Optimal. Leaf size=144 \[ -\frac {a b x}{3 c^5}-\frac {4 b^2 x^2}{45 c^4}+\frac {b^2 x^4}{60 c^2}-\frac {b^2 x \text {ArcTan}(c x)}{3 c^5}+\frac {b x^3 (a+b \text {ArcTan}(c x))}{9 c^3}-\frac {b x^5 (a+b \text {ArcTan}(c x))}{15 c}+\frac {(a+b \text {ArcTan}(c x))^2}{6 c^6}+\frac {1}{6} x^6 (a+b \text {ArcTan}(c x))^2+\frac {23 b^2 \log \left (1+c^2 x^2\right )}{90 c^6} \]
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Rubi [A]
time = 0.22, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 7, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4946, 5036,
272, 45, 4930, 266, 5004} \begin {gather*} \frac {(a+b \text {ArcTan}(c x))^2}{6 c^6}+\frac {b x^3 (a+b \text {ArcTan}(c x))}{9 c^3}+\frac {1}{6} x^6 (a+b \text {ArcTan}(c x))^2-\frac {b x^5 (a+b \text {ArcTan}(c x))}{15 c}-\frac {a b x}{3 c^5}-\frac {b^2 x \text {ArcTan}(c x)}{3 c^5}-\frac {4 b^2 x^2}{45 c^4}+\frac {b^2 x^4}{60 c^2}+\frac {23 b^2 \log \left (c^2 x^2+1\right )}{90 c^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 266
Rule 272
Rule 4930
Rule 4946
Rule 5004
Rule 5036
Rubi steps
\begin {align*} \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{3} (b c) \int \frac {x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {b \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {b \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{15} b^2 \int \frac {x^5}{1+c^2 x^2} \, dx+\frac {b \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac {b \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c^3}\\ &=\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{30} b^2 \text {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right )-\frac {b \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^5}+\frac {b \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^5}-\frac {b^2 \int \frac {x^3}{1+c^2 x^2} \, dx}{9 c^2}\\ &=-\frac {a b x}{3 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{30} b^2 \text {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x}{c^2}+\frac {1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )-\frac {b^2 \int \tan ^{-1}(c x) \, dx}{3 c^5}-\frac {b^2 \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )}{18 c^2}\\ &=-\frac {a b x}{3 c^5}-\frac {b^2 x^2}{30 c^4}+\frac {b^2 x^4}{60 c^2}-\frac {b^2 x \tan ^{-1}(c x)}{3 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {b^2 \log \left (1+c^2 x^2\right )}{30 c^6}+\frac {b^2 \int \frac {x}{1+c^2 x^2} \, dx}{3 c^4}-\frac {b^2 \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{18 c^2}\\ &=-\frac {a b x}{3 c^5}-\frac {4 b^2 x^2}{45 c^4}+\frac {b^2 x^4}{60 c^2}-\frac {b^2 x \tan ^{-1}(c x)}{3 c^5}+\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )}{9 c^3}-\frac {b x^5 \left (a+b \tan ^{-1}(c x)\right )}{15 c}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{6 c^6}+\frac {1}{6} x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {23 b^2 \log \left (1+c^2 x^2\right )}{90 c^6}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 138, normalized size = 0.96 \begin {gather*} \frac {c x \left (30 a^2 c^5 x^5+b^2 c x \left (-16+3 c^2 x^2\right )-4 a b \left (15-5 c^2 x^2+3 c^4 x^4\right )\right )+4 b \left (b c x \left (-15+5 c^2 x^2-3 c^4 x^4\right )+15 a \left (1+c^6 x^6\right )\right ) \text {ArcTan}(c x)+30 b^2 \left (1+c^6 x^6\right ) \text {ArcTan}(c x)^2+46 b^2 \log \left (1+c^2 x^2\right )}{180 c^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 171, normalized size = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {c^{6} x^{6} a^{2}}{6}+\frac {b^{2} c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {b^{2} \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {b^{2} \arctan \left (c x \right ) c^{3} x^{3}}{9}-\frac {b^{2} \arctan \left (c x \right ) c x}{3}+\frac {b^{2} \arctan \left (c x \right )^{2}}{6}+\frac {b^{2} c^{4} x^{4}}{60}-\frac {4 b^{2} c^{2} x^{2}}{45}+\frac {23 b^{2} \ln \left (c^{2} x^{2}+1\right )}{90}+\frac {a b \,c^{6} x^{6} \arctan \left (c x \right )}{3}-\frac {c^{5} x^{5} a b}{15}+\frac {a b \,c^{3} x^{3}}{9}-\frac {a b c x}{3}+\frac {a b \arctan \left (c x \right )}{3}}{c^{6}}\) | \(171\) |
