Integrand size = 14, antiderivative size = 205 \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=-\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \sqrt {1-a^2 x^2}}{35 a^7}+\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^{3/2}}{105 a^7}-\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^{5/2}}{175 a^7}+\frac {d^3 \left (1-a^2 x^2\right )^{7/2}}{49 a^7}+c^3 x \arccos (a x)+c^2 d x^3 \arccos (a x)+\frac {3}{5} c d^2 x^5 \arccos (a x)+\frac {1}{7} d^3 x^7 \arccos (a x) \]
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Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {200, 4756, 12, 1813, 1864} \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=-\frac {3 d^2 \left (1-a^2 x^2\right )^{5/2} \left (7 a^2 c+5 d\right )}{175 a^7}+\frac {d^3 \left (1-a^2 x^2\right )^{7/2}}{49 a^7}+\frac {d \left (1-a^2 x^2\right )^{3/2} \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{105 a^7}-\frac {\sqrt {1-a^2 x^2} \left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right )}{35 a^7}+c^3 x \arccos (a x)+c^2 d x^3 \arccos (a x)+\frac {3}{5} c d^2 x^5 \arccos (a x)+\frac {1}{7} d^3 x^7 \arccos (a x) \]
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Rule 12
Rule 200
Rule 1813
Rule 1864
Rule 4756
Rubi steps \begin{align*} \text {integral}& = c^3 x \arccos (a x)+c^2 d x^3 \arccos (a x)+\frac {3}{5} c d^2 x^5 \arccos (a x)+\frac {1}{7} d^3 x^7 \arccos (a x)+a \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{35 \sqrt {1-a^2 x^2}} \, dx \\ & = c^3 x \arccos (a x)+c^2 d x^3 \arccos (a x)+\frac {3}{5} c d^2 x^5 \arccos (a x)+\frac {1}{7} d^3 x^7 \arccos (a x)+\frac {1}{35} a \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt {1-a^2 x^2}} \, dx \\ & = c^3 x \arccos (a x)+c^2 d x^3 \arccos (a x)+\frac {3}{5} c d^2 x^5 \arccos (a x)+\frac {1}{7} d^3 x^7 \arccos (a x)+\frac {1}{70} a \text {Subst}\left (\int \frac {35 c^3+35 c^2 d x+21 c d^2 x^2+5 d^3 x^3}{\sqrt {1-a^2 x}} \, dx,x,x^2\right ) \\ & = c^3 x \arccos (a x)+c^2 d x^3 \arccos (a x)+\frac {3}{5} c d^2 x^5 \arccos (a x)+\frac {1}{7} d^3 x^7 \arccos (a x)+\frac {1}{70} a \text {Subst}\left (\int \left (\frac {35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3}{a^6 \sqrt {1-a^2 x}}-\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \sqrt {1-a^2 x}}{a^6}+\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x\right )^{3/2}}{a^6}-\frac {5 d^3 \left (1-a^2 x\right )^{5/2}}{a^6}\right ) \, dx,x,x^2\right ) \\ & = -\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \sqrt {1-a^2 x^2}}{35 a^7}+\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^{3/2}}{105 a^7}-\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^{5/2}}{175 a^7}+\frac {d^3 \left (1-a^2 x^2\right )^{7/2}}{49 a^7}+c^3 x \arccos (a x)+c^2 d x^3 \arccos (a x)+\frac {3}{5} c d^2 x^5 \arccos (a x)+\frac {1}{7} d^3 x^7 \arccos (a x) \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.73 \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=-\frac {\sqrt {1-a^2 x^2} \left (240 d^3+24 a^2 d^2 \left (49 c+5 d x^2\right )+2 a^4 d \left (1225 c^2+294 c d x^2+45 d^2 x^4\right )+a^6 \left (3675 c^3+1225 c^2 d x^2+441 c d^2 x^4+75 d^3 x^6\right )\right )}{3675 a^7}+\left (c^3 x+c^2 d x^3+\frac {3}{5} c d^2 x^5+\frac {d^3 x^7}{7}\right ) \arccos (a x) \]
