Optimal. Leaf size=100 \[ \frac {i \left (a+b \tan ^{-1}(c x)\right )}{3 c (1+i c x)^3}+\frac {i b}{24 c (-c x+i)}-\frac {b}{24 c (-c x+i)^2}-\frac {i b}{18 c (-c x+i)^3}-\frac {i b \tan ^{-1}(c x)}{24 c} \]
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Rubi [A] time = 0.05, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {4862, 627, 44, 203} \[ \frac {i \left (a+b \tan ^{-1}(c x)\right )}{3 c (1+i c x)^3}+\frac {i b}{24 c (-c x+i)}-\frac {b}{24 c (-c x+i)^2}-\frac {i b}{18 c (-c x+i)^3}-\frac {i b \tan ^{-1}(c x)}{24 c} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 4862
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{(1+i c x)^4} \, dx &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{3 c (1+i c x)^3}-\frac {1}{3} (i b) \int \frac {1}{(1+i c x)^3 \left (1+c^2 x^2\right )} \, dx\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{3 c (1+i c x)^3}-\frac {1}{3} (i b) \int \frac {1}{(1-i c x) (1+i c x)^4} \, dx\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{3 c (1+i c x)^3}-\frac {1}{3} (i b) \int \left (\frac {1}{2 (-i+c x)^4}+\frac {i}{4 (-i+c x)^3}-\frac {1}{8 (-i+c x)^2}+\frac {1}{8 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac {i b}{18 c (i-c x)^3}-\frac {b}{24 c (i-c x)^2}+\frac {i b}{24 c (i-c x)}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{3 c (1+i c x)^3}-\frac {1}{24} (i b) \int \frac {1}{1+c^2 x^2} \, dx\\ &=-\frac {i b}{18 c (i-c x)^3}-\frac {b}{24 c (i-c x)^2}+\frac {i b}{24 c (i-c x)}-\frac {i b \tan ^{-1}(c x)}{24 c}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{3 c (1+i c x)^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 73, normalized size = 0.73 \[ \frac {-24 a+b \left (-3 i c^2 x^2-9 c x+10 i\right )+3 b \left (-i c^3 x^3-3 c^2 x^2+3 i c x-7\right ) \tan ^{-1}(c x)}{72 c (c x-i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 93, normalized size = 0.93 \[ \frac {-6 i \, b c^{2} x^{2} - 18 \, b c x + {\left (3 \, b c^{3} x^{3} - 9 i \, b c^{2} x^{2} - 9 \, b c x - 21 i \, b\right )} \log \left (-\frac {c x + i}{c x - i}\right ) - 48 \, a + 20 i \, b}{144 \, c^{4} x^{3} - 432 i \, c^{3} x^{2} - 432 \, c^{2} x + 144 i \, c} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 93, normalized size = 0.93 \[ \frac {i a}{3 c \left (i c x +1\right )^{3}}+\frac {i b \arctan \left (c x \right )}{3 c \left (i c x +1\right )^{3}}-\frac {i b \arctan \left (c x \right )}{24 c}-\frac {b}{24 c \left (c x -i\right )^{2}}+\frac {i b}{18 c \left (c x -i\right )^{3}}-\frac {i b}{24 c \left (c x -i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 83, normalized size = 0.83 \[ -\frac {3 i \, b c^{2} x^{2} + 9 \, b c x + {\left (3 i \, b c^{3} x^{3} + 9 \, b c^{2} x^{2} - 9 i \, b c x + 21 \, b\right )} \arctan \left (c x\right ) + 24 \, a - 10 i \, b}{72 \, c^{4} x^{3} - 216 i \, c^{3} x^{2} - 216 \, c^{2} x + 72 i \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{{\left (1+c\,x\,1{}\mathrm {i}\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 4.27, size = 168, normalized size = 1.68 \[ - \frac {i b \log {\left (- i c x + 1 \right )}}{6 c^{4} x^{3} - 18 i c^{3} x^{2} - 18 c^{2} x + 6 i c} + \frac {i b \log {\left (i c x + 1 \right )}}{6 c^{4} x^{3} - 18 i c^{3} x^{2} - 18 c^{2} x + 6 i c} + \frac {b \left (- \frac {\log {\left (b x - \frac {i b}{c} \right )}}{48} + \frac {\log {\left (b x + \frac {i b}{c} \right )}}{48}\right )}{c} + \frac {24 a + 3 i b c^{2} x^{2} + 9 b c x - 10 i b}{- 72 c^{4} x^{3} + 216 i c^{3} x^{2} + 216 c^{2} x - 72 i c} \]
Verification of antiderivative is not currently implemented for this CAS.
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