Optimal. Leaf size=66 \[ \frac {x \text {ArcCos}(a x)}{c \sqrt {c+d x^2}}-\frac {\text {ArcTan}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d}} \]
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Rubi [A]
time = 0.07, 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 = {197, 4756, 12,
455, 65, 223, 209} \begin {gather*} \frac {x \text {ArcCos}(a x)}{c \sqrt {c+d x^2}}-\frac {\text {ArcTan}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{c \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 197
Rule 209
Rule 223
Rule 455
Rule 4756
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 \text {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 {\text {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 {\text {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] Result contains higher order function than in optimal. Order 6 vs. order 3 in
optimal.
time = 0.06, size = 68, normalized size = 1.03 \begin {gather*} \frac {x \left (a x \sqrt {1+\frac {d x^2}{c}} F_1\left (1;\frac {1}{2},\frac {1}{2};2;a^2 x^2,-\frac {d x^2}{c}\right )+2 \text {ArcCos}(a x)\right )}{2 c \sqrt {c+d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, 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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 128 vs.
\(2 (56) = 112\).
time = 2.83, size = 276, normalized size = 4.18 \begin {gather*} \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 ] \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {acos}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.44, size = 75, normalized size = 1.14 \begin {gather*} \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 |}} \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 = -0.02 \begin {gather*} \int \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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