Optimal. Leaf size=182 \[ \frac {3 b^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c d^2 (-c x+i)}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (-c x+i)}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{4 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}-\frac {3 i b^3}{4 c d^2 (-c x+i)}+\frac {3 i b^3 \tan ^{-1}(c x)}{4 c d^2} \]
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Rubi [A] time = 0.22, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4864, 4862, 627, 44, 203, 4884} \[ \frac {3 b^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c d^2 (-c x+i)}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (-c x+i)}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{4 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}-\frac {3 i b^3}{4 c d^2 (-c x+i)}+\frac {3 i b^3 \tan ^{-1}(c x)}{4 c d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 4862
Rule 4864
Rule 4884
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{(d+i c d x)^2} \, dx &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}-\frac {(3 i b) \int \left (-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d (-i+c x)^2}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 d \left (1+c^2 x^2\right )}\right ) \, dx}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}+\frac {(3 i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{2 d^2}-\frac {(3 i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}+\frac {\left (3 i b^2\right ) \int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}+\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{2 d^2}-\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac {3 b^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c d^2 (i-c x)}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{4 c d^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{2 d^2}\\ &=\frac {3 b^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c d^2 (i-c x)}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{4 c d^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}+\frac {\left (3 b^3\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{2 d^2}\\ &=\frac {3 b^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c d^2 (i-c x)}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{4 c d^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}+\frac {\left (3 b^3\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 d^2}\\ &=-\frac {3 i b^3}{4 c d^2 (i-c x)}+\frac {3 b^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c d^2 (i-c x)}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{4 c d^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}+\frac {\left (3 i b^3\right ) \int \frac {1}{1+c^2 x^2} \, dx}{4 d^2}\\ &=-\frac {3 i b^3}{4 c d^2 (i-c x)}+\frac {3 i b^3 \tan ^{-1}(c x)}{4 c d^2}+\frac {3 b^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c d^2 (i-c x)}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2}{4 c d^2}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d^2 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{2 c d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{c d^2 (1+i c x)}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 121, normalized size = 0.66 \[ \frac {4 a^3+3 i b \left (-2 a^2+2 i a b+b^2\right ) (c x+i) \tan ^{-1}(c x)-6 i a^2 b-3 b^2 (b+2 i a) (c x+i) \tan ^{-1}(c x)^2-6 a b^2+2 b^3 (1-i c x) \tan ^{-1}(c x)^3+3 i b^3}{4 c d^2 (c x-i)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 176, normalized size = 0.97 \[ -\frac {{\left (b^{3} c x + i \, b^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 16 \, a^{3} + 24 i \, a^{2} b + 24 \, a b^{2} - 12 i \, b^{3} + {\left (6 \, a b^{2} - 3 i \, b^{3} + 3 \, {\left (-2 i \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - {\left (12 i \, a^{2} b + 12 \, a b^{2} - 6 i \, b^{3} + 6 \, {\left (2 \, a^{2} b - 2 i \, a b^{2} - b^{3}\right )} c x\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{16 \, c^{2} d^{2} x - 16 i \, c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.89, size = 551, normalized size = 3.03 \[ \frac {3 i b^{3}}{4 c \,d^{2} \left (c x -i\right )}-\frac {3 i a^{2} b \arctan \left (c x \right )}{2 c \,d^{2}}-\frac {b^{3} \arctan \left (c x \right )^{3}}{2 c \,d^{2} \left (c x -i\right )}+\frac {i b^{3} \arctan \left (c x \right )^{3}}{c \,d^{2} \left (i c x +1\right )}-\frac {3 b^{3} \arctan \left (c x \right )^{2} x}{4 d^{2} \left (c x -i\right )}+\frac {3 i a \,b^{2} \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{4 c \,d^{2}}-\frac {3 b^{3} \arctan \left (c x \right )}{4 c \,d^{2} \left (c x -i\right )}+\frac {i a^{3}}{c \,d^{2} \left (i c x +1\right )}+\frac {3 i a^{2} b \arctan \left (c x \right )}{c \,d^{2} \left (i c x +1\right )}+\frac {3 i b^{3} \arctan \left (c x \right ) x}{4 d^{2} \left (c x -i\right )}+\frac {3 a \,b^{2} \arctan \left (c x \right ) \ln \left (c x +i\right )}{2 c \,d^{2}}-\frac {3 a \,b^{2} \arctan \left (c x \right ) \ln \left (c x -i\right )}{2 c \,d^{2}}-\frac {3 i b^{3} \arctan \left (c x \right )^{2}}{4 c \,d^{2} \left (c x -i\right )}-\frac {3 i a \,b^{2} \arctan \left (c x \right )}{c \,d^{2} \left (c x -i\right )}-\frac {3 i a \,b^{2} \ln \left (c x +i\right )^{2}}{8 c \,d^{2}}-\frac {3 a \,b^{2} \arctan \left (c x \right )}{2 c \,d^{2}}-\frac {3 a \,b^{2}}{2 c \,d^{2} \left (c x -i\right )}-\frac {3 i a^{2} b}{2 c \,d^{2} \left (c x -i\right )}-\frac {3 i a \,b^{2} \ln \left (-\frac {i \left (-c x +i\right )}{2}\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{4 c \,d^{2}}-\frac {i b^{3} \arctan \left (c x \right )^{3} x}{2 d^{2} \left (c x -i\right )}+\frac {3 i a \,b^{2} \ln \left (-\frac {i \left (-c x +i\right )}{2}\right ) \ln \left (c x +i\right )}{4 c \,d^{2}}+\frac {3 i a \,b^{2} \arctan \left (c x \right )^{2}}{c \,d^{2} \left (i c x +1\right )}-\frac {3 i a \,b^{2} \ln \left (c x -i\right )^{2}}{8 c \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 37.13, size = 627, normalized size = 3.45 \[ - \frac {3 i b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right ) \log {\left (- \frac {3 b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right )}{c} + x \left (- 6 a^{2} b + 6 i a b^{2} + 3 b^{3}\right ) \right )}}{8 c d^{2}} + \frac {3 i b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right ) \log {\left (\frac {3 b \left (a \left (1 - i\right ) - b\right ) \left (a \left (1 - i\right ) - i b\right )}{c} + x \left (- 6 a^{2} b + 6 i a b^{2} + 3 b^{3}\right ) \right )}}{8 c d^{2}} + \frac {\left (- i b^{3} c x + b^{3}\right ) \log {\left (- i c x + 1 \right )}^{3}}{16 i c^{2} d^{2} x + 16 c d^{2}} + \frac {\left (i b^{3} c x - b^{3}\right ) \log {\left (i c x + 1 \right )}^{3}}{16 i c^{2} d^{2} x + 16 c d^{2}} + \frac {\left (- 6 a b^{2} c x - 6 i a b^{2} + 3 i b^{3} c x \log {\left (i c x + 1 \right )} + 3 i b^{3} c x - 3 b^{3} \log {\left (i c x + 1 \right )} - 3 b^{3}\right ) \log {\left (- i c x + 1 \right )}^{2}}{16 i c^{2} d^{2} x + 16 c d^{2}} + \frac {\left (- 24 a^{2} b + 12 a b^{2} c x \log {\left (i c x + 1 \right )} + 12 i a b^{2} \log {\left (i c x + 1 \right )} + 24 i a b^{2} - 3 i b^{3} c x \log {\left (i c x + 1 \right )}^{2} - 6 i b^{3} c x \log {\left (i c x + 1 \right )} + 3 b^{3} \log {\left (i c x + 1 \right )}^{2} + 6 b^{3} \log {\left (i c x + 1 \right )} + 12 b^{3}\right ) \log {\left (- i c x + 1 \right )}}{16 i c^{2} d^{2} x + 16 c d^{2}} + \frac {\left (6 a^{2} b - 6 i a b^{2} - 3 b^{3}\right ) \log {\left (i c x + 1 \right )}}{4 i c^{2} d^{2} x + 4 c d^{2}} + \frac {\left (6 a b^{2} c x + 6 i a b^{2} - 3 i b^{3} c x + 3 b^{3}\right ) \log {\left (i c x + 1 \right )}^{2}}{- 16 i c^{2} d^{2} x - 16 c d^{2}} - \frac {- 4 a^{3} + 6 i a^{2} b + 6 a b^{2} - 3 i b^{3}}{4 c^{2} d^{2} x - 4 i c d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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