Optimal. Leaf size=136 \[ -\frac {a \sqrt {1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \text {ArcCos}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \text {ArcCos}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {2 \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{3 c^2 \sqrt {d}} \]
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Rubi [A]
time = 0.11, antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {198, 197, 4756,
12, 585, 79, 65, 223, 209} \begin {gather*} -\frac {2 \text {ArcTan}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{3 c^2 \sqrt {d}}-\frac {a \sqrt {1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {2 x \text {ArcCos}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {x \text {ArcCos}(a x)}{3 c \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 79
Rule 197
Rule 198
Rule 209
Rule 223
Rule 585
Rule 4756
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)}{\left (c+d x^2\right )^{5/2}} \, dx &=\frac {x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cos ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+a \int \frac {x \left (3 c+2 d x^2\right )}{3 c^2 \sqrt {1-a^2 x^2} \left (c+d x^2\right )^{3/2}} \, dx\\ &=\frac {x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cos ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {a \int \frac {x \left (3 c+2 d x^2\right )}{\sqrt {1-a^2 x^2} \left (c+d x^2\right )^{3/2}} \, dx}{3 c^2}\\ &=\frac {x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cos ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}+\frac {a \text {Subst}\left (\int \frac {3 c+2 d x}{\sqrt {1-a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{6 c^2}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cos ^{-1}(a x)}{3 c^2 \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 )}{3 c^2}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cos ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {2 \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 )}{3 a c^2}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cos ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {2 \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 )}{3 a c^2}\\ &=-\frac {a \sqrt {1-a^2 x^2}}{3 c \left (a^2 c+d\right ) \sqrt {c+d x^2}}+\frac {x \cos ^{-1}(a x)}{3 c \left (c+d x^2\right )^{3/2}}+\frac {2 x \cos ^{-1}(a x)}{3 c^2 \sqrt {c+d x^2}}-\frac {2 \tan ^{-1}\left (\frac {\sqrt {d} \sqrt {1-a^2 x^2}}{a \sqrt {c+d x^2}}\right )}{3 c^2 \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 = 0.14, size = 120, normalized size = 0.88 \begin {gather*} \frac {-\frac {a c \sqrt {1-a^2 x^2} \left (c+d x^2\right )}{a^2 c+d}+a x^2 \left (c+d x^2\right ) \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 )+\left (3 c x+2 d x^3\right ) \text {ArcCos}(a x)}{3 c^2 \left (c+d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.42, size = 0, normalized size = 0.00 \[\int \frac {\arccos \left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {5}{2}}}\, dx\]
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. 280 vs.
\(2 (112) = 224\).
time = 3.85, size = 580, normalized size = 4.26 \begin {gather*} \left [-\frac {{\left (a^{2} c^{3} + {\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\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 ) - 2 \, \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \arccos \left (a x\right ) - {\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt {-a^{2} x^{2} + 1}\right )}}{6 \, {\left (a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\right )}}, -\frac {{\left (a^{2} c^{3} + {\left (a^{2} c d^{2} + d^{3}\right )} x^{4} + c^{2} d + 2 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x^{2}\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 ) - \sqrt {d x^{2} + c} {\left ({\left (2 \, {\left (a^{2} c d^{2} + d^{3}\right )} x^{3} + 3 \, {\left (a^{2} c^{2} d + c d^{2}\right )} x\right )} \arccos \left (a x\right ) - {\left (a c d^{2} x^{2} + a c^{2} d\right )} \sqrt {-a^{2} x^{2} + 1}\right )}}{3 \, {\left (a^{2} c^{5} d + c^{4} d^{2} + {\left (a^{2} c^{3} d^{3} + c^{2} d^{4}\right )} x^{4} + 2 \, {\left (a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} x^{2}\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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 185, normalized size = 1.36 \begin {gather*} \frac {1}{3} \, a {\left (\frac {2 \, a^{2} {\left | d \right |}}{{\left (a^{2} c d - {\left (\sqrt {-a^{2} d} \sqrt {d x^{2} + c} - \sqrt {-{\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}}\right )}^{2} + d^{2}\right )} c \sqrt {-d} {\left | a \right |}} - \frac {{\left | d \right |} \log \left ({\left (\sqrt {-a^{2} d} \sqrt {d x^{2} + c} - \sqrt {-{\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}}\right )}^{2}\right )}{c^{2} \sqrt {-d} d {\left | a \right |}}\right )} + \frac {x {\left (\frac {2 \, d x^{2}}{c^{2}} + \frac {3}{c}\right )} \arccos \left (a x\right )}{3 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}}} \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 \frac {\mathrm {acos}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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