Optimal. Leaf size=122 \[ -\frac {1}{4} c^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2-\frac {1}{2} b c^3 x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {1}{4} x^4 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{6} b c x^3 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )-\frac {2}{3} b^2 c^4 \log (x)+\frac {1}{12} b^2 c^2 x^2-\frac {1}{3} b^2 c^4 \log \left (\frac {c^2}{x^2}+1\right ) \]
[Out]
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Rubi [C] time = 1.79, antiderivative size = 862, normalized size of antiderivative = 7.07, number of steps used = 88, number of rules used = 34, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.125, Rules used = {5035, 2454, 2398, 2411, 2347, 2344, 2301, 2316, 2315, 2314, 31, 2319, 44, 2455, 263, 43, 6742, 30, 2557, 12, 2466, 2448, 2462, 260, 2416, 2394, 2393, 2391, 193, 2410, 2395, 36, 29, 2390} \[ -\frac {1}{16} \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2 c^4+\frac {1}{16} b^2 \log ^2\left (\frac {i c}{x}+1\right ) c^4-\frac {11}{48} b^2 \log \left (i-\frac {c}{x}\right ) c^4-\frac {5}{48} b^2 \log \left (\frac {c}{x}+i\right ) c^4-\frac {5}{48} b^2 \log (c-i x) c^4+\frac {1}{4} i a b \log (c-i x) c^4-\frac {1}{8} b^2 \log \left (1-\frac {i c}{x}\right ) \log (c-i x) c^4-\frac {5}{48} b^2 \log (c+i x) c^4-\frac {1}{8} b^2 \log \left (\frac {i c}{x}+1\right ) \log (c+i x) c^4+\frac {1}{8} b^2 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x) c^4+\frac {1}{8} b^2 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right ) c^4-\frac {11}{24} b^2 \log (x) c^4-\frac {1}{4} i a b \log (x) c^4-\frac {1}{8} b^2 \log (c+i x) \log \left (-\frac {i x}{c}\right ) c^4-\frac {1}{8} b^2 \log (c-i x) \log \left (\frac {i x}{c}\right ) c^4+\frac {1}{8} b^2 \text {PolyLog}\left (2,\frac {c-i x}{2 c}\right ) c^4+\frac {1}{8} b^2 \text {PolyLog}\left (2,\frac {c+i x}{2 c}\right ) c^4+\frac {1}{8} b^2 \text {PolyLog}\left (2,-\frac {i c}{x}\right ) c^4+\frac {1}{8} b^2 \text {PolyLog}\left (2,\frac {i c}{x}\right ) c^4-\frac {1}{8} b^2 \text {PolyLog}\left (2,1-\frac {i x}{c}\right ) c^4-\frac {1}{8} b^2 \text {PolyLog}\left (2,\frac {i x}{c}+1\right ) c^4-\frac {1}{4} a b x c^3-\frac {1}{8} i b^2 x \log \left (1-\frac {i c}{x}\right ) c^3-\frac {1}{8} b \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right ) c^3+\frac {1}{4} i b^2 x \log \left (\frac {i c}{x}+1\right ) c^3+\frac {1}{12} b^2 x^2 c^2-\frac {1}{8} i a b x^2 c^2+\frac {1}{16} b^2 x^2 \log \left (1-\frac {i c}{x}\right ) c^2+\frac {1}{16} i b x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right ) c^2+\frac {1}{12} a b x^3 c+\frac {1}{24} i b^2 x^3 \log \left (1-\frac {i c}{x}\right ) c+\frac {1}{24} b x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right ) c-\frac {1}{12} i b^2 x^3 \log \left (\frac {i c}{x}+1\right ) c+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{16} b^2 x^4 \log ^2\left (\frac {i c}{x}+1\right )-\frac {1}{4} i a b x^4 \log \left (\frac {i c}{x}+1\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (\frac {i c}{x}+1\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 29
Rule 30
Rule 31
Rule 36
Rule 43
Rule 44
Rule 193
Rule 260
Rule 263
Rule 2301
Rule 2314
Rule 2315
Rule 2316
Rule 2319
Rule 2344
Rule 2347
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2395
Rule 2398
Rule 2410
Rule 2411
Rule 2416
Rule 2448
Rule 2454
Rule 2455
Rule 2462
Rule 2466
Rule 2557
Rule 5035
Rule 6742
