Optimal. Leaf size=382 \[ \frac {i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}-\frac {2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac {2 a b f x (d e-c f)}{d^2}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac {i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}+\frac {2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac {b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac {b^2 f^2 x}{3 d^2} \]
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Rubi [A] time = 0.58, antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, 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 203
Rule 260
Rule 321
Rule 2315
Rule 2402
Rule 4847
Rule 4853
Rule 4855
Rule 4865
Rule 4885
Rule 4921
Rule 4985
Rule 5048
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*}
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Mathematica [A] time = 5.37, size = 665, normalized size = 1.74 \[ a^2 e^2 x+a^2 e f x^2+\frac {1}{3} a^2 f^2 x^3+\frac {a b \left (\left (\left (3 c^2-1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (c^2+2 c d x+d^2 x^2+1\right )-2 \left (c^3 f^2-3 c^2 d e f+3 c d^2 e^2-3 c f^2+3 d e f\right ) \tan ^{-1}(c+d x)+2 d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) \cot ^{-1}(c+d x)+d f x (-4 c f+6 d e+d f x)\right )}{3 d^3}+\frac {b^2 f^2 \left (4 i \left (3 c^2-1\right ) \text {Li}_2\left (e^{2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \left ((c+d x)^2+1\right ) \left (3 \left (c^2+1\right ) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right )-(c+d x) \sqrt {\frac {1}{(c+d x)^2}+1} \left ((c+d x)^2+1\right ) \left (\left (3 c^2-1\right ) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right ) \cos \left (3 \cot ^{-1}(c+d x)\right )+2 \left ((c+d x)^2+1\right ) \left (-i \cot ^{-1}(c+d x)^2 \left (\left (3 c^2-1\right ) \cos \left (2 \cot ^{-1}(c+d x)\right )-3 c^2-6 i c+1\right )+2 \cot ^{-1}(c+d x) \left (\left (1-3 c^2\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+\left (3 c^2-1\right ) \cos \left (2 \cot ^{-1}(c+d x)\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+1\right )-6 c \log \left (\frac {1}{(c+d x) \sqrt {\frac {1}{(c+d x)^2}+1}}\right ) \left (\cos \left (2 \cot ^{-1}(c+d x)\right )-1\right )\right )\right )}{12 d^3}+\frac {b^2 e f \left (\left (-c^2-2 i c+d^2 x^2+1\right ) \cot ^{-1}(c+d x)^2-2 i c \text {Li}_2\left (e^{2 i \cot ^{-1}(c+d x)}\right )-2 \log \left (\frac {1}{(c+d x) \sqrt {\frac {1}{(c+d x)^2}+1}}\right )+2 \cot ^{-1}(c+d x) \left (2 c \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+c+d x\right )\right )}{d^2}+\frac {b^2 e^2 \left (i \text {Li}_2\left (e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} f^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} + {\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \operatorname {arccot}\left (d x + c\right )^{2} + 2 \, {\left (a b f^{2} x^{2} + 2 \, a b e f x + a b e^{2}\right )} \operatorname {arccot}\left (d x + c\right ), x\right ) \]
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 \[ \int {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 1832, normalized size = 4.80 \[ \text {result too large to display} \]
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}{12} \, b^{2} f^{2} x^{3} \arctan \left (1, d x + c\right )^{2} + \frac {1}{4} \, b^{2} e f x^{2} \arctan \left (1, d x + c\right )^{2} + \frac {1}{3} \, a^{2} f^{2} x^{3} + \frac {1}{4} \, b^{2} e^{2} x \arctan \left (1, d x + c\right )^{2} + a^{2} e f x^{2} + 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 b e f + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arccot}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} - \frac {2 \, {\left (c^{3} - 3 \, c\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{4}} + \frac {{\left (3 \, c^{2} - 1\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{4}}\right )}\right )} a b f^{2} + a^{2} e^{2} x + \frac {{\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 b e^{2}}{d} - \frac {1}{48} \, {\left (b^{2} f^{2} x^{3} + 3 \, b^{2} e f x^{2} + 3 \, b^{2} e^{2} x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + \int \frac {36 \, b^{2} d^{2} f^{2} x^{4} \arctan \left (1, d x + c\right )^{2} + 8 \, {\left (9 \, b^{2} d^{2} e f \arctan \left (1, d x + c\right )^{2} + {\left (9 \, b^{2} c \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )\right )} d f^{2}\right )} x^{3} + 36 \, {\left (b^{2} c^{2} \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )^{2}\right )} e^{2} + 12 \, {\left (3 \, b^{2} d^{2} e^{2} \arctan \left (1, d x + c\right )^{2} + 2 \, {\left (6 \, b^{2} c \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )\right )} d e f + 3 \, {\left (b^{2} c^{2} \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )^{2}\right )} f^{2}\right )} x^{2} + 3 \, {\left (b^{2} d^{2} f^{2} x^{4} + 2 \, {\left (b^{2} d^{2} e f + b^{2} c d f^{2}\right )} x^{3} + {\left (b^{2} c^{2} + b^{2}\right )} e^{2} + {\left (b^{2} d^{2} e^{2} + 4 \, b^{2} c d e f + {\left (b^{2} c^{2} + b^{2}\right )} f^{2}\right )} x^{2} + 2 \, {\left (b^{2} c d e^{2} + {\left (b^{2} c^{2} + b^{2}\right )} e f\right )} x\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )^{2} + 24 \, {\left ({\left (3 \, b^{2} c \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )\right )} d e^{2} + 3 \, {\left (b^{2} c^{2} \arctan \left (1, d x + c\right )^{2} + b^{2} \arctan \left (1, d x + c\right )^{2}\right )} e f\right )} x + 4 \, {\left (b^{2} d^{2} f^{2} x^{4} + 3 \, b^{2} c d e^{2} x + {\left (3 \, b^{2} d^{2} e f + b^{2} c d f^{2}\right )} x^{3} + 3 \, {\left (b^{2} d^{2} e^{2} + b^{2} c d e f\right )} x^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{48 \, {\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 \[ \int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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