Optimal. Leaf size=124 \[ \frac {b x^3}{6}+\frac {1}{2} x^2 \text {ArcTan}(c-(1-i c) \tan (a+b x))+\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x \text {PolyLog}\left (2,-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i \text {PolyLog}\left (3,-i c e^{2 i a+2 i b x}\right )}{8 b^2} \]
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Rubi [A]
time = 0.15, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5279, 2215,
2221, 2611, 2320, 6724} \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(c-(1-i c) \tan (a+b x))+\frac {i \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{8 b^2}+\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {b x^3}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 2215
Rule 2221
Rule 2320
Rule 2611
Rule 5279
Rule 6724
Rubi steps
\begin {align*} \int x \tan ^{-1}(c+(-1+i c) \tan (a+b x)) \, dx &=\frac {1}{2} x^2 \tan ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{2} (i b) \int \frac {x^2}{i (-1+i c)+c+c e^{2 i a+2 i b x}} \, dx\\ &=\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1-i c) \tan (a+b x))-\frac {1}{2} (b c) \int \frac {e^{2 i a+2 i b x} x^2}{i (-1+i c)+c+c e^{2 i a+2 i b x}} \, dx\\ &=\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1-i c) \tan (a+b x))+\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )-\frac {1}{2} i \int x \log \left (1+\frac {c e^{2 i a+2 i b x}}{i (-1+i c)+c}\right ) \, dx\\ &=\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1-i c) \tan (a+b x))+\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}-\frac {\int \text {Li}_2\left (-\frac {c e^{2 i a+2 i b x}}{i (-1+i c)+c}\right ) \, dx}{4 b}\\ &=\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1-i c) \tan (a+b x))+\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i \text {Subst}\left (\int \frac {\text {Li}_2(-i c x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2}\\ &=\frac {b x^3}{6}+\frac {1}{2} x^2 \tan ^{-1}(c-(1-i c) \tan (a+b x))+\frac {1}{4} i x^2 \log \left (1+i c e^{2 i a+2 i b x}\right )+\frac {x \text {Li}_2\left (-i c e^{2 i a+2 i b x}\right )}{4 b}+\frac {i \text {Li}_3\left (-i c e^{2 i a+2 i b x}\right )}{8 b^2}\\ \end {align*}
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Mathematica [A]
time = 0.27, size = 111, normalized size = 0.90 \begin {gather*} \frac {1}{2} x^2 \text {ArcTan}(c+i (i+c) \tan (a+b x))+\frac {i \left (2 b^2 x^2 \log \left (1-\frac {i e^{-2 i (a+b x)}}{c}\right )+2 i b x \text {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{c}\right )+\text {PolyLog}\left (3,\frac {i e^{-2 i (a+b x)}}{c}\right )\right )}{8 b^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.51, size = 1498, normalized size = 12.08
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1498\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 218 vs. \(2 (86) = 172\).
time = 0.29, size = 218, normalized size = 1.76 \begin {gather*} \frac {\frac {6 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \arctan \left ({\left (i \, c - 1\right )} \tan \left (b x + a\right ) + c\right )}{b} + \frac {{\left (-4 i \, {\left (b x + a\right )}^{3} + 12 i \, {\left (b x + a\right )}^{2} a - 6 i \, b x {\rm Li}_2\left (-i \, c e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, {\left (-i \, {\left (b x + a\right )}^{2} + 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (c \cos \left (2 \, b x + 2 \, a\right ), -c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left (c^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + c^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, c \sin \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\rm Li}_{3}(-i \, c e^{\left (2 i \, b x + 2 i \, a\right )})\right )} {\left (i \, c - 1\right )}}{b {\left (c + i\right )}}}{12 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 271 vs. \(2 (86) = 172\).
time = 0.65, size = 271, normalized size = 2.19 \begin {gather*} \frac {2 \, b^{3} x^{3} + 3 i \, b^{2} x^{2} \log \left (-\frac {{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) + 2 \, a^{3} + 6 \, b x {\rm Li}_2\left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 6 \, b x {\rm Li}_2\left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} + i \, \sqrt {-4 i \, c}}{2 \, c}\right ) + 3 i \, a^{2} \log \left (\frac {2 \, c e^{\left (i \, b x + i \, a\right )} - i \, \sqrt {-4 i \, c}}{2 \, c}\right ) - 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) - 3 \, {\left (-i \, b^{2} x^{2} + i \, a^{2}\right )} \log \left (-\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )} + 1\right ) + 6 i \, {\rm polylog}\left (3, \frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right ) + 6 i \, {\rm polylog}\left (3, -\frac {1}{2} \, \sqrt {-4 i \, c} e^{\left (i \, b x + i \, a\right )}\right )}{12 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: CoercionFailed} \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 = -0.01 \begin {gather*} \int x\,\mathrm {atan}\left (c+\mathrm {tan}\left (a+b\,x\right )\,\left (-1+c\,1{}\mathrm {i}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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