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{2} b x^3 \left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} b^2 x^3 \log ^2\left (1+\frac {i c}{x}\right )\right ) \, dx\\ &=\frac {1}{4} \int x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2 \, dx+\frac {1}{2} b \int x^3 \left (-2 i a+b \log \left (1-\frac {i c}{x}\right )\right ) \log \left (1+\frac {i c}{x}\right ) \, dx-\frac {1}{4} b^2 \int x^3 \log ^2\left (1+\frac {i c}{x}\right ) \, dx\\ &=-\left (\frac {1}{4} \operatorname {Subst}\left (\int \frac {(2 a+i b \log (1-i c x))^2}{x^5} \, dx,x,\frac {1}{x}\right )\right )+\frac {1}{2} b \int \left (-2 i a x^3 \log \left (1+\frac {i c}{x}\right )+b x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )\right ) \, dx+\frac {1}{4} b^2 \operatorname {Subst}\left (\int \frac {\log ^2(1+i c x)}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )-(i a b) \int x^3 \log \left (1+\frac {i c}{x}\right ) \, dx+\frac {1}{2} b^2 \int x^3 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right ) \, dx-\frac {1}{8} (b c) \operatorname {Subst}\left (\int \frac {2 a+i b \log (1-i c x)}{x^4 (1-i c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x^4 (1+i c x)} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{8} (i b) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^4} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{2} b^2 \int \frac {c x^3 \log \left (1-\frac {i c}{x}\right )}{4 (-c+i x)} \, dx-\frac {1}{2} b^2 \int \frac {c x^3 \log \left (1+\frac {i c}{x}\right )}{4 (-c-i x)} \, dx+\frac {1}{4} (a b c) \int \frac {x^2}{1+\frac {i c}{x}} \, dx+\frac {1}{8} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {\log (1+i c x)}{x^4}-\frac {i c \log (1+i c x)}{x^3}-\frac {c^2 \log (1+i c x)}{x^2}+\frac {i c^3 \log (1+i c x)}{x}-\frac {i c^4 \log (1+i c x)}{-i+c x}\right ) \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{8} (i b) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{\left (-\frac {i}{c}+\frac {i x}{c}\right )^4} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{8} (b c) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^3} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{4} (a b c) \int \frac {x^3}{i c+x} \, dx+\frac {1}{8} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x^4} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (b^2 c\right ) \int \frac {x^3 \log \left (1-\frac {i c}{x}\right )}{-c+i x} \, dx-\frac {1}{8} \left (b^2 c\right ) \int \frac {x^3 \log \left (1+\frac {i c}{x}\right )}{-c-i x} \, dx+\frac {1}{8} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x^3} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (i b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x^2} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{x} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (b^2 c^5\right ) \operatorname {Subst}\left (\int \frac {\log (1+i c x)}{-i+c x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{8} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{24} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{8} (b c) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{\left (-\frac {i}{c}+\frac {i x}{c}\right )^3} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{4} (a b c) \int \left (-c^2+\frac {c^3}{c-i x}-i c x+x^2\right ) \, dx-\frac {1}{24} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^3} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{8} \left (b^2 c\right ) \int \left (i c^2 \log \left (1-\frac {i c}{x}\right )-\frac {i c^3 \log \left (1-\frac {i c}{x}\right )}{c-i x}-c x \log \left (1-\frac {i c}{x}\right )-i x^2 \log \left (1-\frac {i c}{x}\right )\right ) \, dx-\frac {1}{8} \left (b^2 c\right ) \int \left (-i c^2 \log \left (1+\frac {i c}{x}\right )+\frac {i c^3 \log \left (1+\frac {i c}{x}\right )}{c+i x}-c x \log \left (1+\frac {i c}{x}\right )+i x^2 \log \left (1+\frac {i c}{x}\right )\right ) \, dx+\frac {1}{8} \left (i b c^2\right ) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{24} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x^3 (1+i c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (i b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x^2 (1+i c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x (1+i c x)} \, dx,x,\frac {1}{x}\right )+\frac {1}{8} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\frac {i c}{x}\right )\\ &=-\frac {1}{4} a b c^3 x-\frac {1}{8} i a b c^2 x^2+\frac {1}{12} a b c x^3+\frac {1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{8} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 