Integrand size = 16, antiderivative size = 146 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=-\frac {1}{2} i b c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^3-\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+b^2 c^2 \left (a+b \arctan \left (c x^3\right )\right ) \log \left (2-\frac {2}{1-i c x^3}\right )-\frac {1}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x^3}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {4948, 4946, 5038, 5044, 4988, 2497, 5004} \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=b^2 c^2 \log \left (2-\frac {2}{1-i c x^3}\right ) \left (a+b \arctan \left (c x^3\right )\right )-\frac {1}{2} i b c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^3-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^3}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}-\frac {1}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-i c x^3}-1\right ) \]
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Rule 2497
Rule 4946
Rule 4948
Rule 4988
Rule 5004
Rule 5038
Rule 5044
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \frac {(a+b \arctan (c x))^3}{x^3} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^2 \left (1+c^2 x^2\right )} \, dx,x,x^3\right ) \\ & = -\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^2} \, dx,x,x^3\right )-\frac {1}{2} \left (b c^3\right ) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx,x,x^3\right ) \\ & = -\frac {b c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^3-\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx,x,x^3\right ) \\ & = -\frac {1}{2} i b c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^3-\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+\left (i b^2 c^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx,x,x^3\right ) \\ & = -\frac {1}{2} i b c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^3-\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+b^2 c^2 \left (a+b \arctan \left (c x^3\right )\right ) \log \left (2-\frac {2}{1-i c x^3}\right )-\left (b^3 c^3\right ) \text {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx,x,x^3\right ) \\ & = -\frac {1}{2} i b c^2 \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b c \left (a+b \arctan \left (c x^3\right )\right )^2}{2 x^3}-\frac {1}{6} c^2 \left (a+b \arctan \left (c x^3\right )\right )^3-\frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{6 x^6}+b^2 c^2 \left (a+b \arctan \left (c x^3\right )\right ) \log \left (2-\frac {2}{1-i c x^3}\right )-\frac {1}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i c x^3}\right ) \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=-\frac {3 b^2 \left (a+a c^2 x^6+b c x^3 \left (1+i c x^3\right )\right ) \arctan \left (c x^3\right )^2+b^3 \left (1+c^2 x^6\right ) \arctan \left (c x^3\right )^3+3 b \arctan \left (c x^3\right ) \left (a \left (a+2 b c x^3+a c^2 x^6\right )-2 b^2 c^2 x^6 \log \left (1-e^{2 i \arctan \left (c x^3\right )}\right )\right )+a \left (a \left (a+3 b c x^3\right )-6 b^2 c^2 x^6 \log \left (\frac {c x^3}{\sqrt {1+c^2 x^6}}\right )\right )+3 i b^3 c^2 x^6 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (c x^3\right )}\right )}{6 x^6} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.88 (sec) , antiderivative size = 11581, normalized size of antiderivative = 79.32
method | result | size |
default | \(\text {Expression too large to display}\) | \(11581\) |
parts | \(\text {Expression too large to display}\) | \(11581\) |
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\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{3}}{x^{7}} \,d x } \]
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\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int \frac {\left (a + b \operatorname {atan}{\left (c x^{3} \right )}\right )^{3}}{x^{7}}\, dx \]
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\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{3}}{x^{7}} \,d x } \]
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\[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int { \frac {{\left (b \arctan \left (c x^{3}\right ) + a\right )}^{3}}{x^{7}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \arctan \left (c x^3\right )\right )^3}{x^7} \, dx=\int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^3\right )\right )}^3}{x^7} \,d x \]
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