Optimal. Leaf size=87 \[ -\frac {b c \left (a+b \text {ArcTan}\left (c x^3\right )\right )}{3 x^3}-\frac {1}{6} c^2 \left (a+b \text {ArcTan}\left (c x^3\right )\right )^2-\frac {\left (a+b \text {ArcTan}\left (c x^3\right )\right )^2}{6 x^6}+b^2 c^2 \log (x)-\frac {1}{6} b^2 c^2 \log \left (1+c^2 x^6\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4948, 4946,
5038, 272, 36, 29, 31, 5004} \begin {gather*} -\frac {1}{6} c^2 \left (a+b \text {ArcTan}\left (c x^3\right )\right )^2-\frac {b c \left (a+b \text {ArcTan}\left (c x^3\right )\right )}{3 x^3}-\frac {\left (a+b \text {ArcTan}\left (c x^3\right )\right )^2}{6 x^6}-\frac {1}{6} b^2 c^2 \log \left (c^2 x^6+1\right )+b^2 c^2 \log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rule 4946
Rule 4948
Rule 5004
Rule 5038
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (c x^3\right )\right )^2}{x^7} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{4 x^7}+\frac {b \left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{2 x^7}-\frac {b^2 \log ^2\left (1+i c x^3\right )}{4 x^7}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{x^7} \, dx+\frac {1}{2} b \int \frac {\left (-2 i a+b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{x^7} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+i c x^3\right )}{x^7} \, dx\\ &=\frac {1}{12} \text {Subst}\left (\int \frac {(2 a+i b \log (1-i c x))^2}{x^3} \, dx,x,x^3\right )+\frac {1}{6} b \text {Subst}\left (\int \frac {(-2 i a+b \log (1-i c x)) \log (1+i c x)}{x^3} \, dx,x,x^3\right )-\frac {1}{12} b^2 \text {Subst}\left (\int \frac {\log ^2(1+i c x)}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 x^6}+\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 x^6}+\frac {b^2 \log ^2\left (1+i c x^3\right )}{24 x^6}+\frac {1}{12} (i b c) \text {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{x^2 (1+i c x)} \, dx,x,x^3\right )+\frac {1}{12} (b c) \text {Subst}\left (\int \frac {2 a+i b \log (1-i c x)}{x^2 (1-i c x)} \, dx,x,x^3\right )-\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{x^2 (1-i c x)} \, dx,x,x^3\right )-\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{x^2 (1+i c x)} \, dx,x,x^3\right )\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 x^6}+\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 x^6}+\frac {b^2 \log ^2\left (1+i c x^3\right )}{24 x^6}+\frac {1}{12} (i b) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-i c x^3\right )+\frac {1}{12} (i b c) \text {Subst}\left (\int \left (\frac {-2 i a+b \log (1-i c x)}{x^2}-\frac {i c (-2 i a+b \log (1-i c x))}{x}+\frac {i c^2 (-2 i a+b \log (1-i c x))}{-i+c x}\right ) \, dx,x,x^3\right )-\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log (1+i c x)}{x^2}-\frac {i c \log (1+i c x)}{x}+\frac {i c^2 \log (1+i c x)}{-i+c x}\right ) \, dx,x,x^3\right )-\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \left (\frac {\log (1+i c x)}{x^2}+\frac {i c \log (1+i c x)}{x}-\frac {i c^2 \log (1+i c x)}{i+c x}\right ) \, dx,x,x^3\right )\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 x^6}+\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 x^6}+\frac {b^2 \log ^2\left (1+i c x^3\right )}{24 x^6}+\frac {1}{12} (i b) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{\left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-i c x^3\right )+\frac {1}{12} (i b c) \text {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{x^2} \, dx,x,x^3\right )-\frac {1}{12} (b c) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )} \, dx,x,1-i c x^3\right )-2 \left (\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{x^2} \, dx,x,x^3\right )\right )+\frac {1}{12} \left (b c^2\right ) \text {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{x} \, dx,x,x^3\right )-\frac {1}{12} \left (b c^3\right ) \text {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{-i+c x} \, dx,x,x^3\right )-\frac {1}{12} \left (b^2 c^3\right ) \text {Subst}\left (\int \frac {\log (1+i c x)}{i+c x} \, dx,x,x^3\right )\\ &=-\frac {1}{2} i a b c^2 \log (x)+\frac {i b c \left (2 i a-b \log \left (1-i c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )}{12 x^3}-\frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 x^6}+\frac {b^2 \log ^2\left (1+i c x^3\right )}{24 x^6}-\frac {1}{12} (b c) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-i c x^3\right )+\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {1}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-i c x^3\right )-\frac {1}{12} \left (i b c^2\right ) \text {Subst}\left (\int \frac {2 a+i b \log (x)}{x} \, dx,x,1-i c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x (1-i c x)} \, dx,x,x^3\right )-2 \left (-\frac {i b^2 c \log \left (1+i c x^3\right )}{12 x^3}-\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x (1+i c x)} \, dx,x,x^3\right )\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+i c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log (1-i c x)}{x} \, dx,x,x^3\right )-\frac {1}{12} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (\frac {1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^3\right )+\frac {1}{12} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^3\right )\\ &=\frac {1}{4} b^2 c^2 \log (x)+\frac {i b c \left (2 i a-b \log \left (1-i c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )}{12 x^3}-\frac {1}{24} c^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 x^6}+\frac {1}{24} b^2 c^2 \log ^2\left (1+i c x^3\right )+\frac {b^2 \log ^2\left (1+i c x^3\right )}{24 x^6}-\frac {1}{12} b^2 c^2 \text {Li}_2\left (i c x^3\right )-\frac {1}{12} \left (i b^2 c\right ) \text {Subst}\left (\int \frac {\log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-i c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-i c x^3\right )+\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+i c x^3\right )+\frac {1}{12} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{1-i c x} \, dx,x,x^3\right )-2 \left (-\frac {i b^2 c \log \left (1+i c x^3\right )}{12 x^3}-\frac {1}{12} \left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )+\frac {1}{12} \left (i b^2 c^3\right ) \text {Subst}\left (\int \frac {1}{1+i c x} \, dx,x,x^3\right )\right )\\ &=\frac {1}{2} b^2 c^2 \log (x)+\frac {i b c \left (2 i a-b \log \left (1-i c x^3\right )\right )}{12 x^3}-\frac {b c \left (1-i c x^3\right ) \left (2 a+i b \log \left (1-i c x^3\right )\right )}{12 x^3}-\frac {1}{24} c^2 \left (2 a+i b \log \left (1-i c x^3\right )\right )^2-\frac {\left (2 a+i b \log \left (1-i c x^3\right )\right )^2}{24 x^6}+\frac {1}{12} b c^2 \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^3\right )\right )-\frac {1}{12} b^2 c^2 \log \left (\frac {1}{2} \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )+\frac {b \left (2 i a-b \log \left (1-i c x^3\right )\right ) \log \left (1+i c x^3\right )}{12 x^6}+\frac {1}{24} b^2 c^2 \log ^2\left (1+i c x^3\right )+\frac {b^2 \log ^2\left (1+i c x^3\right )}{24 x^6}-2 \left (-\frac {1}{4} b^2 c^2 \log (x)+\frac {1}{12} b^2 c^2 \log \left (i-c x^3\right )-\frac {i b^2 c \log \left (1+i c x^3\right )}{12 x^3}\right )-\frac {1}{12} b^2 c^2 \log \left (i+c x^3\right )-\frac {1}{12} b^2 c^2 \text {Li}_2\left (\frac {1}{2} \left (1-i c x^3\right )\right )-\frac {1}{12} b^2 c^2 \text {Li}_2\left (\frac {1}{2} \left (1+i c x^3\right )\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 98, normalized size = 1.13 \begin {gather*} -\frac {a^2+2 a b c x^3+2 b \left (a+b c x^3+a c^2 x^6\right ) \text {ArcTan}\left (c x^3\right )+b^2 \left (1+c^2 x^6\right ) \text {ArcTan}\left (c x^3\right )^2-6 b^2 c^2 x^6 \log (x)+b^2 c^2 x^6 \log \left (1+c^2 x^6\right )}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 118, normalized size = 1.36
method | result | size |
