Optimal. Leaf size=106 \[ -\frac {\left (1-6 a^2\right ) x}{4 b^3}-\frac {a (a+b x)^2}{2 b^4}+\frac {(a+b x)^3}{12 b^4}+\frac {1}{4} x^4 \cot ^{-1}(a+b x)+\frac {\left (1-6 a^2+a^4\right ) \text {ArcTan}(a+b x)}{4 b^4}+\frac {a \left (1-a^2\right ) \log \left (1+(a+b x)^2\right )}{2 b^4} \]
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Rubi [A]
time = 0.08, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5156, 4973,
716, 649, 209, 266} \begin {gather*} \frac {a \left (1-a^2\right ) \log \left ((a+b x)^2+1\right )}{2 b^4}-\frac {\left (1-6 a^2\right ) x}{4 b^3}+\frac {\left (a^4-6 a^2+1\right ) \text {ArcTan}(a+b x)}{4 b^4}+\frac {(a+b x)^3}{12 b^4}-\frac {a (a+b x)^2}{2 b^4}+\frac {1}{4} x^4 \cot ^{-1}(a+b x) \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 716
Rule 4973
Rule 5156
Rubi steps
\begin {align*} \int x^3 \cot ^{-1}(a+b x) \, dx &=\frac {\text {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^3 \cot ^{-1}(x) \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{4} x^4 \cot ^{-1}(a+b x)+\frac {1}{4} \text {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^4}{1+x^2} \, dx,x,a+b x\right )\\ &=\frac {1}{4} x^4 \cot ^{-1}(a+b x)+\frac {1}{4} \text {Subst}\left (\int \left (-\frac {1-6 a^2}{b^4}-\frac {4 a x}{b^4}+\frac {x^2}{b^4}+\frac {1-6 a^2+a^4+4 a \left (1-a^2\right ) x}{b^4 \left (1+x^2\right )}\right ) \, dx,x,a+b x\right )\\ &=-\frac {\left (1-6 a^2\right ) x}{4 b^3}-\frac {a (a+b x)^2}{2 b^4}+\frac {(a+b x)^3}{12 b^4}+\frac {1}{4} x^4 \cot ^{-1}(a+b x)+\frac {\text {Subst}\left (\int \frac {1-6 a^2+a^4+4 a \left (1-a^2\right ) x}{1+x^2} \, dx,x,a+b x\right )}{4 b^4}\\ &=-\frac {\left (1-6 a^2\right ) x}{4 b^3}-\frac {a (a+b x)^2}{2 b^4}+\frac {(a+b x)^3}{12 b^4}+\frac {1}{4} x^4 \cot ^{-1}(a+b x)+\frac {\left (a \left (1-a^2\right )\right ) \text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x\right )}{b^4}+\frac {\left (1-6 a^2+a^4\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,a+b x\right )}{4 b^4}\\ &=-\frac {\left (1-6 a^2\right ) x}{4 b^3}-\frac {a (a+b x)^2}{2 b^4}+\frac {(a+b x)^3}{12 b^4}+\frac {1}{4} x^4 \cot ^{-1}(a+b x)+\frac {\left (1-6 a^2+a^4\right ) \tan ^{-1}(a+b x)}{4 b^4}+\frac {a \left (1-a^2\right ) \log \left (1+(a+b x)^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.05, size = 95, normalized size = 0.90 \begin {gather*} \frac {6 \left (-1+6 a^2\right ) b x-12 a (a+b x)^2+2 (a+b x)^3+6 b^4 x^4 \cot ^{-1}(a+b x)-3 i (-i+a)^4 \log (i-a-b x)+3 i (i+a)^4 \log (i+a+b x)}{24 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 157, normalized size = 1.48
method | result | size |
derivativedivides | \(\frac {\frac {\mathrm {arccot}\left (b x +a \right ) a^{4}}{4}-\mathrm {arccot}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\mathrm {arccot}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\mathrm {arccot}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\mathrm {arccot}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {3 a^{2} \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2} a}{2}+\frac {\left (b x +a \right )^{3}}{12}-\frac {b x}{4}-\frac {a}{4}+\frac {\left (-4 a^{3}+4 a \right ) \ln \left (1+\left (b x +a \right )^{2}\right )}{8}+\frac {\left (a^{4}-6 a^{2}+1\right ) \arctan \left (b x +a \right )}{4}}{b^{4}}\) | \(157\) |
default | \(\frac {\frac {\mathrm {arccot}\left (b x +a \right ) a^{4}}{4}-\mathrm {arccot}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\mathrm {arccot}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\mathrm {arccot}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\mathrm {arccot}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}+\frac {3 a^{2} \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2} a}{2}+\frac {\left (b x +a \right )^{3}}{12}-\frac {b x}{4}-\frac {a}{4}+\frac {\left (-4 a^{3}+4 a \right ) \ln \left (1+\left (b x +a \right )^{2}\right )}{8}+\frac {\left (a^{4}-6 a^{2}+1\right ) \arctan \left (b x +a \right )}{4}}{b^{4}}\) | \(157\) |
