Optimal. Leaf size=157 \[ -\frac {2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {40 x}{27 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \text {ArcTan}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 \text {ArcTan}(a x)}{3 a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \text {ArcTan}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \text {ArcTan}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 157, 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 = {5020, 5018,
197, 198} \begin {gather*} \frac {2 x \text {ArcTan}(a x)^2}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {4 \text {ArcTan}(a x)}{3 a c^2 \sqrt {a^2 c x^2+c}}+\frac {x \text {ArcTan}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \text {ArcTan}(a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac {40 x}{27 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 x}{27 c \left (a^2 c x^2+c\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 5018
Rule 5020
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac {2 \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {2}{9} \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac {2 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 \tan ^{-1}(a x)}{3 a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{27 c}-\frac {4 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {40 x}{27 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \tan ^{-1}(a x)}{9 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {4 \tan ^{-1}(a x)}{3 a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 86, normalized size = 0.55 \begin {gather*} \frac {\sqrt {c+a^2 c x^2} \left (-2 a x \left (21+20 a^2 x^2\right )+6 \left (7+6 a^2 x^2\right ) \text {ArcTan}(a x)+9 a x \left (3+2 a^2 x^2\right ) \text {ArcTan}(a x)^2\right )}{27 a c^3 \left (1+a^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.28, size = 272, normalized size = 1.73
method | result | size |
default | \(-\frac {\left (6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \left (a^{3} x^{3}-3 i a^{2} x^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} a \,c^{3}}+\frac {3 \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{8 c^{3} a \left (a^{2} x^{2}+1\right )}-\frac {\left (-6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i a^{2} x^{2}-3 a x -i\right )}{216 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) a \,c^{3}}\) | \(272\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 111, normalized size = 0.71 \begin {gather*} \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {a^{2} c x^{2} + c} c^{2}} + \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c}\right )} \arctan \left (a x\right )^{2} - \frac {2 \, {\left (20 \, a^{3} x^{3} + 21 \, a x - 3 \, {\left (6 \, a^{2} x^{2} + 7\right )} \arctan \left (a x\right )\right )} a}{27 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )} \sqrt {a^{2} x^{2} + 1} \sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.25, size = 93, normalized size = 0.59 \begin {gather*} -\frac {{\left (40 \, a^{3} x^{3} - 9 \, {\left (2 \, a^{3} x^{3} + 3 \, a x\right )} \arctan \left (a x\right )^{2} + 42 \, a x - 6 \, {\left (6 \, a^{2} x^{2} + 7\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c}}{27 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{3} c^{3} x^{2} + a c^{3}\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 {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \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 {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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