Integrand size = 18, antiderivative size = 337 \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(d e+f-c f) (d e-(1+c) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2}+\frac {3 i b^2 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2} \]
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Time = 0.47 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5156, 4975, 4931, 5041, 4965, 2449, 2352, 5105, 5005, 5115, 6745} \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {3 i b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}-\frac {3 b^2 f \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac {i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac {(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}-\frac {3 b (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}+\frac {3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{i (c+d x)+1}\right )}{2 d^2}+\frac {3 i b^3 f \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )}{2 d^2} \]
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Rule 2352
Rule 2449
Rule 4931
Rule 4965
Rule 4975
Rule 5005
Rule 5041
Rule 5105
Rule 5115
Rule 5156
Rule 6745
Rubi steps \begin{align*} \text {integral}& = \frac {\text {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) \text {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) \text {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) \text {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) \text {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 ) \text {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 ) \text {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)) \text {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)) \text {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 ) \text {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)) \text {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 ) \text {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 ) \text {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 \operatorname {PolyLog}\left (2,1-\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 ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}+\frac {\left (3 i b^3 (d e-c f)\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (2,1-\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 \operatorname {PolyLog}\left (2,1-\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 ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^2}-\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^2} \\ \end{align*}
Time = 5.42 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.87 \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\frac {a^2 (2 a d e+3 b f-2 a c f) (c+d x)+a^3 f (c+d x)^2-3 a^2 b (c+d x) (c f-d (2 e+f x)) \cot ^{-1}(c+d x)-3 a^2 b f \arctan (c+d x)+6 a b^2 f \left ((c+d x) \cot ^{-1}(c+d x)+\frac {1}{2} \left (1+(c+d x)^2\right ) \cot ^{-1}(c+d x)^2-\log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )\right )+3 a^2 b (d e-c f) \log \left (1+(c+d x)^2\right )+6 a b^2 d e \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )-6 a b^2 c f \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )+b^3 f \left (3 (c+d x) \cot ^{-1}(c+d x)^2+\left (1+(c+d x)^2\right ) \cot ^{-1}(c+d x)^3-6 \cot ^{-1}(c+d x) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+3 i \left (\cot ^{-1}(c+d x)^2+\operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )+2 b^3 d e \left (\frac {i \pi ^3}{8}-i \cot ^{-1}(c+d x)^3+(c+d x) \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )-3 i \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )\right )-2 b^3 c f \left (\frac {i \pi ^3}{8}-i \cot ^{-1}(c+d x)^3+(c+d x) \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )-3 i \cot ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac {3}{2} \operatorname {PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )\right )}{2 d^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (316 ) = 632\).
Time = 13.32 (sec) , antiderivative size = 1051, normalized size of antiderivative = 3.12
method | result | size |
parts | \(\text {Expression too large to display}\) | \(1051\) |
derivativedivides | \(\text {Expression too large to display}\) | \(17316\) |
default | \(\text {Expression too large to display}\) | \(17316\) |
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\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )\, dx \]
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\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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\[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
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Timed out. \[ \int (e+f x) \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \]
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