default | \(\frac {\frac {c^{6} x^{6} a^{2}}{6}+\frac {b^{2} c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {b^{2} \arctan \left (c x \right ) c^{5} x^{5}}{15}+\frac {b^{2} \arctan \left (c x \right ) c^{3} x^{3}}{9}-\frac {b^{2} \arctan \left (c x \right ) c x}{3}+\frac {b^{2} \arctan \left (c x \right )^{2}}{6}+\frac {b^{2} c^{4} x^{4}}{60}-\frac {4 b^{2} c^{2} x^{2}}{45}+\frac {23 b^{2} \ln \left (c^{2} x^{2}+1\right )}{90}+\frac {a b \,c^{6} x^{6} \arctan \left (c x \right )}{3}-\frac {c^{5} x^{5} a b}{15}+\frac {a b \,c^{3} x^{3}}{9}-\frac {a b c x}{3}+\frac {a b \arctan \left (c x \right )}{3}}{c^{6}}\) | \(171\) |
risch | \(-\frac {b^{2} \left (c^{6} x^{6}+1\right ) \ln \left (i c x +1\right )^{2}}{24 c^{6}}-\frac {i b \left (30 c^{6} x^{6} a +15 i b \,c^{6} x^{6} \ln \left (-i c x +1\right )-6 b \,c^{5} x^{5}+10 b \,c^{3} x^{3}-30 x b c +15 i b \ln \left (-i c x +1\right )\right ) \ln \left (i c x +1\right )}{180 c^{6}}-\frac {b^{2} x^{6} \ln \left (-i c x +1\right )^{2}}{24}-\frac {i b^{2} x \ln \left (-i c x +1\right )}{6 c^{5}}+\frac {i b^{2} x^{3} \ln \left (-i c x +1\right )}{18 c^{3}}+\frac {a^{2} x^{6}}{6}-\frac {x^{5} a b}{15 c}-\frac {i b^{2} x^{5} \ln \left (-i c x +1\right )}{30 c}+\frac {b^{2} x^{4}}{60 c^{2}}+\frac {a b \,x^{3}}{9 c^{3}}+\frac {i a b \,x^{6} \ln \left (-i c x +1\right )}{6}-\frac {4 b^{2} x^{2}}{45 c^{4}}-\frac {b^{2} \ln \left (-i c x +1\right )^{2}}{24 c^{6}}-\frac {a b x}{3 c^{5}}+\frac {a b \arctan \left (c x \right )}{3 c^{6}}+\frac {23 b^{2} \ln \left (c^{2} x^{2}+1\right )}{90 c^{6}}\) | \(304\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 163, normalized size = 1.13 \begin {gather*} \frac {1}{6} \, b^{2} x^{6} \arctan \left (c x\right )^{2} + \frac {1}{6} \, a^{2} x^{6} + \frac {1}{45} \, {\left (15 \, x^{6} \arctan \left (c x\right ) - c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )}\right )} a b - \frac {1}{180} \, {\left (4 \, c {\left (\frac {3 \, c^{4} x^{5} - 5 \, c^{2} x^{3} + 15 \, x}{c^{6}} - \frac {15 \, \arctan \left (c x\right )}{c^{7}}\right )} \arctan \left (c x\right ) - \frac {3 \, c^{4} x^{4} - 16 \, c^{2} x^{2} - 30 \, \arctan \left (c x\right )^{2} + 46 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )} b^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.83, size = 152, normalized size = 1.06 \begin {gather*} \frac {30 \, a^{2} c^{6} x^{6} - 12 \, a b c^{5} x^{5} + 3 \, b^{2} c^{4} x^{4} + 20 \, a b c^{3} x^{3} - 16 \, b^{2} c^{2} x^{2} - 60 \, a b c x + 30 \, {\left (b^{2} c^{6} x^{6} + b^{2}\right )} \arctan \left (c x\right )^{2} + 46 \, b^{2} \log \left (c^{2} x^{2} + 1\right ) + 4 \, {\left (15 \, a b c^{6} x^{6} - 3 \, b^{2} c^{5} x^{5} + 5 \, b^{2} c^{3} x^{3} - 15 \, b^{2} c x + 15 \, a b\right )} \arctan \left (c x\right )}{180 \, c^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.46, size = 199, normalized size = 1.38 \begin {gather*} \begin {cases} \frac {a^{2} x^{6}}{6} + \frac {a b x^{6} \operatorname {atan}{\left (c x \right )}}{3} - \frac {a b x^{5}}{15 c} + \frac {a b x^{3}}{9 c^{3}} - \frac {a b x}{3 c^{5}} + \frac {a b \operatorname {atan}{\left (c x \right )}}{3 c^{6}} + \frac {b^{2} x^{6} \operatorname {atan}^{2}{\left (c x \right )}}{6} - \frac {b^{2} x^{5} \operatorname {atan}{\left (c x \right )}}{15 c} + \frac {b^{2} x^{4}}{60 c^{2}} + \frac {b^{2} x^{3} \operatorname {atan}{\left (c x \right )}}{9 c^{3}} - \frac {4 b^{2} x^{2}}{45 c^{4}} - \frac {b^{2} x \operatorname {atan}{\left (c x \right )}}{3 c^{5}} + \frac {23 b^{2} \log {\left (x^{2} + \frac {1}{c^{2}} \right )}}{90 c^{6}} + \frac {b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{6 c^{6}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{6}}{6} & \text {otherwise} \end {cases} \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 [B]
time = 0.67, size = 171, normalized size = 1.19 \begin {gather*} \frac {30\,b^2\,{\mathrm {atan}\left (c\,x\right )}^2+46\,b^2\,\ln \left (c^2\,x^2+1\right )+30\,a^2\,c^6\,x^6-16\,b^2\,c^2\,x^2+3\,b^2\,c^4\,x^4+60\,a\,b\,\mathrm {atan}\left (c\,x\right )+20\,b^2\,c^3\,x^3\,\mathrm {atan}\left (c\,x\right )-12\,b^2\,c^5\,x^5\,\mathrm {atan}\left (c\,x\right )-60\,b^2\,c\,x\,\mathrm {atan}\left (c\,x\right )+30\,b^2\,c^6\,x^6\,{\mathrm {atan}\left (c\,x\right )}^2+20\,a\,b\,c^3\,x^3-12\,a\,b\,c^5\,x^5-60\,a\,b\,c\,x+60\,a\,b\,c^6\,x^6\,\mathrm {atan}\left (c\,x\right )}{180\,c^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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