Optimal. Leaf size=146 \[ -\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {b c^2 (c d-e) (c d+e) \text {ArcTan}(c x)}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {a+b \text {ArcTan}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c^3 d \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2} \]
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Rubi [A]
time = 0.09, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4972, 724, 815,
649, 209, 266} \begin {gather*} -\frac {a+b \text {ArcTan}(c x)}{2 e (d+e x)^2}+\frac {b c^2 \text {ArcTan}(c x) (c d-e) (c d+e)}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {b c^3 d \log \left (c^2 x^2+1\right )}{2 \left (c^2 d^2+e^2\right )^2}+\frac {b c^3 d \log (d+e x)}{\left (c^2 d^2+e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 266
Rule 649
Rule 724
Rule 815
Rule 4972
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{(d+e x)^3} \, dx &=-\frac {a+b \tan ^{-1}(c x)}{2 e (d+e x)^2}+\frac {(b c) \int \frac {1}{(d+e x)^2 \left (1+c^2 x^2\right )} \, dx}{2 e}\\ &=-\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (b c^3\right ) \int \frac {d-e x}{(d+e x) \left (1+c^2 x^2\right )} \, dx}{2 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{2 e (d+e x)^2}+\frac {\left (b c^3\right ) \int \left (\frac {2 d e^2}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac {c^2 d^2-e^2-2 c^2 d e x}{\left (c^2 d^2+e^2\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{2 e \left (c^2 d^2+e^2\right )}\\ &=-\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b c^3\right ) \int \frac {c^2 d^2-e^2-2 c^2 d e x}{1+c^2 x^2} \, dx}{2 e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}-\frac {a+b \tan ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}-\frac {\left (b c^5 d\right ) \int \frac {x}{1+c^2 x^2} \, dx}{\left (c^2 d^2+e^2\right )^2}+\frac {\left (b c^3 (c d-e) (c d+e)\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 e \left (c^2 d^2+e^2\right )^2}\\ &=-\frac {b c}{2 \left (c^2 d^2+e^2\right ) (d+e x)}+\frac {b c^2 (c d-e) (c d+e) \tan ^{-1}(c x)}{2 e \left (c^2 d^2+e^2\right )^2}-\frac {a+b \tan ^{-1}(c x)}{2 e (d+e x)^2}+\frac {b c^3 d \log (d+e x)}{\left (c^2 d^2+e^2\right )^2}-\frac {b c^3 d \log \left (1+c^2 x^2\right )}{2 \left (c^2 d^2+e^2\right )^2}\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 192, normalized size = 1.32 \begin {gather*} -\frac {2 (a+b \text {ArcTan}(c x))+\frac {b c (d+e x) \left (2 e \left (c^2 d^2+e^2\right )-\left (c^2 d \left (\sqrt {-c^2} d-2 e\right )-\sqrt {-c^2} e^2\right ) (d+e x) \log \left (1-\sqrt {-c^2} x\right )-\left (\sqrt {-c^2} e^2-c^2 d \left (\sqrt {-c^2} d+2 e\right )\right ) (d+e x) \log \left (1+\sqrt {-c^2} x\right )-4 c^2 d e (d+e x) \log (d+e x)\right )}{\left (c^2 d^2+e^2\right )^2}}{4 e (d+e x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 188, normalized size = 1.29
method | result | size |
derivativedivides | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}-\frac {b \,c^{3} \arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {b \,c^{5} \arctan \left (c x \right ) d^{2}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {b \,c^{4} d \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {b \,c^{3} e \arctan \left (c x \right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {b \,c^{3}}{2 \left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {b \,c^{4} d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{c}\) | \(188\) |
default | \(\frac {-\frac {a \,c^{3}}{2 \left (c e x +c d \right )^{2} e}-\frac {b \,c^{3} \arctan \left (c x \right )}{2 \left (c e x +c d \right )^{2} e}+\frac {b \,c^{5} \arctan \left (c x \right ) d^{2}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {b \,c^{4} d \ln \left (c^{2} x^{2}+1\right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {b \,c^{3} e \arctan \left (c x \right )}{2 \left (c^{2} d^{2}+e^{2}\right )^{2}}-\frac {b \,c^{3}}{2 \left (c^{2} d^{2}+e^{2}\right ) \left (c e x +c d \right )}+\frac {b \,c^{4} d \ln \left (c e x +c d \right )}{\left (c^{2} d^{2}+e^{2}\right )^{2}}}{c}\) | \(188\) |
risch | \(\text {Expression too large to display}\) | \(2102\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 206, normalized size = 1.41 \begin {gather*} -\frac {1}{2} \, {\left ({\left (\frac {c^{2} d \log \left (c^{2} x^{2} + 1\right )}{c^{4} d^{4} + 2 \, c^{2} d^{2} e^{2} + e^{4}} - \frac {2 \, c^{2} d \log \left (x e + d\right )}{c^{4} d^{4} + 2 \, c^{2} d^{2} e^{2} + e^{4}} - \frac {{\left (c^{4} d^{2} - c^{2} e^{2}\right )} \arctan \left (c x\right )}{{\left (c^{4} d^{4} e + 2 \, c^{2} d^{2} e^{3} + e^{5}\right )} c} + \frac {1}{c^{2} d^{3} + {\left (c^{2} d^{2} e + e^{3}\right )} x + d e^{2}}\right )} c + \frac {\arctan \left (c x\right )}{x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e}\right )} b - \frac {a}{2 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 299 vs.
