Optimal. Leaf size=181 \[ \frac {(c+d x)^3}{3 (a+i b) d}+\frac {b (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {i b d (c+d x) \text {PolyLog}\left (2,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f^2}+\frac {b d^2 \text {PolyLog}\left (3,-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3} \]
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Rubi [A]
time = 0.20, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3813, 2221,
2611, 2320, 6724} \begin {gather*} -\frac {i b d (c+d x) \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{f^2 \left (a^2+b^2\right )}+\frac {b (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{f \left (a^2+b^2\right )}+\frac {b d^2 \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 f^3 \left (a^2+b^2\right )}+\frac {(c+d x)^3}{3 d (a+i b)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3813
Rule 6724
Rubi steps
\begin {align*} \int \frac {(c+d x)^2}{a+b \tan (e+f x)} \, dx &=\frac {(c+d x)^3}{3 (a+i b) d}+(2 i b) \int \frac {e^{2 i (e+f x)} (c+d x)^2}{(a+i b)^2+\left (a^2+b^2\right ) e^{2 i (e+f x)}} \, dx\\ &=\frac {(c+d x)^3}{3 (a+i b) d}+\frac {b (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {(2 b d) \int (c+d x) \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f}\\ &=\frac {(c+d x)^3}{3 (a+i b) d}+\frac {b (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {i b d (c+d x) \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f^2}+\frac {\left (i b d^2\right ) \int \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right ) \, dx}{\left (a^2+b^2\right ) f^2}\\ &=\frac {(c+d x)^3}{3 (a+i b) d}+\frac {b (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {i b d (c+d x) \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f^2}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {\left (a^2+b^2\right ) x}{(a+i b)^2}\right )}{x} \, dx,x,e^{2 i (e+f x)}\right )}{2 \left (a^2+b^2\right ) f^3}\\ &=\frac {(c+d x)^3}{3 (a+i b) d}+\frac {b (c+d x)^2 \log \left (1+\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f}-\frac {i b d (c+d x) \text {Li}_2\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{\left (a^2+b^2\right ) f^2}+\frac {b d^2 \text {Li}_3\left (-\frac {\left (a^2+b^2\right ) e^{2 i (e+f x)}}{(a+i b)^2}\right )}{2 \left (a^2+b^2\right ) f^3}\\ \end {align*}
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Mathematica [A]
time = 1.70, size = 321, normalized size = 1.77 \begin {gather*} \frac {b \left (2 f^2 \left (-2 i (a-i b) e^{2 i e} f x \left (3 c^2+3 c d x+d^2 x^2\right )+3 \left (-i b \left (-1+e^{2 i e}\right )+a \left (1+e^{2 i e}\right )\right ) (c+d x)^2 \log \left (1+\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )\right )+6 d \left (b-b e^{2 i e}-i a \left (1+e^{2 i e}\right )\right ) f (c+d x) \text {PolyLog}\left (2,-\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )+3 d^2 \left (-i b \left (-1+e^{2 i e}\right )+a \left (1+e^{2 i e}\right )\right ) \text {PolyLog}\left (3,-\frac {(a-i b) e^{2 i (e+f x)}}{a+i b}\right )\right )}{6 \left (a^2+b^2\right ) \left (-i b \left (-1+e^{2 i e}\right )+a \left (1+e^{2 i e}\right )\right ) f^3}+\frac {x \left (3 c^2+3 c d x+d^2 x^2\right ) \cos (e)}{3 (a \cos (e)+b \sin (e))} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 939 vs. \(2 (166 ) = 332\).
