Optimal. Leaf size=66 \[ \frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d}} \]
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Rubi [A] time = 0.10, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {191, 4666, 12, 444, 63, 217, 203} \[ \frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 63
Rule 191
Rule 203
Rule 217
Rule 444
Rule 4666
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}+a \int \frac {x}{c \sqrt {1-a^2 x^2} \sqrt {c+d x^2}} \, dx\\ &=\frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \int \frac {x}{\sqrt {1-a^2 x^2} \sqrt {c+d x^2}} \, dx}{c}\\ &=\frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-a^2 x} \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {c+\frac {d}{a^2}-\frac {d x^2}{a^2}}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{a c}\\ &=\frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{1+\frac {d x^2}{a^2}} \, dx,x,\frac {\sqrt {1-a^2 x^2}}{\sqrt {c+d x^2}}\right )}{a c}\\ &=\frac {x \cos ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d}}\\ \end {align*}
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Mathematica [C] time = 0.08, size = 68, normalized size = 1.03 \[ \frac {x \left (a x \sqrt {\frac {d x^2}{c}+1} F_1\left (1;\frac {1}{2},\frac {1}{2};2;a^2 x^2,-\frac {d x^2}{c}\right )+2 \cos ^{-1}(a x)\right )}{2 c \sqrt {c+d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.52, size = 276, normalized size = 4.18 \[ \left [\frac {4 \, \sqrt {d x^{2} + c} d x \arccos \left (a x\right ) - {\left (d x^{2} + c\right )} \sqrt {-d} \log \left (8 \, a^{4} d^{2} x^{4} + a^{4} c^{2} - 6 \, a^{2} c d + 8 \, {\left (a^{4} c d - a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a^{3} d x^{2} + a^{3} c - a d\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {d x^{2} + c} \sqrt {-d} + d^{2}\right )}{4 \, {\left (c d^{2} x^{2} + c^{2} d\right )}}, \frac {2 \, \sqrt {d x^{2} + c} d x \arccos \left (a x\right ) - {\left (d x^{2} + c\right )} \sqrt {d} \arctan \left (\frac {{\left (2 \, a^{2} d x^{2} + a^{2} c - d\right )} \sqrt {-a^{2} x^{2} + 1} \sqrt {d x^{2} + c} \sqrt {d}}{2 \, {\left (a^{3} d^{2} x^{4} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (c d^{2} x^{2} + c^{2} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.36, size = 75, normalized size = 1.14 \[ \frac {x \arccos \left (a x\right )}{\sqrt {d x^{2} + c} c} + \frac {a \log \left ({\left | -\sqrt {-a^{2} x^{2} + 1} \sqrt {-d} + \sqrt {a^{2} c + {\left (a^{2} x^{2} - 1\right )} d + d} \right |}\right )}{c \sqrt {-d} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.43, size = 0, normalized size = 0.00 \[ \int \frac {\arccos \left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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