\(2 (137) = 274\).
time = 1.36, size = 299, normalized size = 2.05 \begin {gather*} -\frac {a c^{4} d^{4} + b c^{3} d^{3} e + b c d e^{3} - {\left (2 \, b c^{4} d^{3} x e - 2 \, b c^{2} d x e^{3} - {\left (b c^{2} x^{2} + b\right )} e^{4} + {\left (b c^{4} d^{2} x^{2} - 3 \, b c^{2} d^{2}\right )} e^{2}\right )} \arctan \left (c x\right ) + {\left (b c x + a\right )} e^{4} + {\left (b c^{3} d^{2} x + 2 \, a c^{2} d^{2}\right )} e^{2} + {\left (b c^{3} d x^{2} e^{3} + 2 \, b c^{3} d^{2} x e^{2} + b c^{3} d^{3} e\right )} \log \left (c^{2} x^{2} + 1\right ) - 2 \, {\left (b c^{3} d x^{2} e^{3} + 2 \, b c^{3} d^{2} x e^{2} + b c^{3} d^{3} e\right )} \log \left (x e + d\right )}{2 \, {\left (2 \, c^{4} d^{5} x e^{2} + c^{4} d^{6} e + 4 \, c^{2} d^{3} x e^{4} + x^{2} e^{7} + 2 \, d x e^{6} + {\left (2 \, c^{2} d^{2} x^{2} + d^{2}\right )} e^{5} + {\left (c^{4} d^{4} x^{2} + 2 \, c^{2} d^{4}\right )} e^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 3.42, size = 2866, normalized size = 19.63 \begin {gather*} \text {Too large to display} \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 = 4.43, size = 591, normalized size = 4.05 \begin {gather*} \frac {\frac {x\,\left (a\,c^2\,d^2+\frac {b\,c\,d\,e}{2}+a\,e^2\right )}{d\,\left (c^2\,d^2+e^2\right )}-\frac {b\,\mathrm {atan}\left (c\,x\right )}{2\,e}+\frac {x^2\,\left (\frac {a\,c^2\,d^2\,e}{2}+\frac {b\,c\,d\,e^2}{2}+\frac {a\,e^3}{2}\right )}{d^2\,\left (c^2\,d^2+e^2\right )}+\frac {x^4\,\left (\frac {a\,c^4\,d^2\,e}{2}+\frac {b\,c^3\,d\,e^2}{2}+\frac {a\,c^2\,e^3}{2}\right )}{d^2\,\left (c^2\,d^2+e^2\right )}+\frac {x^3\,\left (a\,c^4\,d^2+\frac {b\,c^3\,d\,e}{2}+a\,c^2\,e^2\right )}{d\,\left (c^2\,d^2+e^2\right )}-\frac {b\,c^2\,x^2\,\mathrm {atan}\left (c\,x\right )}{2\,e}}{c^2\,d^2\,x^2+2\,c^2\,d\,e\,x^3+c^2\,e^2\,x^4+d^2+2\,d\,e\,x+e^2\,x^2}+\frac {b\,c^3\,d\,\ln \left (d+e\,x\right )}{c^4\,d^4+2\,c^2\,d^2\,e^2+e^4}-\frac {b\,c^3\,d\,\ln \left (c^2\,x^2+1\right )}{2\,\left (c^4\,d^4+2\,c^2\,d^2\,e^2+e^4\right )}+\frac {\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {c^2}}\right )\,{\left (c^2\right )}^{7/2}\,\left (c^4\,d^4+8\,c^2\,d^2\,e^2+2\,e^4\right )\,\left (3\,c^6\,d^4+26\,c^4\,d^2\,e^2+4\,c^2\,e^4\right )\,\left (27\,b\,c^{10}\,d^{10}+23\,b\,c^8\,d^8\,e^2-34\,b\,c^6\,d^6\,e^4-26\,b\,c^4\,d^4\,e^6+7\,b\,c^2\,d^2\,e^8+3\,b\,e^{10}\right )}{2\,c\,\left (81\,c^{26}\,d^{20}\,e+1662\,c^{24}\,d^{18}\,e^3+11515\,c^{22}\,d^{16}\,e^5+32306\,c^{20}\,d^{14}\,e^7+43705\,c^{18}\,d^{12}\,e^9+28142\,c^{16}\,d^{10}\,e^{11}+4857\,c^{14}\,d^8\,e^{13}-3650\,c^{12}\,d^6\,e^{15}-2054\,c^{10}\,d^4\,e^{17}-380\,c^8\,d^2\,e^{19}-24\,c^6\,e^{21}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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