Optimal. Leaf size=343 \[ \frac {i f^2 \text {Li}_2\left (-i e^{i (c+d x)}\right )}{3 a d^3}-\frac {i f^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{3 a d^3}-\frac {2 i f^2 \text {Li}_2\left (-e^{2 i (c+d x)}\right )}{3 a d^3}+\frac {f^2 \tan (c+d x)}{3 a d^3}-\frac {f^2 \sec (c+d x)}{3 a d^3}+\frac {4 f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{3 a d^2}-\frac {2 i f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{3 a d^2}-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}+\frac {f (e+f x) \tan (c+d x) \sec (c+d x)}{3 a d^2}+\frac {2 (e+f x)^2 \tan (c+d x)}{3 a d}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 a d}-\frac {2 i (e+f x)^2}{3 a d} \]
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Rubi [A] time = 0.38, antiderivative size = 343, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4531, 4186, 3767, 8, 4184, 3719, 2190, 2279, 2391, 4409, 4185, 4181} \[ \frac {i f^2 \text {PolyLog}\left (2,-i e^{i (c+d x)}\right )}{3 a d^3}-\frac {i f^2 \text {PolyLog}\left (2,i e^{i (c+d x)}\right )}{3 a d^3}-\frac {2 i f^2 \text {PolyLog}\left (2,-e^{2 i (c+d x)}\right )}{3 a d^3}+\frac {4 f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{3 a d^2}-\frac {2 i f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{3 a d^2}-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}+\frac {f (e+f x) \tan (c+d x) \sec (c+d x)}{3 a d^2}+\frac {f^2 \tan (c+d x)}{3 a d^3}-\frac {f^2 \sec (c+d x)}{3 a d^3}+\frac {2 (e+f x)^2 \tan (c+d x)}{3 a d}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {(e+f x)^2 \tan (c+d x) \sec ^2(c+d x)}{3 a d}-\frac {2 i (e+f x)^2}{3 a d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 3767
Rule 4181
Rule 4184
Rule 4185
Rule 4186
Rule 4409
Rule 4531
Rubi steps
\begin {align*} \int \frac {(e+f x)^2 \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x)^2 \sec ^4(c+d x) \, dx}{a}-\frac {\int (e+f x)^2 \sec ^3(c+d x) \tan (c+d x) \, dx}{a}\\ &=-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {2 \int (e+f x)^2 \sec ^2(c+d x) \, dx}{3 a}+\frac {(2 f) \int (e+f x) \sec ^3(c+d x) \, dx}{3 a d}+\frac {f^2 \int \sec ^2(c+d x) \, dx}{3 a d^2}\\ &=-\frac {f^2 \sec (c+d x)}{3 a d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {2 (e+f x)^2 \tan (c+d x)}{3 a d}+\frac {f (e+f x) \sec (c+d x) \tan (c+d x)}{3 a d^2}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {f \int (e+f x) \sec (c+d x) \, dx}{3 a d}-\frac {(4 f) \int (e+f x) \tan (c+d x) \, dx}{3 a d}-\frac {f^2 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 a d^3}\\ &=-\frac {2 i (e+f x)^2}{3 a d}-\frac {2 i f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{3 a d^2}-\frac {f^2 \sec (c+d x)}{3 a d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {f^2 \tan (c+d x)}{3 a d^3}+\frac {2 (e+f x)^2 \tan (c+d x)}{3 a d}+\frac {f (e+f x) \sec (c+d x) \tan (c+d x)}{3 a d^2}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {(8 i f) \int \frac {e^{2 i (c+d x)} (e+f x)}{1+e^{2 i (c+d x)}} \, dx}{3 a d}-\frac {f^2 \int \log \left (1-i e^{i (c+d x)}\right ) \, dx}{3 a d^2}+\frac {f^2 \int \log \left (1+i e^{i (c+d x)}\right ) \, dx}{3 a d^2}\\ &=-\frac {2 i (e+f x)^2}{3 a d}-\frac {2 i f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{3 a d^2}+\frac {4 f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{3 a d^2}-\frac {f^2 \sec (c+d x)}{3 a d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {f^2 \tan (c+d x)}{3 a d^3}+\frac {2 (e+f x)^2 \tan (c+d x)}{3 a d}+\frac {f (e+f x) \sec (c+d x) \tan (c+d x)}{3 a d^2}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {\left (i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{3 a d^3}-\frac {\left (i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i (c+d x)}\right )}{3 a d^3}-\frac {\left (4 f^2\right ) \int \log \left (1+e^{2 i (c+d x)}\right ) \, dx}{3 a d^2}\\ &=-\frac {2 i (e+f x)^2}{3 a d}-\frac {2 i f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{3 a d^2}+\frac {4 f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{3 a d^2}+\frac {i f^2 \text {Li}_2\left (-i e^{i (c+d x)}\right )}{3 a d^3}-\frac {i f^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{3 a d^3}-\frac {f^2 \sec (c+d x)}{3 a d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {f^2 \tan (c+d x)}{3 a d^3}+\frac {2 (e+f x)^2 \tan (c+d x)}{3 a d}+\frac {f (e+f x) \sec (c+d x) \tan (c+d x)}{3 a d^2}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {\left (2 i f^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{3 