Optimal. Leaf size=565 \[ \frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \text {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \text {PolyLog}\left (3,1-\frac {2}{1+i (c+d x)}\right )}{2 d^3} \]
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Rubi [A]
time = 0.69, antiderivative size = 565, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 14, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used =
{5156, 4975, 4931, 5041, 4965, 2449, 2352, 4947, 5037, 266, 5005, 5105, 5115, 6745}
\begin {gather*} \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 ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}-\frac {6 b^2 f (d e-c f) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}+\frac {a b^2 f^2 x}{d^2}+\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 )^3}{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 )^3}{3 d^3 f}-\frac {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 )^2}{d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (c+d x) (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {b^3 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_3\left (1-\frac {2}{i (c+d x)+1}\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{i (c+d x)+1}\right )}{d^3}+\frac {b^3 f^2 \log \left ((c+d x)^2+1\right )}{2 d^3}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 2352
Rule 2449
Rule 4931
Rule 4947
Rule 4965
Rule 4975
Rule 5005
Rule 5037
Rule 5041
Rule 5105
Rule 5115
Rule 5156
Rule 6745
Rubi steps
\begin {align*} \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \cot ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {b \text {Subst}\left (\int \left (\frac {3 f^2 (d e-c f) \left (a+b \cot ^{-1}(x)\right )^2}{d^3}+\frac {f^3 x \left (a+b \cot ^{-1}(x)\right )^2}{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 )^2}{d^3 \left (1+x^2\right )}\right ) \, dx,x,c+d x\right )}{f}\\ &=\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {b \text {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 )^2}{1+x^2} \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b f^2\right ) \text {Subst}\left (\int x \left (a+b \cot ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}+\frac {(3 b f (d e-c f)) \text {Subst}\left (\int \left (a+b \cot ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d^3}\\ &=\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {b \text {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 )^2}{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 )^2}{1+x^2}\right ) \, dx,x,c+d x\right )}{d^3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \cot ^{-1}(x)\right )}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 b^2 f (d e-c 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^3}\\ &=\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{1+x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {a+b \cot ^{-1}(x)}{i-x} \, dx,x,c+d x\right )}{d^3}+\frac {\left (b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {x \left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (b (d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{1+x^2} \, dx,x,c+d x\right )}{d^3 f}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 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 )^3}{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 )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \cot ^{-1}(x) \, dx,x,c+d x\right )}{d^3}-\frac {\left (6 b^3 f (d e-c 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^3}-\frac {\left (b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\left (a+b \cot ^{-1}(x)\right )^2}{i-x} \, dx,x,c+d x\right )}{d^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 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 )^3}{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 )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {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 )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {\left (b^3 f^2\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,c+d x\right )}{d^3}+\frac {\left (6 i b^3 f (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i (c+d x)}\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 ) \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^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 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 )^3}{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 )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {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 )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\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 ) \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {\left (i b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right )\right ) \text {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^3}\\ &=\frac {a b^2 f^2 x}{d^2}+\frac {b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac {3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 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 )^3}{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 )^3}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac {6 b^2 f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {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 )^2 \log \left (\frac {2}{1+i (c+d x)}\right )}{d^3}+\frac {b^3 f^2 \log \left (1+(c+d x)^2\right )}{2 d^3}+\frac {3 i b^3 f (d e-c f) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\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 ) \left (a+b \cot ^{-1}(c+d x)\right ) \text {Li}_2\left (1-\frac {2}{1+i (c+d x)}\right )}{d^3}-\frac {b^3 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \text {Li}_3\left (1-\frac {2}{1+i (c+d x)}\right )}{2 d^3}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(2336\) vs. \(2(565)=1130\).
time = 9.74, size = 2336, normalized size = 4.13 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 52.64, size = 12407, normalized size = 21.96
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(12407\) |
default | \(\text {Expression too large to display}\) | \(12407\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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