Integrand size = 18, antiderivative size = 157 \[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=\frac {a (c+d x)^3}{3 d}-\frac {2 i a (c+d x)^2 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {2 a d^2 \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {2 a d^2 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3} \]
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Time = 0.17 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {4275, 4266, 2611, 2320, 6724} \[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=-\frac {2 i a (c+d x)^2 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}+\frac {a (c+d x)^3}{3 d}-\frac {2 a d^2 \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {2 a d^2 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3} \]
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Rule 2320
Rule 2611
Rule 4266
Rule 4275
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (a (c+d x)^2+a (c+d x)^2 \sec (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^3}{3 d}+a \int (c+d x)^2 \sec (e+f x) \, dx \\ & = \frac {a (c+d x)^3}{3 d}-\frac {2 i a (c+d x)^2 \arctan \left (e^{i (e+f x)}\right )}{f}-\frac {(2 a d) \int (c+d x) \log \left (1-i e^{i (e+f x)}\right ) \, dx}{f}+\frac {(2 a d) \int (c+d x) \log \left (1+i e^{i (e+f x)}\right ) \, dx}{f} \\ & = \frac {a (c+d x)^3}{3 d}-\frac {2 i a (c+d x)^2 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {\left (2 i a d^2\right ) \int \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right ) \, dx}{f^2}+\frac {\left (2 i a d^2\right ) \int \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right ) \, dx}{f^2} \\ & = \frac {a (c+d x)^3}{3 d}-\frac {2 i a (c+d x)^2 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {\left (2 a d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^3}+\frac {\left (2 a d^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i (e+f x)}\right )}{f^3} \\ & = \frac {a (c+d x)^3}{3 d}-\frac {2 i a (c+d x)^2 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )}{f^2}-\frac {2 i a d (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )}{f^2}-\frac {2 a d^2 \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )}{f^3}+\frac {2 a d^2 \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )}{f^3} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.96 \[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=a \left (\frac {(c+d x)^3}{3 d}-\frac {2 i (c+d x)^2 \arctan \left (e^{i (e+f x)}\right )}{f}+\frac {2 i d \left (f (c+d x) \operatorname {PolyLog}\left (2,-i e^{i (e+f x)}\right )+i d \operatorname {PolyLog}\left (3,-i e^{i (e+f x)}\right )\right )}{f^3}+\frac {2 d \left (-i f (c+d x) \operatorname {PolyLog}\left (2,i e^{i (e+f x)}\right )+d \operatorname {PolyLog}\left (3,i e^{i (e+f x)}\right )\right )}{f^3}\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (138 ) = 276\).
Time = 1.34 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.80
method | result | size |
risch | \(\frac {a \,d^{2} x^{3}}{3}+a d c \,x^{2}+a \,c^{2} x +\frac {a \,c^{3}}{3 d}-\frac {a \,e^{2} d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {2 a c d \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}+\frac {2 a \,d^{2} \operatorname {polylog}\left (3, i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {2 a c d \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}+\frac {2 i a c d \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {2 i a \,d^{2} \operatorname {polylog}\left (2, -i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}+\frac {4 i a c d e \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}-\frac {2 i a \,d^{2} e^{2} \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}+\frac {a \,e^{2} d^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {2 a \,d^{2} \operatorname {polylog}\left (3, -i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{3}}-\frac {2 i a c d \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (f x +e \right )}\right )}{f^{2}}+\frac {a \,d^{2} \ln \left (1-i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{f}-\frac {a \,d^{2} \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) x^{2}}{f}-\frac {2 i a \,c^{2} \arctan \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{f}-\frac {2 a c d \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f}-\frac {2 a c d \ln \left (1+i {\mathrm e}^{i \left (f x +e \right )}\right ) e}{f^{2}}-\frac {2 i a \,d^{2} \operatorname {polylog}\left (2, i {\mathrm e}^{i \left (f x +e \right )}\right ) x}{f^{2}}\) | \(440\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (129) = 258\).