c^2 x^2 \log \left (1+\frac {i c}{x}\right )-\frac {1}{24} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{16} b^2 c^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{4} i a b c^4 \log (c-i x)+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )-\frac {1}{24} \left (i b^2 c\right ) \operatorname {Subst}\left (\int \left (\frac {i c^3}{(-1+x)^3}-\frac {i c^3}{(-1+x)^2}+\frac {i c^3}{-1+x}-\frac {i c^3}{x}\right ) \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{8} \left (i b^2 c\right ) \int x^2 \log \left (1-\frac {i c}{x}\right ) \, dx-\frac {1}{8} \left (i b^2 c\right ) \int x^2 \log \left (1+\frac {i c}{x}\right ) \, dx+\frac {1}{8} \left (i b c^2\right ) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{\left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{24} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^3}-\frac {i c}{x^2}-\frac {c^2}{x}+\frac {c^3}{-i+c x}\right ) \, dx,x,\frac {1}{x}\right )+\frac {1}{16} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )^2} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{8} \left (b^2 c^2\right ) \int x \log \left (1-\frac {i c}{x}\right ) \, dx+\frac {1}{8} \left (b^2 c^2\right ) \int x \log \left (1+\frac {i c}{x}\right ) \, dx-\frac {1}{8} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{x \left (-\frac {i}{c}+\frac {i x}{c}\right )} \, dx,x,1-\frac {i c}{x}\right )+\frac {1}{16} \left (i b^2 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^2}-\frac {i c}{x}+\frac {i c^2}{-i+c x}\right ) \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (i b^2 c^3\right ) \int \log \left (1-\frac {i c}{x}\right ) \, dx+\frac {1}{8} \left (i b^2 c^3\right ) \int \log \left (1+\frac {i c}{x}\right ) \, dx+\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log \left (1-\frac {i c}{x}\right )}{c-i x} \, dx-\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log \left (1+\frac {i c}{x}\right )}{c+i x} \, dx+\frac {1}{8} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{8} \left (i b^2 c^5\right ) \operatorname {Subst}\left (\int \frac {1}{1+i c x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4} a b c^3 x-\frac {1}{16} i b^2 c^3 x-\frac {1}{8} i a b c^2 x^2+\frac {1}{24} b^2 c^2 x^2+\frac {1}{12} a b c x^3-\frac {11}{48} b^2 c^4 \log \left (i-\frac {c}{x}\right )-\frac {1}{24} b^2 c^4 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} i b^2 c^3 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{24} i b^2 c x^3 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} b c^3 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{4} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{16} b^2 c^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{4} i a b c^4 \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {13}{48} b^2 c^4 \log (x)+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x}{1-\frac {i c}{x}} \, dx+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x}{1+\frac {i c}{x}} \, dx+\frac {1}{16} \left (b^2 c^2\right ) \operatorname {Subst}\left (\int \left (-\frac {c^2}{(-1+x)^2}+\frac {c^2}{-1+x}-\frac {c^2}{x}\right ) \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{8} \left (b c^3\right ) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{16} \left (i b^2 c^3\right ) \int \frac {1}{1-\frac {i c}{x}} \, dx+\frac {1}{16} \left (i b^2 c^3\right ) \int \frac {1}{1+\frac {i c}{x}} \, dx+\frac {1}{8} \left (i b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{8} \left (i b c^4\right ) \operatorname {Subst}\left (\int \frac {2 a+i b \log (x)}{x} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {1}{\left (1-\frac {i c}{x}\right ) x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {1}{\left (1+\frac {i c}{x}\right ) x} \, dx+\frac {1}{8} \left (i b^2 c^5\right ) \int \frac {\log (c-i x)}{\left (1-\frac {i c}{x}\right ) x^2} \, dx-\frac {1}{8} \left (i b^2 c^5\right ) \int \frac {\log (c+i x)}{\left (1+\frac {i c}{x}\right ) x^2} \, dx\\ &=-\frac {1}{4} a b c^3 x-\frac {1}{8} i a b c^2 