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Time = 0.36 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {\arccos \left (a x \right ) c^{3} a x +a \arccos \left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \arccos \left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \arccos \left (a x \right ) d^{3} x^{7}}{7}+\frac {5 d^{3} \left (-\frac {x^{6} a^{6} \sqrt {-a^{2} x^{2}+1}}{7}-\frac {6 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {8 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-a^{2} x^{2}+1}}{35}\right )-35 c^{3} a^{6} \sqrt {-a^{2} x^{2}+1}+35 c^{2} a^{4} d \left (-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3}\right )+21 c \,a^{2} d^{2} \left (-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15}\right )}{35 a^{6}}}{a}\) | \(270\) |
default | \(\frac {\arccos \left (a x \right ) c^{3} a x +a \arccos \left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \arccos \left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \arccos \left (a x \right ) d^{3} x^{7}}{7}+\frac {5 d^{3} \left (-\frac {x^{6} a^{6} \sqrt {-a^{2} x^{2}+1}}{7}-\frac {6 a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {8 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-a^{2} x^{2}+1}}{35}\right )-35 c^{3} a^{6} \sqrt {-a^{2} x^{2}+1}+35 c^{2} a^{4} d \left (-\frac {a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3}\right )+21 c \,a^{2} d^{2} \left (-\frac {a^{4} x^{4} \sqrt {-a^{2} x^{2}+1}}{5}-\frac {4 a^{2} x^{2} \sqrt {-a^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15}\right )}{35 a^{6}}}{a}\) | \(270\) |
parts | \(\frac {d^{3} x^{7} \arccos \left (a x \right )}{7}+\frac {3 c \,d^{2} x^{5} \arccos \left (a x \right )}{5}+c^{2} d \,x^{3} \arccos \left (a x \right )+c^{3} x \arccos \left (a x \right )+\frac {a \left (5 d^{3} \left (-\frac {x^{6} \sqrt {-a^{2} x^{2}+1}}{7 a^{2}}+\frac {-\frac {6 x^{4} \sqrt {-a^{2} x^{2}+1}}{35 a^{2}}+\frac {6 \left (-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}\right )}{7 a^{2}}}{a^{2}}\right )-\frac {35 c^{3} \sqrt {-a^{2} x^{2}+1}}{a^{2}}+21 c \,d^{2} \left (-\frac {x^{4} \sqrt {-a^{2} x^{2}+1}}{5 a^{2}}+\frac {-\frac {4 x^{2} \sqrt {-a^{2} x^{2}+1}}{15 a^{2}}-\frac {8 \sqrt {-a^{2} x^{2}+1}}{15 a^{4}}}{a^{2}}\right )+35 c^{2} d \left (-\frac {x^{2} \sqrt {-a^{2} x^{2}+1}}{3 a^{2}}-\frac {2 \sqrt {-a^{2} x^{2}+1}}{3 a^{4}}\right )\right )}{35}\) | \(281\) |
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Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.82 \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=\frac {105 \, {\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \arccos \left (a x\right ) - {\left (75 \, a^{6} d^{3} x^{6} + 3675 \, a^{6} c^{3} + 2450 \, a^{4} c^{2} d + 1176 \, a^{2} c d^{2} + 9 \, {\left (49 \, a^{6} c d^{2} + 10 \, a^{4} d^{3}\right )} x^{4} + 240 \, d^{3} + {\left (1225 \, a^{6} c^{2} d + 588 \, a^{4} c d^{2} + 120 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {-a^{2} x^{2} + 1}}{3675 \, a^{7}} \]