default | \(-\frac {a^{2}}{6 x^{6}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{6 x^{6}}-\frac {b^{2} c \arctan \left (c \,x^{3}\right )}{3 x^{3}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2} c^{2}}{6}+b^{2} c^{2} \ln \left (x \right )-\frac {b^{2} c^{2} \ln \left (c^{2} x^{6}+1\right )}{6}-\frac {a b \arctan \left (c \,x^{3}\right )}{3 x^{6}}-\frac {a b c}{3 x^{3}}-\frac {a b \arctan \left (c \,x^{3}\right ) c^{2}}{3}\) | \(118\) |
risch | \(\frac {b^{2} \left (c^{2} x^{6}+1\right ) \ln \left (i c \,x^{3}+1\right )^{2}}{24 x^{6}}+\frac {i b \left (i b \,c^{2} x^{6} \ln \left (-i c \,x^{3}+1\right )+2 b c \,x^{3}+2 a +i b \ln \left (-i c \,x^{3}+1\right )\right ) \ln \left (i c \,x^{3}+1\right )}{12 x^{6}}-\frac {4 i \ln \left (\left (-7 i b c +a c \right ) x^{3}+7 b +i a \right ) a b \,c^{2} x^{6}-4 i \ln \left (\left (7 i b c +a c \right ) x^{3}+7 b -i a \right ) a b \,c^{2} x^{6}-b^{2} c^{2} x^{6} \ln \left (-i c \,x^{3}+1\right )^{2}-24 b^{2} c^{2} \ln \left (x \right ) x^{6}+4 \ln \left (\left (-7 i b c +a c \right ) x^{3}+7 b +i a \right ) b^{2} c^{2} x^{6}+4 \ln \left (\left (7 i b c +a c \right ) x^{3}+7 b -i a \right ) b^{2} c^{2} x^{6}+4 i b^{2} c \,x^{3} \ln \left (-i c \,x^{3}+1\right )+8 a b c \,x^{3}+4 i b \ln \left (-i c \,x^{3}+1\right ) a -b^{2} \ln \left (-i c \,x^{3}+1\right )^{2}+4 a^{2}}{24 x^{6}}\) | \(332\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 110, normalized size = 1.26 \begin {gather*} -\frac {1}{3} \, {\left ({\left (c \arctan \left (c x^{3}\right ) + \frac {1}{x^{3}}\right )} c + \frac {\arctan \left (c x^{3}\right )}{x^{6}}\right )} a b + \frac {1}{6} \, {\left ({\left (\arctan \left (c x^{3}\right )^{2} - \log \left (c^{2} x^{6} + 1\right ) + 6 \, \log \left (x\right )\right )} c^{2} - 2 \, {\left (c \arctan \left (c x^{3}\right ) + \frac {1}{x^{3}}\right )} c \arctan \left (c x^{3}\right )\right )} b^{2} - \frac {b^{2} \arctan \left (c x^{3}\right )^{2}}{6 \, x^{6}} - \frac {a^{2}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.24, size = 102, normalized size = 1.17 \begin {gather*} -\frac {b^{2} c^{2} x^{6} \log \left (c^{2} x^{6} + 1\right ) - 6 \, b^{2} c^{2} x^{6} \log \left (x\right ) + 2 \, a b c x^{3} + {\left (b^{2} c^{2} x^{6} + b^{2}\right )} \arctan \left (c x^{3}\right )^{2} + a^{2} + 2 \, {\left (a b c^{2} x^{6} + b^{2} c x^{3} + a b\right )} \arctan \left (c x^{3}\right )}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (80) = 160\).
time = 72.19, size = 207, normalized size = 2.38 \begin {gather*} \begin {cases} - \frac {a^{2}}{6 x^{6}} - \frac {a b c^{2} \operatorname {atan}{\left (c x^{3} \right )}}{3} - \frac {a b c}{3 x^{3}} - \frac {a b \operatorname {atan}{\left (c x^{3} \right )}}{3 x^{6}} + \frac {b^{2} c^{3} \sqrt {- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{3} + b^{2} c^{2} \log {\left (x \right )} - \frac {b^{2} c^{2} \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{3} - \frac {b^{2} c^{2} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{3} - \frac {b^{2} c^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6} - \frac {b^{2} c \operatorname {atan}{\left (c x^{3} \right )}}{3 x^{3}} - \frac {b^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{6 x^{6}} & \text {for}\: c \neq 0 \\- \frac {a^{2}}{6 x^{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 = 0.69, size = 152, normalized size = 1.75 \begin {gather*} b^2\,c^2\,\ln \left (x\right )-\frac {b^2\,c^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6}-\frac {b^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{6\,x^6}-\frac {b^2\,c^2\,\ln \left (c^2\,x^6+1\right )}{6}-\frac {a^2}{6\,x^6}-\frac {b^2\,c\,\mathrm {atan}\left (c\,x^3\right )}{3\,x^3}-\frac {a\,b\,c}{3\,x^3}-\frac {a\,b\,c^2\,\mathrm {atan}\left (\frac {a^2\,c\,x^3}{a^2+49\,b^2}+\frac {49\,b^2\,c\,x^3}{a^2+49\,b^2}\right )}{3}-\frac {a\,b\,\mathrm {atan}\left (c\,x^3\right )}{3\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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