time = 0.39, size = 940, normalized size = 5.19
method | result | size |
risch | \(-\frac {d^{2} x^{3}}{3 \left (i b -a \right )}-\frac {d c \,x^{2}}{i b -a}-\frac {c^{2} x}{i b -a}-\frac {c^{3}}{3 d \left (i b -a \right )}+\frac {i b \,d^{2} e^{2} \ln \left (1-\frac {\left (-i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b -a}\right )}{f^{3} \left (i a +b \right ) \left (-i b -a \right )}-\frac {b \,d^{2} \polylog \left (2, \frac {\left (-i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b -a}\right ) x}{f^{2} \left (i a +b \right ) \left (-i b -a \right )}+\frac {i b \,d^{2} e^{2} \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{f^{3} \left (i a +b \right ) \left (i b +a \right )}-\frac {2 i b \,d^{2} e^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3} \left (i a +b \right ) \left (i b +a \right )}-\frac {2 i b c d e \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{f^{2} \left (i a +b \right ) \left (i b +a \right )}+\frac {i b \,c^{2} \ln \left (i {\mathrm e}^{2 i \left (f x +e \right )} b -a \,{\mathrm e}^{2 i \left (f x +e \right )}-i b -a \right )}{f \left (i a +b \right ) \left (i b +a \right )}-\frac {2 i b c d \ln \left (1-\frac {\left (-i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b -a}\right ) e}{f^{2} \left (i a +b \right ) \left (-i b -a \right )}-\frac {2 i b \,c^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f \left (i a +b \right ) \left (i b +a \right )}-\frac {2 i b c d \ln \left (1-\frac {\left (-i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b -a}\right ) x}{f \left (i a +b \right ) \left (-i b -a \right )}+\frac {4 i b c d e \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2} \left (i a +b \right ) \left (i b +a \right )}-\frac {2 b \,d^{2} x^{3}}{3 \left (i a +b \right ) \left (-i b -a \right )}+\frac {2 b \,d^{2} e^{2} x}{f^{2} \left (i a +b \right ) \left (-i b -a \right )}+\frac {4 b \,d^{2} e^{3}}{3 f^{3} \left (i a +b \right ) \left (-i b -a \right )}-\frac {i b \,d^{2} \ln \left (1-\frac {\left (-i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b -a}\right ) x^{2}}{f \left (i a +b \right ) \left (-i b -a \right )}-\frac {i b \,d^{2} \polylog \left (3, \frac {\left (-i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b -a}\right )}{2 f^{3} \left (i a +b \right ) \left (-i b -a \right )}-\frac {2 b c d \,x^{2}}{\left (i a +b \right ) \left (-i b -a \right )}-\frac {4 b c d e x}{f \left (i a +b \right ) \left (-i b -a \right )}-\frac {2 b c d \,e^{2}}{f^{2} \left (i a +b \right ) \left (-i b -a \right )}-\frac {b c d \polylog \left (2, \frac {\left (-i b +a \right ) {\mathrm e}^{2 i \left (f x +e \right )}}{-i b -a}\right )}{f^{2} \left (i a +b \right ) \left (-i b -a \right )}\) | \(940\) |
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. 755 vs. \(2 (163) = 326\).
time = 0.65, size = 755, normalized size = 4.17 \begin {gather*} -\frac {6 \, c d {\left (\frac {2 \, {\left (f x + e\right )} a}{{\left (a^{2} + b^{2}\right )} f} + \frac {2 \, b \log \left (b \tan \left (f x + e\right ) + a\right )}{{\left (a^{2} + b^{2}\right )} f} - \frac {b \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{{\left (a^{2} + b^{2}\right )} f}\right )} e - 3 \, {\left (\frac {2 \, {\left (f x + e\right )} a}{a^{2} + b^{2}} + \frac {2 \, b \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{2} + b^{2}} - \frac {b \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{2} + b^{2}}\right )} c^{2} - \frac {2 \, {\left (f x + e\right )}^{3} {\left (a - i \, b\right )} d^{2} + 6 i \, b d^{2} \arctan \left (-b \cos \left (2 \, f x + 2 \, e\right ) + a \sin \left (2 \, f x + 2 \, e\right ) + b, a \cos \left (2 \, f x + 2 \, e\right ) + b \sin \left (2 \, f x + 2 \, e\right ) + a\right ) e^{2} + 3 \, b d^{2} e^{2} \log \left ({\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )\right ) + 6 \, {\left (f x + e\right )} {\left (a e^{2} - i \, b e^{2}\right )} d^{2} + 3 \, b d^{2} {\rm Li}_{3}(\frac {{\left (i \, a + b\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{-i \, a + b}) + 6 \, {\left ({\left (a - i \, b\right )} c d f - {\left (a e - i \, b e\right )} d^{2}\right )} {\left (f x + e\right )}^{2} - 6 \, {\left (i \, {\left (f x + e\right )}^{2} b d^{2} + 2 \, {\left (i \, b c d f - i \, b d^{2} e\right )} {\left (f x + e\right )}\right )} \arctan \left (\frac {2 \, a b \cos \left (2 \, f x + 2 \, e\right ) - {\left (a^{2} - b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}, \frac {2 \, a b \sin \left (2 \, f x + 2 \, e\right ) + a^{2} + b^{2} + {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right ) - 6 \, {\left (i \, {\left (f x + e\right )} b d^{2} + i \, b c d f - i \, b d^{2} e\right )} {\rm Li}_2\left (\frac {{\left (i \, a + b\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}{-i \, a + b}\right ) + 3 \, {\left ({\left (f x + e\right )}^{2} b d^{2} + 2 \, {\left (b c d f - b d^{2} e\right )} {\left (f x + e\right )}\right )} \log \left (\frac {{\left (a^{2} + b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a b \sin \left (2 \, f x + 2 \, e\right ) + {\left (a^{2} + b^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + a^{2} + b^{2} + 2 \, {\left (a^{2} - b^{2}\right )} \cos \left (2 \, f x + 2 \, e\right )}{a^{2} + b^{2}}\right )}{{\left (a^{2} + b^{2}\right )} f^{2}}}{6 \, f} \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. 858 vs. \(2 (163) = 326\).
time = 0.39, size = 858, normalized size = 4.74 \begin {gather*} \frac {4 \, a d^{2} f^{3} x^{3} + 12 \, a c d f^{3} x^{2} + 12 \, a c^{2} f^{3} x + 3 \, b d^{2} {\rm polylog}\left (3, \frac {{\left (a^{2} + 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - 2 i \, a b + b^{2} - 2 \, {\left (-i \, a^{2} + 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 3 \, b d^{2} {\rm polylog}\left (3, \frac {{\left (a^{2} - 2 i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + 2 i \, a b + b^{2} - 2 \, {\left (i \, a^{2} + 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) - 6 \, {\left (-i \, b d^{2} f x - i \, b c d f\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) - 6 \, {\left (i \, b d^{2} f x + i \, b c d f\right )} {\rm Li}_2\left (\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}} + 1\right ) + 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + 2 \, b c d f e - b d^{2} e^{2}\right )} \log \left (-\frac {2 \, {\left ({\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} - i \, a b + {\left (i \, a^{2} - 2 \, a b - i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 6 \, {\left (b d^{2} f^{2} x^{2} + 2 \, b c d f^{2} x + 2 \, b c d f e - b d^{2} e^{2}\right )} \log \left (-\frac {2 \, {\left ({\left (-i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (-i \, a^{2} - 2 \, a b + i \, b^{2}\right )} \tan \left (f x + e\right )\right )}}{{\left (a^{2} + b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + b^{2}}\right ) + 6 \, {\left (b c^{2} f^{2} - 2 \, b c d f e + b d^{2} e^{2}\right )} \log \left (\frac {{\left (i \, a b + b^{2}\right )} \tan \left (f x + e\right )^{2} - a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right ) + 6 \, {\left (b c^{2} f^{2} - 2 \, b c d f e + b d^{2} e^{2}\right )} \log \left (\frac {{\left (i \, a b - b^{2}\right )} \tan \left (f x + e\right )^{2} + a^{2} + i \, a b + {\left (i \, a^{2} + i \, b^{2}\right )} \tan \left (f x + e\right )}{\tan \left (f x + e\right )^{2} + 1}\right )}{12 \, {\left (a^{2} + b^{2}\right )} f^{3}} \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 \frac {\left (c + d x\right )^{2}}{a + b \tan {\left (e + f x \right )}}\, 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 = -0.01 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^2}{a+b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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