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Time = 0.68 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.59 \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=\begin {cases} c^{3} x \operatorname {acos}{\left (a x \right )} + c^{2} d x^{3} \operatorname {acos}{\left (a x \right )} + \frac {3 c d^{2} x^{5} \operatorname {acos}{\left (a x \right )}}{5} + \frac {d^{3} x^{7} \operatorname {acos}{\left (a x \right )}}{7} - \frac {c^{3} \sqrt {- a^{2} x^{2} + 1}}{a} - \frac {c^{2} d x^{2} \sqrt {- a^{2} x^{2} + 1}}{3 a} - \frac {3 c d^{2} x^{4} \sqrt {- a^{2} x^{2} + 1}}{25 a} - \frac {d^{3} x^{6} \sqrt {- a^{2} x^{2} + 1}}{49 a} - \frac {2 c^{2} d \sqrt {- a^{2} x^{2} + 1}}{3 a^{3}} - \frac {4 c d^{2} x^{2} \sqrt {- a^{2} x^{2} + 1}}{25 a^{3}} - \frac {6 d^{3} x^{4} \sqrt {- a^{2} x^{2} + 1}}{245 a^{3}} - \frac {8 c d^{2} \sqrt {- a^{2} x^{2} + 1}}{25 a^{5}} - \frac {8 d^{3} x^{2} \sqrt {- a^{2} x^{2} + 1}}{245 a^{5}} - \frac {16 d^{3} \sqrt {- a^{2} x^{2} + 1}}{245 a^{7}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{3} x + c^{2} d x^{3} + \frac {3 c d^{2} x^{5}}{5} + \frac {d^{3} x^{7}}{7}\right )}{2} & \text {otherwise} \end {cases} \]
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Time = 0.28 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.30 \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=-\frac {1}{3675} \, {\left (\frac {75 \, \sqrt {-a^{2} x^{2} + 1} d^{3} x^{6}}{a^{2}} + \frac {441 \, \sqrt {-a^{2} x^{2} + 1} c d^{2} x^{4}}{a^{2}} + \frac {1225 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d x^{2}}{a^{2}} + \frac {90 \, \sqrt {-a^{2} x^{2} + 1} d^{3} x^{4}}{a^{4}} + \frac {3675 \, \sqrt {-a^{2} x^{2} + 1} c^{3}}{a^{2}} + \frac {588 \, \sqrt {-a^{2} x^{2} + 1} c d^{2} x^{2}}{a^{4}} + \frac {2450 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d}{a^{4}} + \frac {120 \, \sqrt {-a^{2} x^{2} + 1} d^{3} x^{2}}{a^{6}} + \frac {1176 \, \sqrt {-a^{2} x^{2} + 1} c d^{2}}{a^{6}} + \frac {240 \, \sqrt {-a^{2} x^{2} + 1} d^{3}}{a^{8}}\right )} a + \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \arccos \left (a x\right ) \]
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Time = 0.30 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.32 \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=\frac {1}{7} \, d^{3} x^{7} \arccos \left (a x\right ) + \frac {3}{5} \, c d^{2} x^{5} \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} d^{3} x^{6}}{49 \, a} + c^{2} d x^{3} \arccos \left (a x\right ) - \frac {3 \, \sqrt {-a^{2} x^{2} + 1} c d^{2} x^{4}}{25 \, a} + c^{3} x \arccos \left (a x\right ) - \frac {\sqrt {-a^{2} x^{2} + 1} c^{2} d x^{2}}{3 \, a} - \frac {6 \, \sqrt {-a^{2} x^{2} + 1} d^{3} x^{4}}{245 \, a^{3}} - \frac {\sqrt {-a^{2} x^{2} + 1} c^{3}}{a} - \frac {4 \, \sqrt {-a^{2} x^{2} + 1} c d^{2} x^{2}}{25 \, a^{3}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} c^{2} d}{3 \, a^{3}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} d^{3} x^{2}}{245 \, a^{5}} - \frac {8 \, \sqrt {-a^{2} x^{2} + 1} c d^{2}}{25 \, a^{5}} - \frac {16 \, \sqrt {-a^{2} x^{2} + 1} d^{3}}{245 \, a^{7}} \]
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Timed out. \[ \int \left (c+d x^2\right )^3 \arccos (a x) \, dx=\int \mathrm {acos}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^3 \,d x \]
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