a d^3}\\ &=-\frac {2 i (e+f x)^2}{3 a d}-\frac {2 i f (e+f x) \tan ^{-1}\left (e^{i (c+d x)}\right )}{3 a d^2}+\frac {4 f (e+f x) \log \left (1+e^{2 i (c+d x)}\right )}{3 a d^2}+\frac {i f^2 \text {Li}_2\left (-i e^{i (c+d x)}\right )}{3 a d^3}-\frac {i f^2 \text {Li}_2\left (i e^{i (c+d x)}\right )}{3 a d^3}-\frac {2 i f^2 \text {Li}_2\left (-e^{2 i (c+d x)}\right )}{3 a d^3}-\frac {f^2 \sec (c+d x)}{3 a d^3}-\frac {f (e+f x) \sec ^2(c+d x)}{3 a d^2}-\frac {(e+f x)^2 \sec ^3(c+d x)}{3 a d}+\frac {f^2 \tan (c+d x)}{3 a d^3}+\frac {2 (e+f x)^2 \tan (c+d x)}{3 a d}+\frac {f (e+f x) \sec (c+d x) \tan (c+d x)}{3 a d^2}+\frac {(e+f x)^2 \sec ^2(c+d x) \tan (c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 7.04, size = 637, normalized size = 1.86 \[ \frac {\frac {d^2 e^2 \sin (2 (c+d x))+2 d^2 e^2 \cos (c+d x)-4 d^2 e^2 \cos (c+2 d x)+2 d^2 e f x \sin (2 (c+d x))+4 d^2 e f x \cos (c+d x)-8 d^2 e f x \cos (c+2 d x)+d^2 f^2 x^2 \sin (2 (c+d x))+2 d^2 f^2 x^2 \cos (c+d x)-4 d^2 f^2 x^2 \cos (c+2 d x)-2 d e f \cos (2 c+d x)+2 f^2 \sin (2 (c+d x))-2 f^2 \sin (2 c+d x)-2 d f^2 x \cos (2 c+d x)+4 f^2 \cos (c+d x)-2 f^2 \cos (c+2 d x)-2 f^2 \cos (c)+8 d^2 e^2 \sin (d x)+16 d^2 e f x \sin (d x)+8 d^2 f^2 x^2 \sin (d x)-2 d f \cos (d x) (e+f x)+2 f^2 \sin (d x)}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {12 d^2 f (\cos (c)+i \sin (c)) \left (\frac {f (\cos (c)-i (\sin (c)-1)) \text {Li}_2(i \cos (c+d x)+\sin (c+d x))}{d^2}+\frac {(-\sin (c)-i \cos (c)+1) (e+f x) \log (-\sin (c+d x)-i \cos (c+d x)+1)}{d}+\frac {(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right )}{\cos (c)+i (\sin (c)-1)}-\frac {20 d^2 f (\cos (c)+i \sin (c)) \left (\frac {f (\cos (c)-i (\sin (c)+1)) \text {Li}_2(-i \cos (c+d x)-\sin (c+d x))}{d^2}-\frac {(\sin (c)+i \cos (c)+1) (e+f x) \log (\sin (c+d x)+i \cos (c+d x)+1)}{d}+\frac {(\cos (c)-i \sin (c)) (e+f x)^2}{2 f}\right )}{\cos (c)+i (\sin (c)+1)}}{12 a d^3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.56, size = 855, normalized size = 2.49 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (f x + e\right )}^{2} \sec \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 573, normalized size = 1.67 \[ -\frac {2 \left (4 d^{2} e^{2} {\mathrm e}^{i \left (d x +c \right )}+2 i d^{2} e^{2}+f^{2} {\mathrm e}^{3 i \left (d x +c \right )}+i f^{2} {\mathrm e}^{2 i \left (d x +c \right )}+i f^{2}+4 d^{2} f^{2} x^{2} {\mathrm e}^{i \left (d x +c \right )}+2 i d^{2} f^{2} x^{2}+f^{2} {\mathrm e}^{i \left (d x +c \right )}+8 d^{2} e f x \,{\mathrm e}^{i \left (d x +c \right )}+i d \,f^{2} x \,{\mathrm e}^{i \left (d x +c \right )}+i d e f \,{\mathrm e}^{i \left (d x +c \right )}+i d \,f^{2} x \,{\mathrm e}^{3 i \left (d x +c \right )}+i d e f \,{\mathrm e}^{3 i \left (d x +c \right )}+4 i d^{2} e f x \right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} d^{3} a}+\frac {e f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a \,d^{2}}+\frac {5 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) e}{3 a \,d^{2}}-\frac {8 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right ) e}{3 a \,d^{2}}-\frac {f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{a \,d^{3}}-\frac {5 f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{3 a \,d^{3}}+\frac {8 f^{2} c \ln \left ({\mathrm e}^{i \left (d x +c \right )}\right )}{3 a \,d^{3}}-\frac {5 i f^{2} \polylog \left (2, i {\mathrm e}^{i \left (d x +c \right )}\right )}{3 a \,d^{3}}-\frac {i f^{2} \polylog \left (2, -i {\mathrm e}^{i \left (d x +c \right )}\right )}{a \,d^{3}}-\frac {4 i f^{2} c^{2}}{3 a \,d^{3}}+\frac {f^{2} \ln \left (1+i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{a \,d^{2}}+\frac {f^{2} \ln \left (1+i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{a \,d^{3}}-\frac {8 i f^{2} c x}{3 a \,d^{2}}+\frac {5 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) x}{3 a \,d^{2}}+\frac {5 f^{2} \ln \left (1-i {\mathrm e}^{i \left (d x +c \right )}\right ) c}{3 a \,d^{3}}-\frac {4 i f^{2} x^{2}}{3 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.87, size = 1332, normalized size = 3.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
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 \[ \frac {\int \frac {e^{2} \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f^{2} x^{2} \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {2 e f x \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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