x^2+\frac {1}{24} b^2 c^2 x^2+\frac {1}{12} a b c x^3-\frac {11}{48} b^2 c^4 \log \left (i-\frac {c}{x}\right )-\frac {5}{48} b^2 c^4 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} i b^2 c^3 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{24} i b^2 c x^3 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} b c^3 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{4} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{16} b^2 c^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{4} i a b c^4 \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{4} i a b c^4 \log (x)-\frac {11}{24} b^2 c^4 \log (x)+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x^2}{-i c+x} \, dx+\frac {1}{24} \left (b^2 c^2\right ) \int \frac {x^2}{i c+x} \, dx-\frac {1}{16} \left (i b^2 c^3\right ) \int \frac {x}{-i c+x} \, dx+\frac {1}{16} \left (i b^2 c^3\right ) \int \frac {x}{i c+x} \, dx-\frac {1}{8} \left (i b^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{-\frac {i}{c}+\frac {i x}{c}} \, dx,x,1-\frac {i c}{x}\right )-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {1}{-i c+x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {1}{i c+x} \, dx+\frac {1}{8} \left (i b^2 c^5\right ) \int \left (\frac {\log (c-i x)}{c (c+i x)}+\frac {i \log (c-i x)}{c x}\right ) \, dx-\frac {1}{8} \left (i b^2 c^5\right ) \int \left (\frac {\log (c+i x)}{c (c-i x)}-\frac {i \log (c+i x)}{c x}\right ) \, dx\\ &=-\frac {1}{4} a b c^3 x-\frac {1}{8} i a b c^2 x^2+\frac {1}{24} b^2 c^2 x^2+\frac {1}{12} a b c x^3-\frac {11}{48} b^2 c^4 \log \left (i-\frac {c}{x}\right )-\frac {5}{48} b^2 c^4 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} i b^2 c^3 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{24} i b^2 c x^3 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} b c^3 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{4} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{16} b^2 c^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{4} i a b c^4 \log (c-i x)-\frac {1}{8} b^2 c^4 \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {1}{8} b^2 c^4 \log (c+i x)-\frac {1}{8} b^2 c^4 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)-\frac {1}{4} i a b c^4 \log (x)-\frac {11}{24} b^2 c^4 \log (x)+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {i c}{x}\right )+\frac {1}{24} \left (b^2 c^2\right ) \int \left (-i c+\frac {i c^2}{c-i x}+x\right ) \, dx+\frac {1}{24} \left (b^2 c^2\right ) \int \left (i c-\frac {i c^2}{c+i x}+x\right ) \, dx+\frac {1}{16} \left (i b^2 c^3\right ) \int \left (1-\frac {c}{c-i x}\right ) \, dx-\frac {1}{16} \left (i b^2 c^3\right ) \int \left (1-\frac {c}{c+i x}\right ) \, dx+\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log (c-i x)}{c+i x} \, dx-\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log (c+i x)}{c-i x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log (c-i x)}{x} \, dx-\frac {1}{8} \left (b^2 c^4\right ) \int \frac {\log (c+i x)}{x} \, dx\\ &=-\frac {1}{4} a b c^3 x-\frac {1}{8} i a b c^2 x^2+\frac {1}{12} b^2 c^2 x^2+\frac {1}{12} a b c x^3-\frac {11}{48} b^2 c^4 \log \left (i-\frac {c}{x}\right )-\frac {5}{48} b^2 c^4 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} i b^2 c^3 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{24} i b^2 c x^3 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} b c^3 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{4} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{16} b^2 c^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{4} i a b c^4 \log (c-i x)-\frac {5}{48} b^2 c^4 \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {5}{48} b^2 c^4 \log (c+i x)-\frac {1}{8} b^2 c^4 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)+\frac {1}{8} b^2 c^4 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)+\frac {1}{8} b^2 c^4 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{4} i a b c^4 \log (x)-\frac {11}{24} b^2 c^4 \log (x)-\frac {1}{8} b^2 c^4 \log (c+i x) \log \left (-\frac {i x}{c}\right )-\frac {1}{8} b^2 c^4 \log (c-i x) \log \left (\frac {i x}{c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log \left (\frac {c-i x}{2 c}\right )}{c+i x} \, dx+\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log \left (\frac {c+i x}{2 c}\right )}{c-i x} \, dx+\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log \left (-\frac {i x}{c}\right )}{c+i x} \, dx-\frac {1}{8} \left (i b^2 c^4\right ) \int \frac {\log \left (\frac {i x}{c}\right )}{c-i x} \, dx\\ &=-\frac {1}{4} a b c^3 x-\frac {1}{8} i a b c^2 x^2+\frac {1}{12} b^2 c^2 x^2+\frac {1}{12} a b c x^3-\frac {11}{48} b^2 c^4 \log \left (i-\frac {c}{x}\right )-\frac {5}{48} b^2 c^4 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} i b^2 c^3 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{24} i b^2 c x^3 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} b c^3 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{4} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{16} b^2 c^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{4} i a b c^4 \log (c-i x)-\frac {5}{48} b^2 c^4 \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {5}{48} b^2 c^4 \log (c+i x)-\frac {1}{8} b^2 c^4 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)+\frac {1}{8} b^2 c^4 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)+\frac {1}{8} b^2 c^4 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{4} i a b c^4 \log (x)-\frac {11}{24} b^2 c^4 \log (x)-\frac {1}{8} b^2 c^4 \log (c+i x) \log \left (-\frac {i x}{c}\right )-\frac {1}{8} b^2 c^4 \log (c-i x) \log \left (\frac {i x}{c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (1-\frac {i x}{c}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (1+\frac {i x}{c}\right )-\frac {1}{8} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c-i x\right )-\frac {1}{8} \left (b^2 c^4\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{2 c}\right )}{x} \, dx,x,c+i x\right )\\ &=-\frac {1}{4} a b c^3 x-\frac {1}{8} i a b c^2 x^2+\frac {1}{12} b^2 c^2 x^2+\frac {1}{12} a b c x^3-\frac {11}{48} b^2 c^4 \log \left (i-\frac {c}{x}\right )-\frac {5}{48} b^2 c^4 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} i b^2 c^3 x \log \left (1-\frac {i c}{x}\right )+\frac {1}{16} b^2 c^2 x^2 \log \left (1-\frac {i c}{x}\right )+\frac {1}{24} i b^2 c x^3 \log \left (1-\frac {i c}{x}\right )-\frac {1}{8} b c^3 \left (1-\frac {i c}{x}\right ) x \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{16} i b c^2 x^2 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )+\frac {1}{24} b c x^3 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )-\frac {1}{16} c^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{16} x^4 \left (2 a+i b \log \left (1-\frac {i c}{x}\right )\right )^2+\frac {1}{4} i b^2 c^3 x \log \left (1+\frac {i c}{x}\right )-\frac {1}{12} i b^2 c x^3 \log \left (1+\frac {i c}{x}\right )-\frac {1}{4} i a b x^4 \log \left (1+\frac {i c}{x}\right )+\frac {1}{8} b^2 x^4 \log \left (1-\frac {i c}{x}\right ) \log \left (1+\frac {i c}{x}\right )+\frac {1}{16} b^2 c^4 \log ^2\left (1+\frac {i c}{x}\right )-\frac {1}{16} b^2 x^4 \log ^2\left (1+\frac {i c}{x}\right )+\frac {1}{4} i a b c^4 \log (c-i x)-\frac {5}{48} b^2 c^4 \log (c-i x)-\frac {1}{8} b^2 c^4 \log \left (1-\frac {i c}{x}\right ) \log (c-i x)-\frac {5}{48} b^2 c^4 \log (c+i x)-\frac {1}{8} b^2 c^4 \log \left (1+\frac {i c}{x}\right ) \log (c+i x)+\frac {1}{8} b^2 c^4 \log \left (\frac {c-i x}{2 c}\right ) \log (c+i x)+\frac {1}{8} b^2 c^4 \log (c-i x) \log \left (\frac {c+i x}{2 c}\right )-\frac {1}{4} i a b c^4 \log (x)-\frac {11}{24} b^2 c^4 \log (x)-\frac {1}{8} b^2 c^4 \log (c+i x) \log \left (-\frac {i x}{c}\right )-\frac {1}{8} b^2 c^4 \log (c-i x) \log \left (\frac {i x}{c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c-i x}{2 c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {c+i x}{2 c}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (-\frac {i c}{x}\right )+\frac {1}{8} b^2 c^4 \text {Li}_2\left (\frac {i c}{x}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (1-\frac {i x}{c}\right )-\frac {1}{8} b^2 c^4 \text {Li}_2\left (1+\frac {i x}{c}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 111, normalized size = 0.91 \[ \frac {1}{12} \left (x \left (3 a^2 x^3+2 a b c \left (x^2-3 c^2\right )+b^2 c^2 x\right )+2 b \tan ^{-1}\left (\frac {c}{x}\right ) \left (3 a \left (x^4-c^4\right )+b c x \left (x^2-3 c^2\right )\right )+3 b^2 \left (x^4-c^4\right ) \tan ^{-1}\left (\frac {c}{x}\right )^2-4 b^2 c^4 \log \left (c^2+x^2\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 125, normalized size = 1.02 \[ \frac {1}{2} \, a b c^{4} \arctan \left (\frac {x}{c}\right ) - \frac {1}{3} \, b^{2} c^{4} \log \left (c^{2} + x^{2}\right ) - \frac {1}{2} \, a b c^{3} x + \frac {1}{12} \, b^{2} c^{2} x^{2} + \frac {1}{6} \, a b c x^{3} + \frac {1}{4} \, a^{2} x^{4} - \frac {1}{4} \, {\left (b^{2} c^{4} - b^{2} x^{4}\right )} \arctan \left (\frac {c}{x}\right )^{2} - \frac {1}{6} \, {\left (3 \, b^{2} c^{3} x - b^{2} c x^{3} - 3 \, a b x^{4}\right )} \arctan \left (\frac {c}{x}\right ) \]
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 \[ \int {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{2} x^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 157, normalized size = 1.29 \[ \frac {a^{2} x^{4}}{4}+\frac {b^{2} x^{4} \arctan \left (\frac {c}{x}\right )^{2}}{4}+\frac {c \,b^{2} \arctan \left (\frac {c}{x}\right ) x^{3}}{6}-\frac {c^{3} b^{2} \arctan \left (\frac {c}{x}\right ) x}{2}-\frac {c^{4} b^{2} \arctan \left (\frac {c}{x}\right )^{2}}{4}+\frac {b^{2} c^{2} x^{2}}{12}+\frac {2 c^{4} b^{2} \ln \left (\frac {c}{x}\right )}{3}-\frac {b^{2} c^{4} \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{3}+\frac {a b \,x^{4} \arctan \left (\frac {c}{x}\right )}{2}+\frac {a b c \,x^{3}}{6}-\frac {c^{3} a b x}{2}+\frac {c^{4} a b \arctan \left (\frac {x}{c}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 134, normalized size = 1.10 \[ \frac {1}{4} \, b^{2} x^{4} \arctan \left (\frac {c}{x}\right )^{2} + \frac {1}{4} \, a^{2} x^{4} + \frac {1}{6} \, {\left (3 \, x^{4} \arctan \left (\frac {c}{x}\right ) + {\left (3 \, c^{3} \arctan \left (\frac {x}{c}\right ) - 3 \, c^{2} x + x^{3}\right )} c\right )} a b + \frac {1}{12} \, {\left ({\left (3 \, c^{2} \arctan \left (\frac {x}{c}\right )^{2} - 4 \, c^{2} \log \left (c^{2} + x^{2}\right ) + x^{2}\right )} c^{2} + 2 \, {\left (3 \, c^{3} \arctan \left (\frac {x}{c}\right ) - 3 \, c^{2} x + x^{3}\right )} c \arctan \left (\frac {c}{x}\right )\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.47, size = 140, normalized size = 1.15 \[ \frac {a^2\,x^4}{4}-\frac {b^2\,c^4\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{4}-\frac {b^2\,c^4\,\ln \left (c^2+x^2\right )}{3}+\frac {b^2\,x^4\,{\mathrm {atan}\left (\frac {c}{x}\right )}^2}{4}+\frac {b^2\,c^2\,x^2}{12}+\frac {b^2\,c\,x^3\,\mathrm {atan}\left (\frac {c}{x}\right )}{6}-\frac {b^2\,c^3\,x\,\mathrm {atan}\left (\frac {c}{x}\right )}{2}+\frac {a\,b\,c\,x^3}{6}-\frac {a\,b\,c^3\,x}{2}-\frac {a\,b\,c^4\,\mathrm {atan}\left (\frac {c}{x}\right )}{2}+\frac {a\,b\,x^4\,\mathrm {atan}\left (\frac {c}{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 144, normalized size = 1.18 \[ \frac {a^{2} x^{4}}{4} - \frac {a b c^{4} \operatorname {atan}{\left (\frac {c}{x} \right )}}{2} - \frac {a b c^{3} x}{2} + \frac {a b c x^{3}}{6} + \frac {a b x^{4} \operatorname {atan}{\left (\frac {c}{x} \right )}}{2} - \frac {b^{2} c^{4} \log {\left (c^{2} + x^{2} \right )}}{3} - \frac {b^{2} c^{4} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{4} - \frac {b^{2} c^{3} x \operatorname {atan}{\left (\frac {c}{x} \right )}}{2} + \frac {b^{2} c^{2} x^{2}}{12} + \frac {b^{2} c x^{3} \operatorname {atan}{\left (\frac {c}{x} \right )}}{6} + \frac {b^{2} x^{4} \operatorname {atan}^{2}{\left (\frac {c}{x} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
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