \arccos \left (a x\right ) - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c d^{3} x^{6}}{49 \, a} + \frac {4}{3} \, c^{3} d x^{3} \arccos \left (a x\right ) - \frac {6 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d^{2} x^{4}}{25 \, a} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} d^{4} x^{6}}{567 \, a^{3}} + c^{4} x \arccos \left (a x\right ) - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c^{3} d x^{2}}{9 \, a} - \frac {24 \, \sqrt {-a^{2} x^{2} + 1} c d^{3} x^{4}}{245 \, a^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{4}}{a} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d^{2} x^{2}}{25 \, a^{3}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} d^{4} x^{4}}{945 \, a^{5}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} c^{3} d}{9 \, a^{3}} - \frac {32 \, \sqrt {-a^{2} x^{2} + 1} c d^{3} x^{2}}{245 \, a^{5}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d^{2}}{25 \, a^{5}} - \frac {64 \, \sqrt {-a^{2} x^{2} + 1} d^{4} x^{2}}{2835 \, a^{7}} - \frac {64 \, \sqrt {-a^{2} x^{2} + 1} c d^{3}}{245 \, a^{7}} - \frac {128 \, \sqrt {-a^{2} x^{2} + 1} d^{4}}{2835 \, a^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 393, normalized size = 1.35 \[ \frac {\frac {a \arccos \left (a x \right ) d^{4} x^{9}}{9}+\frac {4 a \arccos \left (a x \right ) c \,d^{3} x^{7}}{7}+\frac {6 a \arccos \left (a x \right ) c^{2} d^{2} x^{5}}{5}+\frac {4 a \arccos \left (a x \right ) c^{3} d \,x^{3}}{3}+\arccos \left (a x \right ) c^{4} a x +\frac {35 d^{4} \left (-\frac {a^{8} x^{8} \sqrt {-a^{2} x^{2}+1}}{9}-\frac {8 a^{6} x^{6} \sqrt {-a^{2} x^{2}+1}}{63}-\frac {16 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{105}-\frac {64 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-a^{2} x^{2}+1}}{315}\right )+180 a^{2} c \,d^{3} \left (-\frac {a^{6} x^{6} \sqrt {-a^{2} x^{2}+1}}{7}-\frac {6 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {8 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-a^{2} x^{2}+1}}{35}\right )+378 a^{4} c^{2} d^{2} \left (-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15}\right )+420 a^{6} c^{3} d \left (-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3}\right )-315 a^{8} c^{4} \sqrt {-a^{2} x^{2}+1}}{315 a^{8}}}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 400, normalized size = 1.37 \[ -\frac {1}{99225} \, {\left (\frac {1225 \, \sqrt {-a^{2} x^{2} + 1} d^{4} x^{8}}{a^{2}} + \frac {8100 \, \sqrt {-a^{2} x^{2} + 1} c d^{3} x^{6}}{a^{2}} + \frac {23814 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d^{2} x^{4}}{a^{2}} + \frac {1400 \, \sqrt {-a^{2} x^{2} + 1} d^{4} x^{6}}{a^{4}} + \frac {44100 \, \sqrt {-a^{2} x^{2} + 1} c^{3} d x^{2}}{a^{2}} + \frac {9720 \, \sqrt {-a^{2} x^{2} + 1} c d^{3} x^{4}}{a^{4}} + \frac {99225 \, \sqrt {-a^{2} x^{2} + 1} c^{4}}{a^{2}} + \frac {31752 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d^{2} x^{2}}{a^{4}} + \frac {1680 \, \sqrt {-a^{2} x^{2} + 1} d^{4} x^{4}}{a^{6}} + \frac {88200 \, \sqrt {-a^{2} x^{2} + 1} c^{3} d}{a^{4}} + \frac {12960 \, \sqrt {-a^{2} x^{2} + 1} c d^{3} x^{2}}{a^{6}} + \frac {63504 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d^{2}}{a^{6}} + \frac {2240 \, \sqrt {-a^{2} x^{2} + 1} d^{4} x^{2}}{a^{8}} + \frac {25920 \, \sqrt {-a^{2} x^{2} + 1} c d^{3}}{a^{8}} + \frac {4480 \, \sqrt {-a^{2} x^{2} + 1} d^{4}}{a^{10}}\right )} a + \frac {1}{315} \, {\left (35 \, d^{4} x^{9} + 180 \, c d^{3} x^{7} + 378 \, c^{2} d^{2} x^{5} + 420 \, c^{3} d x^{3} + 315 \, c^{4} x\right )} \arccos \left (a x\right ) \]
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 \[ \int \mathrm {acos}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.87, size = 502, normalized size = 1.72 \[ \begin {cases} c^{4} x \operatorname {acos}{\left (a x \right )} + \frac {4 c^{3} d x^{3} \operatorname {acos}{\left (a x \right )}}{3} + \frac {6 c^{2} d^{2} x^{5} \operatorname {acos}{\left (a x \right )}}{5} + \frac {4 c d^{3} x^{7} \operatorname {acos}{\left (a x \right )}}{7} + \frac {d^{4} x^{9} \operatorname {acos}{\left (a x \right )}}{9} - \frac {c^{4} \sqrt {- a^{2} x^{2} + 1}}{a} - \frac {4 c^{3} d x^{2} \sqrt {- a^{2} x^{2} + 1}}{9 a} - \frac {6 c^{2} d^{2} x^{4} \sqrt {- a^{2} x^{2} + 1}}{25 a} - \frac {4 c d^{3} x^{6} \sqrt {- a^{2} x^{2} + 1}}{49 a} - \frac {d^{4} x^{8} \sqrt {- a^{2} x^{2} + 1}}{81 a} - \frac {8 c^{3} d \sqrt {- a^{2} x^{2} + 1}}{9 a^{3}} - \frac {8 c^{2} d^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{25 a^{3}} - \frac {24 c d^{3} x^{4} \sqrt {- a^{2} x^{2} + 1}}{245 a^{3}} - \frac {8 d^{4} x^{6} \sqrt {- a^{2} x^{2} + 1}}{567 a^{3}} - \frac {16 c^{2} d^{2} \sqrt {- a^{2} x^{2} + 1}}{25 a^{5}} - \frac {32 c d^{3} x^{2} \sqrt {- a^{2} x^{2} + 1}}{245 a^{5}} - \frac {16 d^{4} x^{4} \sqrt {- a^{2} x^{2} + 1}}{945 a^{5}} - \frac {64 c d^{3} \sqrt {- a^{2} x^{2} + 1}}{245 a^{7}} - \frac {64 d^{4} x^{2} \sqrt {- a^{2} x^{2} + 1}}{2835 a^{7}} - \frac {128 d^{4} \sqrt {- a^{2} x^{2} + 1}}{2835 a^{9}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{4} x + \frac {4 c^{3} d x^{3}}{3} + \frac {6 c^{2} d^{2} x^{5}}{5} + \frac {4 c d^{3} x^{7}}{7} + \frac {d^{4} x^{9}}{9}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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