Optimal. Leaf size=215 \[ -\frac {40}{9 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \tan ^{-1}(a x)^2}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {40 x \tan ^{-1}(a x)}{9 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{27 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.18, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4900, 4898, 4894, 4896} \[ -\frac {40}{9 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \tan ^{-1}(a x)^2}{a c^2 \sqrt {a^2 c x^2+c}}-\frac {40 x \tan ^{-1}(a x)}{9 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{27 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \tan ^{-1}(a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\tan ^{-1}(a x)^2}{3 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {2 x \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4894
Rule 4896
Rule 4898
Rule 4900
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {\tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2}{3} \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac {2 \int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {2}{27 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \tan ^{-1}(a x)^2}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {4 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{9 c}-\frac {4 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac {2}{27 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac {40}{9 a c^2 \sqrt {c+a^2 c x^2}}-\frac {2 x \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {40 x \tan ^{-1}(a x)}{9 c^2 \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^2}{3 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \tan ^{-1}(a x)^2}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^3}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \tan ^{-1}(a x)^3}{3 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 104, normalized size = 0.48 \[ \frac {\sqrt {a^2 c x^2+c} \left (-2 \left (60 a^2 x^2+61\right )+9 a x \left (2 a^2 x^2+3\right ) \tan ^{-1}(a x)^3+9 \left (6 a^2 x^2+7\right ) \tan ^{-1}(a x)^2-6 a x \left (20 a^2 x^2+21\right ) \tan ^{-1}(a x)\right )}{27 a c^3 \left (a^2 x^2+1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 111, normalized size = 0.52 \[ -\frac {\sqrt {a^{2} c x^{2} + c} {\left (120 \, a^{2} x^{2} - 9 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{3} - 9 \, {\left (6 \, a^{2} x^{2} + 7\right )} \arctan \left (a x\right )^{2} + 6 \, {\left (20 \, a^{3} x^{3} + 21 \, a x\right )} \arctan \left (a x\right ) + 122\right )}}{27 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.53, size = 308, normalized size = 1.43 \[ -\frac {\left (9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}-2 i-6 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i x^{2} a^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a \,c^{3}}+\frac {3 \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right )}{8 c^{3} a \left (a^{2} x^{2}+1\right )}-\frac {\left (-9 i \arctan \left (a x \right )^{2}+9 \arctan \left (a x \right )^{3}+2 i-6 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i x^{2} a^{2}-3 a x -i\right )}{216 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) a \,c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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