Optimal. Leaf size=90 \[ \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{4 c^2}-\frac {a b x^2}{2 c}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2+\frac {b^2 \log \left (c^2 x^4+1\right )}{4 c^2}-\frac {b^2 x^2 \tan ^{-1}\left (c x^2\right )}{2 c} \]
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Rubi [C] time = 1.04, antiderivative size = 612, normalized size of antiderivative = 6.80, number of steps used = 44, number of rules used = 16, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5035, 2454, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2395, 43, 2439, 2416, 2394, 2393, 2391} \[ \frac {b^2 \text {PolyLog}\left (2,\frac {1}{2} \left (1-i c x^2\right )\right )}{8 c^2}+\frac {b^2 \text {PolyLog}\left (2,\frac {1}{2} \left (1+i c x^2\right )\right )}{8 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 c^2}+\frac {i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{16 c^2}-\frac {b \log \left (\frac {1}{2} \left (1+i c x^2\right )\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )}{8 c^2}-\frac {3 a b x^2}{4 c}+\frac {1}{16} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )-\frac {1}{8} b x^4 \log \left (1+i c x^2\right ) \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac {b^2 \left (1-i c x^2\right )^2}{32 c^2}+\frac {b^2 \left (1+i c x^2\right )^2}{32 c^2}+\frac {b^2 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{16 c^2}-\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c^2}-\frac {b^2 \log \left (-c x^2+i\right )}{16 c^2}+\frac {3 b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{8 c^2}-\frac {b^2 \left (1+i c x^2\right )^2 \log \left (1+i c x^2\right )}{16 c^2}+\frac {3 b^2 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{8 c^2}-\frac {b^2 \log \left (c x^2+i\right )}{16 c^2}-\frac {1}{16} b^2 x^4 \log \left (1+i c x^2\right )+\frac {b^2 x^4}{16} \]
Warning: Unable to verify antiderivative.
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Rule 43
Rule 2295
Rule 2296
Rule 2304
Rule 2305
Rule 2389
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2401
Rule 2416
Rule 2439
Rule 2454
Rule 5035
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac {1}{2} b x^3 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac {1}{4} b^2 x^3 \log ^2\left (1+i c x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \, dx+\frac {1}{2} b \int x^3 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right ) \, dx-\frac {1}{4} b^2 \int x^3 \log ^2\left (1+i c x^2\right ) \, dx\\ &=\frac {1}{8} \operatorname {Subst}\left (\int x (2 a+i b \log (1-i c x))^2 \, dx,x,x^2\right )+\frac {1}{4} b \operatorname {Subst}\left (\int x (-2 i a+b \log (1-i c x)) \log (1+i c x) \, dx,x,x^2\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int x \log ^2(1+i c x) \, dx,x,x^2\right )\\ &=-\frac {1}{8} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )+\frac {1}{8} \operatorname {Subst}\left (\int \left (-\frac {i (2 a+i b \log (1-i c x))^2}{c}+\frac {i (1-i c x) (2 a+i b \log (1-i c x))^2}{c}\right ) \, dx,x,x^2\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int \left (\frac {i \log ^2(1+i c x)}{c}-\frac {i (1+i c x) \log ^2(1+i c x)}{c}\right ) \, dx,x,x^2\right )-\frac {1}{8} (i b c) \operatorname {Subst}\left (\int \frac {x^2 (-2 i a+b \log (1-i c x))}{1+i c x} \, dx,x,x^2\right )+\frac {1}{8} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2 \log (1+i c x)}{1-i c x} \, dx,x,x^2\right )\\ &=-\frac {1}{8} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac {i \operatorname {Subst}\left (\int (2 a+i b \log (1-i c x))^2 \, dx,x,x^2\right )}{8 c}+\frac {i \operatorname {Subst}\left (\int (1-i c x) (2 a+i b \log (1-i c x))^2 \, dx,x,x^2\right )}{8 c}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \log ^2(1+i c x) \, dx,x,x^2\right )}{8 c}+\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int (1+i c x) \log ^2(1+i c x) \, dx,x,x^2\right )}{8 c}-\frac {1}{8} (i b c) \operatorname {Subst}\left (\int \left (\frac {-2 i a+b \log (1-i c x)}{c^2}-\frac {i x (-2 i a+b \log (1-i c x))}{c}+\frac {i (-2 i a+b \log (1-i c x))}{c^2 (-i+c x)}\right ) \, dx,x,x^2\right )+\frac {1}{8} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {\log (1+i c x)}{c^2}+\frac {i x \log (1+i c x)}{c}-\frac {i \log (1+i c x)}{c^2 (i+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{8} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac {1}{8} b \operatorname {Subst}\left (\int x (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )-\frac {1}{8} b^2 \operatorname {Subst}\left (\int x \log (1+i c x) \, dx,x,x^2\right )+\frac {\operatorname {Subst}\left (\int (2 a+i b \log (x))^2 \, dx,x,1-i c x^2\right )}{8 c^2}-\frac {\operatorname {Subst}\left (\int x (2 a+i b \log (x))^2 \, dx,x,1-i c x^2\right )}{8 c^2}-\frac {b^2 \operatorname {Subst}\left (\int \log ^2(x) \, dx,x,1+i c x^2\right )}{8 c^2}+\frac {b^2 \operatorname {Subst}\left (\int x \log ^2(x) \, dx,x,1+i c x^2\right )}{8 c^2}-\frac {(i b) \operatorname {Subst}\left (\int (-2 i a+b \log (1-i c x)) \, dx,x,x^2\right )}{8 c}+\frac {b \operatorname {Subst}\left (\int \frac {-2 i a+b \log (1-i c x)}{-i+c x} \, dx,x,x^2\right )}{8 c}+\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \log (1+i c x) \, dx,x,x^2\right )}{8 c}+\frac {b^2 \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{i+c x} \, dx,x,x^2\right )}{8 c}\\ &=-\frac {a b x^2}{4 c}+\frac {1}{16} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}-\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{8 c^2}-\frac {1}{16} b^2 x^4 \log \left (1+i c x^2\right )+\frac {b^2 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{8 c^2}-\frac {1}{8} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c^2}+\frac {b^2 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{16 c^2}+\frac {(i b) \operatorname {Subst}\left (\int x (2 a+i b \log (x)) \, dx,x,1-i c x^2\right )}{8 c^2}-\frac {(i b) \operatorname {Subst}\left (\int (2 a+i b \log (x)) \, dx,x,1-i c x^2\right )}{4 c^2}+\frac {b^2 \operatorname {Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{8 c^2}-\frac {b^2 \operatorname {Subst}\left (\int x \log (x) \, dx,x,1+i c x^2\right )}{8 c^2}+\frac {b^2 \operatorname {Subst}\left (\int \log (x) \, dx,x,1+i c x^2\right )}{4 c^2}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \log (1-i c x) \, dx,x,x^2\right )}{8 c}+\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {1}{2} i (-i+c x)\right )}{1-i c x} \, dx,x,x^2\right )}{8 c}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int \frac {\log \left (-\frac {1}{2} i (i+c x)\right )}{1+i c x} \, dx,x,x^2\right )}{8 c}-\frac {1}{16} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-i c x} \, dx,x,x^2\right )+\frac {1}{16} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+i c x} \, dx,x,x^2\right )\\ &=-\frac {3 a b x^2}{4 c}-\frac {3 i b^2 x^2}{8 c}+\frac {b^2 \left (1-i c x^2\right )^2}{32 c^2}+\frac {b^2 \left (1+i c x^2\right )^2}{32 c^2}+\frac {1}{16} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac {i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{16 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}-\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{8 c^2}-\frac {1}{16} b^2 x^4 \log \left (1+i c x^2\right )+\frac {3 b^2 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^2}-\frac {b^2 \left (1+i c x^2\right )^2 \log \left (1+i c x^2\right )}{16 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{8 c^2}-\frac {1}{8} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c^2}+\frac {b^2 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{16 c^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1-i c x^2\right )}{8 c^2}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2}\right )}{x} \, dx,x,1+i c x^2\right )}{8 c^2}+\frac {b^2 \operatorname {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{8 c^2}+\frac {b^2 \operatorname {Subst}\left (\int \log (x) \, dx,x,1-i c x^2\right )}{4 c^2}+\frac {1}{16} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {i x}{c}+\frac {i}{c^2 (-i+c x)}\right ) \, dx,x,x^2\right )-\frac {1}{16} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}+\frac {i x}{c}-\frac {i}{c^2 (i+c x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {3 a b x^2}{4 c}+\frac {b^2 x^4}{16}+\frac {b^2 \left (1-i c x^2\right )^2}{32 c^2}+\frac {b^2 \left (1+i c x^2\right )^2}{32 c^2}-\frac {b^2 \log \left (i-c x^2\right )}{16 c^2}+\frac {3 b^2 \left (1-i c x^2\right ) \log \left (1-i c x^2\right )}{8 c^2}+\frac {1}{16} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right )+\frac {i b \left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{16 c^2}+\frac {\left (1-i c x^2\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{8 c^2}-\frac {\left (1-i c x^2\right )^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{16 c^2}-\frac {b \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (\frac {1}{2} \left (1+i c x^2\right )\right )}{8 c^2}-\frac {1}{16} b^2 x^4 \log \left (1+i c x^2\right )+\frac {3 b^2 \left (1+i c x^2\right ) \log \left (1+i c x^2\right )}{8 c^2}-\frac {b^2 \left (1+i c x^2\right )^2 \log \left (1+i c x^2\right )}{16 c^2}+\frac {b^2 \log \left (\frac {1}{2} \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{8 c^2}-\frac {1}{8} b x^4 \left (2 i a-b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac {b^2 \left (1+i c x^2\right ) \log ^2\left (1+i c x^2\right )}{8 c^2}+\frac {b^2 \left (1+i c x^2\right )^2 \log ^2\left (1+i c x^2\right )}{16 c^2}-\frac {b^2 \log \left (i+c x^2\right )}{16 c^2}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1-i c x^2\right )\right )}{8 c^2}+\frac {b^2 \text {Li}_2\left (\frac {1}{2} \left (1+i c x^2\right )\right )}{8 c^2}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 85, normalized size = 0.94 \[ \frac {2 b \tan ^{-1}\left (c x^2\right ) \left (a c^2 x^4+a-b c x^2\right )+a c x^2 \left (a c x^2-2 b\right )+b^2 \log \left (c^2 x^4+1\right )+b^2 \left (c^2 x^4+1\right ) \tan ^{-1}\left (c x^2\right )^2}{4 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 100, normalized size = 1.11 \[ \frac {a^{2} c^{2} x^{4} - 2 \, a b c x^{2} + {\left (b^{2} c^{2} x^{4} + b^{2}\right )} \arctan \left (c x^{2}\right )^{2} - 2 \, a b \arctan \left (\frac {1}{c x^{2}}\right ) + b^{2} \log \left (c^{2} x^{4} + 1\right ) + 2 \, {\left (a b c^{2} x^{4} - b^{2} c x^{2}\right )} \arctan \left (c x^{2}\right )}{4 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.33, size = 100, normalized size = 1.11 \[ \frac {a^{2} c x^{4} + \frac {2 \, {\left (c^{2} x^{4} \arctan \left (c x^{2}\right ) - c x^{2} + \arctan \left (c x^{2}\right )\right )} a b}{c} + \frac {{\left (c^{2} x^{4} \arctan \left (c x^{2}\right )^{2} - 2 \, c x^{2} \arctan \left (c x^{2}\right ) + \arctan \left (c x^{2}\right )^{2} + \log \left (c^{2} x^{4} + 1\right )\right )} b^{2}}{c}}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 113, normalized size = 1.26 \[ \frac {a^{2} x^{4}}{4}+\frac {b^{2} x^{4} \arctan \left (c \,x^{2}\right )^{2}}{4}-\frac {b^{2} x^{2} \arctan \left (c \,x^{2}\right )}{2 c}+\frac {b^{2} \arctan \left (c \,x^{2}\right )^{2}}{4 c^{2}}+\frac {b^{2} \ln \left (c^{2} x^{4}+1\right )}{4 c^{2}}+\frac {a b \,x^{4} \arctan \left (c \,x^{2}\right )}{2}-\frac {a b \,x^{2}}{2 c}+\frac {a b \arctan \left (c \,x^{2}\right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 126, normalized size = 1.40 \[ \frac {1}{4} \, b^{2} x^{4} \arctan \left (c x^{2}\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{2} \, {\left (x^{4} \arctan \left (c x^{2}\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\arctan \left (c x^{2}\right )}{c^{3}}\right )}\right )} a b - \frac {1}{4} \, {\left (2 \, c {\left (\frac {x^{2}}{c^{2}} - \frac {\arctan \left (c x^{2}\right )}{c^{3}}\right )} \arctan \left (c x^{2}\right ) + \frac {\arctan \left (c x^{2}\right )^{2} - \log \left (4 \, c^{5} x^{4} + 4 \, c^{3}\right )}{c^{2}}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 112, normalized size = 1.24 \[ \frac {a^2\,x^4}{4}+\frac {b^2\,{\mathrm {atan}\left (c\,x^2\right )}^2}{4\,c^2}+\frac {b^2\,x^4\,{\mathrm {atan}\left (c\,x^2\right )}^2}{4}+\frac {b^2\,\ln \left (c^2\,x^4+1\right )}{4\,c^2}-\frac {b^2\,x^2\,\mathrm {atan}\left (c\,x^2\right )}{2\,c}-\frac {a\,b\,x^2}{2\,c}+\frac {a\,b\,\mathrm {atan}\left (c\,x^2\right )}{2\,c^2}+\frac {a\,b\,x^4\,\mathrm {atan}\left (c\,x^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 34.91, size = 155, normalized size = 1.72 \[ \begin {cases} \frac {a^{2} x^{4}}{4} + \frac {a b x^{4} \operatorname {atan}{\left (c x^{2} \right )}}{2} - \frac {a b x^{2}}{2 c} + \frac {a b \operatorname {atan}{\left (c x^{2} \right )}}{2 c^{2}} + \frac {b^{2} x^{4} \operatorname {atan}^{2}{\left (c x^{2} \right )}}{4} - \frac {b^{2} x^{2} \operatorname {atan}{\left (c x^{2} \right )}}{2 c} + \frac {i b^{2} \sqrt {\frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{2} \right )}}{2 c} + \frac {b^{2} \log {\left (x^{2} + i \sqrt {\frac {1}{c^{2}}} \right )}}{2 c^{2}} + \frac {b^{2} \operatorname {atan}^{2}{\left (c x^{2} \right )}}{4 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{4}}{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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