Time = 0.34 (sec) , antiderivative size = 675, normalized size of antiderivative = 4.30 \[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=\frac {2 \, a d^{2} f^{3} x^{3} + 6 \, a c d f^{3} x^{2} + 6 \, a c^{2} f^{3} x - 6 \, a d^{2} {\rm polylog}\left (3, i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 6 \, a d^{2} {\rm polylog}\left (3, i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 6 \, a d^{2} {\rm polylog}\left (3, -i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) + 6 \, a d^{2} {\rm polylog}\left (3, -i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 6 \, {\left (i \, a d^{2} f x + i \, a c d f\right )} {\rm Li}_2\left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) - 6 \, {\left (i \, a d^{2} f x + i \, a c d f\right )} {\rm Li}_2\left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) - 6 \, {\left (-i \, a d^{2} f x - i \, a c d f\right )} {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right )\right ) - 6 \, {\left (-i \, a d^{2} f x - i \, a c d f\right )} {\rm Li}_2\left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right )\right ) + 3 \, {\left (a d^{2} e^{2} - 2 \, a c d e f + a c^{2} f^{2}\right )} \log \left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - 3 \, {\left (a d^{2} e^{2} - 2 \, a c d e f + a c^{2} f^{2}\right )} \log \left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right ) + 3 \, {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x - a d^{2} e^{2} + 2 \, a c d e f\right )} \log \left (i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x - a d^{2} e^{2} + 2 \, a c d e f\right )} \log \left (i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x - a d^{2} e^{2} + 2 \, a c d e f\right )} \log \left (-i \, \cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (a d^{2} f^{2} x^{2} + 2 \, a c d f^{2} x - a d^{2} e^{2} + 2 \, a c d e f\right )} \log \left (-i \, \cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right ) + 3 \, {\left (a d^{2} e^{2} - 2 \, a c d e f + a c^{2} f^{2}\right )} \log \left (-\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right ) + i\right ) - 3 \, {\left (a d^{2} e^{2} - 2 \, a c d e f + a c^{2} f^{2}\right )} \log \left (-\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right ) + i\right )}{6 \, f^{3}} \]
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\[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=a \left (\int c^{2}\, dx + \int c^{2} \sec {\left (e + f x \right )}\, dx + \int d^{2} x^{2}\, dx + \int 2 c d x\, dx + \int d^{2} x^{2} \sec {\left (e + f x \right )}\, dx + \int 2 c d x \sec {\left (e + f x \right )}\, dx\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 516 vs. \(2 (129) = 258\).
Time = 0.40 (sec) , antiderivative size = 516, normalized size of antiderivative = 3.29 \[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=\frac {6 \, {\left (f x + e\right )} a c^{2} + \frac {2 \, {\left (f x + e\right )}^{3} a d^{2}}{f^{2}} - \frac {6 \, {\left (f x + e\right )}^{2} a d^{2} e}{f^{2}} + \frac {6 \, {\left (f x + e\right )} a d^{2} e^{2}}{f^{2}} + \frac {6 \, {\left (f x + e\right )}^{2} a c d}{f} - \frac {12 \, {\left (f x + e\right )} a c d e}{f} + 6 \, a c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + \frac {6 \, a d^{2} e^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right )}{f^{2}} - \frac {12 \, a c d e \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right )}{f} + \frac {3 \, {\left (4 \, a d^{2} {\rm Li}_{3}(i \, e^{\left (i \, f x + i \, e\right )}) - 4 \, a d^{2} {\rm Li}_{3}(-i \, e^{\left (i \, f x + i \, e\right )}) - 2 \, {\left (i \, {\left (f x + e\right )}^{2} a d^{2} + 2 \, {\left (-i \, a d^{2} e + i \, a c d f\right )} {\left (f x + e\right )}\right )} \arctan \left (\cos \left (f x + e\right ), \sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (i \, {\left (f x + e\right )}^{2} a d^{2} + 2 \, {\left (-i \, a d^{2} e + i \, a c d f\right )} {\left (f x + e\right )}\right )} \arctan \left (\cos \left (f x + e\right ), -\sin \left (f x + e\right ) + 1\right ) - 4 \, {\left (i \, {\left (f x + e\right )} a d^{2} - i \, a d^{2} e + i \, a c d f\right )} {\rm Li}_2\left (i \, e^{\left (i \, f x + i \, e\right )}\right ) - 4 \, {\left (-i \, {\left (f x + e\right )} a d^{2} + i \, a d^{2} e - i \, a c d f\right )} {\rm Li}_2\left (-i \, e^{\left (i \, f x + i \, e\right )}\right ) + {\left ({\left (f x + e\right )}^{2} a d^{2} - 2 \, {\left (a d^{2} e - a c d f\right )} {\left (f x + e\right )}\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right ) - {\left ({\left (f x + e\right )}^{2} a d^{2} - 2 \, {\left (a d^{2} e - a c d f\right )} {\left (f x + e\right )}\right )} \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right )\right )}}{f^{2}}}{6 \, f} \]
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\[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=\int { {\left (d x + c\right )}^{2} {\left (a \sec \left (f x + e\right ) + a\right )} \,d x } \]
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Timed out. \[ \int (c+d x)^2 (a+a \sec (e+f x)) \, dx=\int \left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )\,{\left (c+d\,x\right )}^2 \,d x \]
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