Optimal. Leaf size=337 \[ \frac {3 i b^2 (d e-c f) \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}-\frac {3 b^2 f \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}-\frac {3 b (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}+\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^3 (d e-c f) \text {Li}_3\left (1-\frac {2}{i (c+d x)+1}\right )}{2 d^2}+\frac {3 i b^3 f \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{2 d^2} \]
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Rubi [A] time = 0.66, antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5048, 4865, 4847, 4921, 4855, 2402, 2315, 4985, 4885, 4995, 6610} \[ \frac {3 i b^2 (d e-c f) \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}-\frac {3 b^3 (d e-c f) \text {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^3 f \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}-\frac {3 b^2 f \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}-\frac {3 b (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}+\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 4847
Rule 4855
Rule 4865
Rule 4885
Rule 4921
Rule 4985
Rule 4995
Rule 5048
Rule 6610
Rubi steps
\begin {align*} \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (a+b \cot ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}+\frac {(3 b) \operatorname {Subst}\left (\int \left (\frac {f^2 \left (a+b \cot ^{-1}(x)\right )^2}{d^2}+\frac {((d e-f-c f) (d e+f-c f)+2 f (d e-c f) x) \left (a+b \cot ^{-1}(x)\right )^2}{d^2 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{2 f}\\ &=\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {((d e-f-c f) (d e+f-c f)+2 f (d e-c f) x) \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{2 d^2 f}+\frac {(3 b f) \operatorname {Subst}\left (\int \left (a+b \cot ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{2 d^2}\\ &=\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}+\frac {(3 b) \operatorname {Subst}\left (\int \left (\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2}-\frac {2 f (-d e+c f) x \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{2 d^2 f}+\frac {\left (3 b^2 f\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \cot ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {\left (3 b^2 f\right ) \operatorname {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d^2}+\frac {(3 b (d e-c f)) \operatorname {Subst}\left (\int \frac {x \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d^2}+\frac {(3 b (d e+f-c f) (d e-(1+c) f)) \operatorname {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{2 d^2 f}\\ &=\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {\left (3 b^3 f\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}-\frac {(3 b (d e-c f)) \operatorname {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{i-x} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {\left (3 i b^3 f\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\right )}{d^2}-\frac {\left (6 b^2 (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right ) \log \left (\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 i b^3 f \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {\left (3 i b^3 (d e-c f)\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{d^2}\\ &=\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 i b^3 f \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b^3 (d e-c f) \text {Li}_3\left (1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 630, normalized size = 1.87 \[ \frac {a^3 f (c+d x)^2+a^2 (c+d x) (-2 a c f+2 a d e+3 b f)+3 a^2 b (d e-c f) \log \left ((c+d x)^2+1\right )-3 a^2 b (c+d x) \cot ^{-1}(c+d x) (c f-d (2 e+f x))-3 a^2 b f \tan ^{-1}(c+d x)+6 a b^2 d e \left (i \text {Li}_2\left (e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )-6 a b^2 c f \left (i \text {Li}_2\left (e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )+6 a b^2 f \left (-\log \left (\frac {1}{(c+d x) \sqrt {\frac {1}{(c+d x)^2}+1}}\right )+\frac {1}{2} \left ((c+d x)^2+1\right ) \cot ^{-1}(c+d x)^2+(c+d x) \cot ^{-1}(c+d x)\right )+2 b^3 d e \left (-3 i \cot ^{-1}(c+d x) \text {Li}_2\left (e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \text {Li}_3\left (e^{-2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )+\frac {i \pi ^3}{8}\right )+b^3 f \left (3 i \left (\cot ^{-1}(c+d x)^2+\text {Li}_2\left (e^{2 i \cot ^{-1}(c+d x)}\right )\right )+\left ((c+d x)^2+1\right ) \cot ^{-1}(c+d x)^3+3 (c+d x) \cot ^{-1}(c+d x)^2-6 \cot ^{-1}(c+d x) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )-2 b^3 c f \left (-3 i \cot ^{-1}(c+d x) \text {Li}_2\left (e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \text {Li}_3\left (e^{-2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )+\frac {i \pi ^3}{8}\right )}{2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{3} f x + a^{3} e + {\left (b^{3} f x + b^{3} e\right )} \operatorname {arccot}\left (d x + c\right )^{3} + 3 \, {\left (a b^{2} f x + a b^{2} e\right )} \operatorname {arccot}\left (d x + c\right )^{2} + 3 \, {\left (a^{2} b f x + a^{2} b e\right )} \operatorname {arccot}\left (d x + c\right ), x\right ) \]
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 \[ \int {\left (f x + e\right )} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.20, size = 1570, normalized size = 4.66 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{16} \, b^{3} f x^{2} \arctan \left (1, d x + c\right )^{3} + \frac {1}{8} \, b^{3} e x \arctan \left (1, d x + c\right )^{3} + \frac {1}{2} \, a^{3} f x^{2} + \frac {3}{2} \, {\left (x^{2} \operatorname {arccot}\left (d x + c\right ) + d {\left (\frac {x}{d^{2}} + \frac {{\left (c^{2} - 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{3}} - \frac {c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} a^{2} b f + a^{3} e x + \frac {3 \, {\left (2 \, {\left (d x + c\right )} \operatorname {arccot}\left (d x + c\right ) + \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} a^{2} b e}{2 \, d} - \frac {3}{64} \, {\left (b^{3} f x^{2} \arctan \left (1, d x + c\right ) + 2 \, b^{3} e x \arctan \left (1, d x + c\right )\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + \int \frac {8 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} d^{2} f x^{3} + 4 \, {\left (2 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} d^{2} e + {\left (3 \, b^{3} \arctan \left (1, d x + c\right )^{2} + 4 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c\right )} d f\right )} x^{2} + 3 \, {\left (2 \, b^{3} d^{2} f x^{3} \arctan \left (1, d x + c\right ) + {\left (2 \, b^{3} d^{2} e \arctan \left (1, d x + c\right ) + {\left (4 \, b^{3} c \arctan \left (1, d x + c\right ) - b^{3}\right )} d f\right )} x^{2} + 2 \, {\left (b^{3} c^{2} \arctan \left (1, d x + c\right ) + b^{3} \arctan \left (1, d x + c\right )\right )} e + 2 \, {\left ({\left (2 \, b^{3} c \arctan \left (1, d x + c\right ) - b^{3}\right )} d e + {\left (b^{3} c^{2} \arctan \left (1, d x + c\right ) + b^{3} \arctan \left (1, d x + c\right )\right )} f\right )} x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + 8 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2} + {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c^{2}\right )} e + 8 \, {\left ({\left (3 \, b^{3} \arctan \left (1, d x + c\right )^{2} + 2 \, {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c\right )} d e + {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2} + {\left (7 \, b^{3} \arctan \left (1, d x + c\right )^{3} + 24 \, a b^{2} \arctan \left (1, d x + c\right )^{2}\right )} c^{2}\right )} f\right )} x + 12 \, {\left (b^{3} d^{2} f x^{3} \arctan \left (1, d x + c\right ) + 2 \, b^{3} c d e x \arctan \left (1, d x + c\right ) + {\left (2 \, b^{3} d^{2} e \arctan \left (1, d x + c\right ) + b^{3} c d f \arctan \left (1, d x + c\right )\right )} x^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{64 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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