Optimal. Leaf size=103 \[ \frac {2 a^2 (3 c+2 d) \tan (e+f x)}{3 f}+\frac {a^2 (3 c+2 d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {a^2 (3 c+2 d) \tan (e+f x) \sec (e+f x)}{6 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f} \]
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Rubi [A] time = 0.12, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {4001, 3788, 3767, 8, 4046, 3770} \[ \frac {2 a^2 (3 c+2 d) \tan (e+f x)}{3 f}+\frac {a^2 (3 c+2 d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {a^2 (3 c+2 d) \tan (e+f x) \sec (e+f x)}{6 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^2}{3 f} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3788
Rule 4001
Rule 4046
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \, dx &=\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac {1}{3} (3 c+2 d) \int \sec (e+f x) (a+a \sec (e+f x))^2 \, dx\\ &=\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac {1}{3} (3 c+2 d) \int \sec (e+f x) \left (a^2+a^2 \sec ^2(e+f x)\right ) \, dx+\frac {1}{3} \left (2 a^2 (3 c+2 d)\right ) \int \sec ^2(e+f x) \, dx\\ &=\frac {a^2 (3 c+2 d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 f}+\frac {1}{2} \left (a^2 (3 c+2 d)\right ) \int \sec (e+f x) \, dx-\frac {\left (2 a^2 (3 c+2 d)\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 f}\\ &=\frac {a^2 (3 c+2 d) \tanh ^{-1}(\sin (e+f x))}{2 f}+\frac {2 a^2 (3 c+2 d) \tan (e+f x)}{3 f}+\frac {a^2 (3 c+2 d) \sec (e+f x) \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^2 \tan (e+f x)}{3 f}\\ \end {align*}
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Mathematica [B] time = 6.36, size = 481, normalized size = 4.67 \[ \frac {a^2 \cos ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (\sec (e+f x)+1)^2 (c+d \sec (e+f x)) \left (\frac {4 (6 c+5 d) \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {4 (6 c+5 d) \sin \left (\frac {f x}{2}\right )}{\left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {(3 c+7 d) \cos \left (\frac {e}{2}\right )-(3 c+5 d) \sin \left (\frac {e}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}-\frac {(3 c+5 d) \sin \left (\frac {e}{2}\right )+(3 c+7 d) \cos \left (\frac {e}{2}\right )}{\left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2}-6 (3 c+2 d) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+6 (3 c+2 d) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )+\frac {2 d \sin \left (\frac {f x}{2}\right )}{\left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {2 d \sin \left (\frac {f x}{2}\right )}{\left (\sin \left (\frac {e}{2}\right )+\cos \left (\frac {e}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}\right )}{48 f (c \cos (e+f x)+d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 138, normalized size = 1.34 \[ \frac {3 \, {\left (3 \, a^{2} c + 2 \, a^{2} d\right )} \cos \left (f x + e\right )^{3} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (3 \, a^{2} c + 2 \, a^{2} d\right )} \cos \left (f x + e\right )^{3} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (2 \, a^{2} d + 2 \, {\left (6 \, a^{2} c + 5 \, a^{2} d\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (a^{2} c + 2 \, a^{2} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{12 \, f \cos \left (f x + e\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.16, size = 141, normalized size = 1.37 \[ \frac {3 a^{2} c \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}+\frac {5 a^{2} d \tan \left (f x +e \right )}{3 f}+\frac {2 a^{2} c \tan \left (f x +e \right )}{f}+\frac {a^{2} d \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}+\frac {a^{2} d \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {a^{2} c \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {a^{2} d \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 167, normalized size = 1.62 \[ \frac {4 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} d - 3 \, a^{2} c {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 6 \, a^{2} d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 12 \, a^{2} c \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 24 \, a^{2} c \tan \left (f x + e\right ) + 12 \, a^{2} d \tan \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.52, size = 161, normalized size = 1.56 \[ \frac {2\,a^2\,\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,c}{2}+d\right )}{6\,c+4\,d}\right )\,\left (\frac {3\,c}{2}+d\right )}{f}-\frac {\left (3\,a^2\,c+2\,a^2\,d\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-8\,a^2\,c-\frac {16\,a^2\,d}{3}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (5\,a^2\,c+6\,a^2\,d\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
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 \[ a^{2} \left (\int c \sec {\left (e + f x \right )}\, dx + \int 2 c \sec ^{2}{\left (e + f x \right )}\, dx + \int c \sec ^{3}{\left (e + f x \right )}\, dx + \int d \sec ^{2}{\left (e + f x \right )}\, dx + \int 2 d \sec ^{3}{\left (e + f x \right )}\, dx + \int d \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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