Optimal. Leaf size=63 \[ -\frac {b}{2 d e^3 (c+d x)}-\frac {b \text {ArcTan}(c+d x)}{2 d e^3}-\frac {a+b \text {ArcTan}(c+d x)}{2 d e^3 (c+d x)^2} \]
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Rubi [A]
time = 0.03, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5151, 12, 4946,
331, 209} \begin {gather*} -\frac {a+b \text {ArcTan}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b \text {ArcTan}(c+d x)}{2 d e^3}-\frac {b}{2 d e^3 (c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 209
Rule 331
Rule 4946
Rule 5151
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c+d x)}{(c e+d e x)^3} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{e^3 x^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \tan ^{-1}(x)}{x^3} \, dx,x,c+d x\right )}{d e^3}\\ &=-\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}+\frac {b \text {Subst}\left (\int \frac {1}{x^2 \left (1+x^2\right )} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b}{2 d e^3 (c+d x)}-\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}-\frac {b \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,c+d x\right )}{2 d e^3}\\ &=-\frac {b}{2 d e^3 (c+d x)}-\frac {b \tan ^{-1}(c+d x)}{2 d e^3}-\frac {a+b \tan ^{-1}(c+d x)}{2 d e^3 (c+d x)^2}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.02, size = 51, normalized size = 0.81 \begin {gather*} -\frac {a+b \text {ArcTan}(c+d x)+b (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-(c+d x)^2\right )}{2 d e^3 (c+d x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 63, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b \arctan \left (d x +c \right )}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b}{2 e^{3} \left (d x +c \right )}-\frac {b \arctan \left (d x +c \right )}{2 e^{3}}}{d}\) | \(63\) |
default | \(\frac {-\frac {a}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b \arctan \left (d x +c \right )}{2 e^{3} \left (d x +c \right )^{2}}-\frac {b}{2 e^{3} \left (d x +c \right )}-\frac {b \arctan \left (d x +c \right )}{2 e^{3}}}{d}\) | \(63\) |
risch | \(\frac {i b \ln \left (1+i \left (d x +c \right )\right )}{4 d \,e^{3} \left (d x +c \right )^{2}}-\frac {i \ln \left (-d x -c -i\right ) b \,d^{2} x^{2}-i \ln \left (-d x -c +i\right ) b \,d^{2} x^{2}+2 i \ln \left (-d x -c -i\right ) b c d x -2 i \ln \left (-d x -c +i\right ) b c d x +i \ln \left (-d x -c -i\right ) b \,c^{2}-i \ln \left (-d x -c +i\right ) b \,c^{2}+i b \ln \left (1-i \left (d x +c \right )\right )+2 b d x +2 b c +2 a}{4 e^{3} \left (d x +c \right )^{2} d}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs.
\(2 (54) = 108\).
time = 0.51, size = 111, normalized size = 1.76 \begin {gather*} -\frac {1}{2} \, {\left (d {\left (\frac {\arctan \left (\frac {d^{2} x + c d}{d}\right ) e^{\left (-3\right )}}{d^{2}} + \frac {1}{d^{3} x e^{3} + c d^{2} e^{3}}\right )} + \frac {\arctan \left (d x + c\right )}{d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}}\right )} b - \frac {a}{2 \, {\left (d^{3} x^{2} e^{3} + 2 \, c d^{2} x e^{3} + c^{2} d e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.47, size = 63, normalized size = 1.00 \begin {gather*} -\frac {{\left (b d x + b c + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + b\right )} \arctan \left (d x + c\right ) + a\right )} e^{\left (-3\right )}}{2 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs.
\(2 (54) = 108\).
time = 3.72, size = 314, normalized size = 4.98 \begin {gather*} \begin {cases} - \frac {a}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b c^{2} \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {2 b c d x \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b c}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b d^{2} x^{2} \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b d x}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} - \frac {b \operatorname {atan}{\left (c + d x \right )}}{2 c^{2} d e^{3} + 4 c d^{2} e^{3} x + 2 d^{3} e^{3} x^{2}} & \text {for}\: d \neq 0 \\\frac {x \left (a + b \operatorname {atan}{\left (c \right )}\right )}{c^{3} e^{3}} & \text {otherwise} \end {cases} \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 [B]
time = 0.75, size = 103, normalized size = 1.63 \begin {gather*} -\frac {\frac {a+b\,c}{d}+b\,x}{2\,c^2\,e^3+4\,c\,d\,e^3\,x+2\,d^2\,e^3\,x^2}-\frac {b\,\mathrm {atan}\left (\frac {b\,c+b\,d\,x}{b}\right )}{2\,d\,e^3}-\frac {b\,\mathrm {atan}\left (c+d\,x\right )}{2\,d^3\,e^3\,\left (x^2+\frac {c^2}{d^2}+\frac {2\,c\,x}{d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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