risch | \(\frac {i x^{4} \ln \left (1+i \left (b x +a \right )\right )}{8}-\frac {i x^{4} \ln \left (1-i \left (b x +a \right )\right )}{8}+\frac {\pi \,x^{4}}{8}+\frac {x^{3}}{12 b}+\frac {a^{4} \arctan \left (b x +a \right )}{4 b^{4}}-\frac {a \,x^{2}}{4 b^{2}}-\frac {a^{3} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{4}}+\frac {3 x \,a^{2}}{4 b^{3}}-\frac {3 a^{2} \arctan \left (b x +a \right )}{2 b^{4}}+\frac {a \ln \left (b^{2} x^{2}+2 a b x +a^{2}+1\right )}{2 b^{4}}-\frac {x}{4 b^{3}}+\frac {\arctan \left (b x +a \right )}{4 b^{4}}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 104, normalized size = 0.98 \begin {gather*} \frac {1}{4} \, x^{4} \operatorname {arccot}\left (b x + a\right ) + \frac {1}{12} \, b {\left (\frac {b^{2} x^{3} - 3 \, a b x^{2} + 3 \, {\left (3 \, a^{2} - 1\right )} x}{b^{4}} + \frac {3 \, {\left (a^{4} - 6 \, a^{2} + 1\right )} \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b^{5}} - \frac {6 \, {\left (a^{3} - a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{b^{5}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.56, size = 92, normalized size = 0.87 \begin {gather*} \frac {3 \, b^{4} x^{4} \operatorname {arccot}\left (b x + a\right ) + b^{3} x^{3} - 3 \, a b^{2} x^{2} + 3 \, {\left (3 \, a^{2} - 1\right )} b x + 3 \, {\left (a^{4} - 6 \, a^{2} + 1\right )} \arctan \left (b x + a\right ) - 6 \, {\left (a^{3} - a\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{12 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.75, size = 155, normalized size = 1.46 \begin {gather*} \begin {cases} - \frac {a^{4} \operatorname {acot}{\left (a + b x \right )}}{4 b^{4}} - \frac {a^{3} \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{4}} + \frac {3 a^{2} x}{4 b^{3}} + \frac {3 a^{2} \operatorname {acot}{\left (a + b x \right )}}{2 b^{4}} - \frac {a x^{2}}{4 b^{2}} + \frac {a \log {\left (a^{2} + 2 a b x + b^{2} x^{2} + 1 \right )}}{2 b^{4}} + \frac {x^{4} \operatorname {acot}{\left (a + b x \right )}}{4} + \frac {x^{3}}{12 b} - \frac {x}{4 b^{3}} - \frac {\operatorname {acot}{\left (a + b x \right )}}{4 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \operatorname {acot}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 617 vs.
\(2 (92) = 184\).
time = 0.79, size = 617, normalized size = 5.82 \begin {gather*} \frac {96 \, a^{3} \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{5} + 72 \, a^{2} \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{6} + 24 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{7} + 3 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{8} + 96 \, a^{3} \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 96 \, a^{3} \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + 144 \, a^{2} \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 144 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{5} - 72 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{5} - 24 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{6} - 12 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{6} - 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{7} - 96 \, a \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2}}{\tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 72 \, a^{2} \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} + 144 \, a^{2} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + 72 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} - 48 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} - 30 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4} + 30 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{5} - 24 \, a \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right ) - 24 \, a \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 12 \, \arctan \left (\frac {1}{b x + a}\right ) \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{2} - 30 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{3} + 3 \, \arctan \left (\frac {1}{b x + a}\right ) + 2 \, \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )}{192 \, b^{4} \tan \left (\frac {1}{2} \, \arctan \left (\frac {1}{b x + a}\right )\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.77, size = 133, normalized size = 1.25 \begin {gather*} \frac {\mathrm {atan}\left (a+b\,x\right )}{4\,b^4}+\frac {x^4\,\mathrm {acot}\left (a+b\,x\right )}{4}-\frac {x}{4\,b^3}+\frac {x^3}{12\,b}-\frac {a^3\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2\,b^4}-\frac {3\,a^2\,\mathrm {atan}\left (a+b\,x\right )}{2\,b^4}+\frac {a^4\,\mathrm {atan}\left (a+b\,x\right )}{4\,b^4}-\frac {a\,x^2}{4\,b^2}+\frac {3\,a^2\,x}{4\,b^3}+\frac {a\,\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}{2\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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