Optimal. Leaf size=382 \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{3 d^3}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}-\frac{2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]
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Rubi [A] time = 0.582392, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {5048, 4865, 4847, 260, 4853, 321, 203, 4985, 4885, 4921, 4855, 2402, 2315} \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{3 d^3}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}-\frac{2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]
Antiderivative was successfully verified.
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Rule 5048
Rule 4865
Rule 4847
Rule 260
Rule 4853
Rule 321
Rule 203
Rule 4985
Rule 4885
Rule 4921
Rule 4855
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^2 \left (a+b \cot ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac{(2 b) \operatorname{Subst}\left (\int \left (\frac{3 f^2 (d e-c f) \left (a+b \cot ^{-1}(x)\right )}{d^3}+\frac{f^3 x \left (a+b \cot ^{-1}(x)\right )}{d^3}+\frac{\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) \left (a+b \cot ^{-1}(x)\right )}{d^3 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f}\\ &=\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\left ((d e-c f) \left (d^2 e^2-2 c d e f-3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x\right ) \left (a+b \cot ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{3 d^3 f}+\frac{\left (2 b f^2\right ) \operatorname{Subst}\left (\int x \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d^3}+\frac{(2 b f (d e-c f)) \operatorname{Subst}\left (\int \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{2 a b f (d e-c f) x}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac{(2 b) \operatorname{Subst}\left (\int \left (\frac{(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(x)\right )}{1+x^2}+\frac{f \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) x \left (a+b \cot ^{-1}(x)\right )}{1+x^2}\right ) \, dx,x,c+d x\right )}{3 d^3 f}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{x^2}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \cot ^{-1}(x) \, dx,x,c+d x\right )}{d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac{\left (2 b^2 f (d e-c f)\right ) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac{\left (2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{x \left (a+b \cot ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac{\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{a+b \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{3 d^3 f}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}-\frac{\left (2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{a+b \cot ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}-\frac{2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+i (c+d x)}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}-\frac{\left (2 b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1+i x}\right )}{1+x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}-\frac{2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+i (c+d x)}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac{\left (2 i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i (c+d x)}\right )}{3 d^3}\\ &=\frac{b^2 f^2 x}{3 d^2}+\frac{2 a b f (d e-c f) x}{d^2}+\frac{2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}-\frac{2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac{2}{1+i (c+d x)}\right )}{3 d^3}+\frac{b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac{i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \text{Li}_2\left (1-\frac{2}{1+i (c+d x)}\right )}{3 d^3}\\ \end{align*}
Mathematica [A] time = 4.98049, size = 665, normalized size = 1.74 \[ \frac{b^2 e f \left (-2 i c \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+\left (-c^2-2 i c+d^2 x^2+1\right ) \cot ^{-1}(c+d x)^2-2 \log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right )+2 \cot ^{-1}(c+d x) \left (2 c \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+c+d x\right )\right )}{d^2}+\frac{b^2 f^2 \left (4 i \left (3 c^2-1\right ) \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \left ((c+d x)^2+1\right ) \left (3 \left (c^2+1\right ) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right )-(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left ((c+d x)^2+1\right ) \left (\left (3 c^2-1\right ) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right ) \cos \left (3 \cot ^{-1}(c+d x)\right )+2 \left ((c+d x)^2+1\right ) \left (-i \cot ^{-1}(c+d x)^2 \left (\left (3 c^2-1\right ) \cos \left (2 \cot ^{-1}(c+d x)\right )-3 c^2-6 i c+1\right )+2 \cot ^{-1}(c+d x) \left (\left (1-3 c^2\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+\left (3 c^2-1\right ) \cos \left (2 \cot ^{-1}(c+d x)\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+1\right )-6 c \log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right ) \left (\cos \left (2 \cot ^{-1}(c+d x)\right )-1\right )\right )\right )}{12 d^3}+\frac{b^2 e^2 \left (i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )}{d}+a^2 e^2 x+a^2 e f x^2+\frac{1}{3} a^2 f^2 x^3+\frac{a b \left (\left (\left (3 c^2-1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (c^2+2 c d x+d^2 x^2+1\right )-2 \left (-3 c^2 d e f+c^3 f^2+3 c d^2 e^2-3 c f^2+3 d e f\right ) \tan ^{-1}(c+d x)+2 d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) \cot ^{-1}(c+d x)+d f x (-4 c f+6 d e+d f x)\right )}{3 d^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.153, size = 1832, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (a^{2} f^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} +{\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \operatorname{arccot}\left (d x + c\right )^{2} + 2 \,{\left (a b f^{2} x^{2} + 2 \, a b e f x + a b e^{2}\right )} \operatorname{arccot}\left (d x + c\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acot}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (f x + e\right )}^{2}{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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