Optimal. Leaf size=176 \[ \frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {a^2 \left (c^3-8 c^2 d-20 c d^2-8 d^3\right ) \tan (e+f x)}{6 d f}-\frac {a^2 \left (2 c (c-8 d)-21 d^2\right ) \tan (e+f x) \sec (e+f x)}{24 f}+\frac {a^2 \tan (e+f x) (c+d \sec (e+f x))^3}{4 d f}-\frac {a^2 (c-8 d) \tan (e+f x) (c+d \sec (e+f x))^2}{12 d f} \]
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Rubi [A] time = 0.26, antiderivative size = 234, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3987, 90, 80, 50, 63, 217, 203} \[ \frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)}{8 f}+\frac {a^3 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {\left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x) \left (a^2 \sec (e+f x)+a^2\right )}{24 f}+\frac {d (5 c+2 d) \tan (e+f x) (a \sec (e+f x)+a)^2}{12 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))}{4 f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 80
Rule 90
Rule 203
Rule 217
Rule 3987
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^2 (c+d \sec (e+f x))^2 \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{3/2} (c+d x)^2}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \tan (e+f x)}{4 f}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(a+a x)^{3/2} \left (-a^2 \left (4 c^2+2 c d+d^2\right )-a^2 d (5 c+2 d) x\right )}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d (5 c+2 d) (a+a \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {d (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \tan (e+f x)}{4 f}-\frac {\left (a^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{3/2}}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{12 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {d (5 c+2 d) (a+a \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {\left (12 c^2+16 c d+7 d^2\right ) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \tan (e+f x)}{4 f}-\frac {\left (a^3 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+a x}}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)}{8 f}+\frac {d (5 c+2 d) (a+a \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {\left (12 c^2+16 c d+7 d^2\right ) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \tan (e+f x)}{4 f}-\frac {\left (a^4 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a-a x} \sqrt {a+a x}} \, dx,x,\sec (e+f x)\right )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)}{8 f}+\frac {d (5 c+2 d) (a+a \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {\left (12 c^2+16 c d+7 d^2\right ) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \tan (e+f x)}{4 f}+\frac {\left (a^3 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 a-x^2}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)}{8 f}+\frac {d (5 c+2 d) (a+a \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {\left (12 c^2+16 c d+7 d^2\right ) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \tan (e+f x)}{4 f}+\frac {\left (a^3 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^2 \left (12 c^2+16 c d+7 d^2\right ) \tan (e+f x)}{8 f}+\frac {a^3 \left (12 c^2+16 c d+7 d^2\right ) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {d (5 c+2 d) (a+a \sec (e+f x))^2 \tan (e+f x)}{12 f}+\frac {\left (12 c^2+16 c d+7 d^2\right ) \left (a^2+a^2 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^2 (c+d \sec (e+f x)) \tan (e+f x)}{4 f}\\ \end {align*}
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Mathematica [B] time = 1.00, size = 479, normalized size = 2.72 \[ -\frac {a^2 \sec ^4(e+f x) \left (12 \left (12 c^2+16 c d+7 d^2\right ) \cos (2 (e+f x)) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )+3 \left (12 c^2+16 c d+7 d^2\right ) \cos (4 (e+f x)) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-24 c^2 \sin (e+f x)-96 c^2 \sin (2 (e+f x))-24 c^2 \sin (3 (e+f x))-48 c^2 \sin (4 (e+f x))+108 c^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-108 c^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-96 c d \sin (e+f x)-224 c d \sin (2 (e+f x))-96 c d \sin (3 (e+f x))-80 c d \sin (4 (e+f x))+144 c d \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-144 c d \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )-90 d^2 \sin (e+f x)-128 d^2 \sin (2 (e+f x))-42 d^2 \sin (3 (e+f x))-32 d^2 \sin (4 (e+f x))+63 d^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-63 d^2 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )}{192 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 209, normalized size = 1.19 \[ \frac {3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} \log \left (\sin \left (f x + e\right ) + 1\right ) - 3 \, {\left (12 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{4} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (6 \, a^{2} d^{2} + 16 \, {\left (3 \, a^{2} c^{2} + 5 \, a^{2} c d + 2 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, a^{2} c^{2} + 16 \, a^{2} c d + 7 \, a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + 16 \, {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{48 \, f \cos \left (f x + e\right )^{4}} \]
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 = 1.42, size = 268, normalized size = 1.52 \[ \frac {3 a^{2} c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}+\frac {10 a^{2} c d \tan \left (f x +e \right )}{3 f}+\frac {7 a^{2} d^{2} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}+\frac {7 a^{2} d^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8 f}+\frac {2 a^{2} c^{2} \tan \left (f x +e \right )}{f}+\frac {2 a^{2} c d \sec \left (f x +e \right ) \tan \left (f x +e \right )}{f}+\frac {2 a^{2} c d \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}+\frac {4 a^{2} d^{2} \tan \left (f x +e \right )}{3 f}+\frac {2 a^{2} d^{2} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f}+\frac {a^{2} c^{2} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {2 a^{2} c d \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f}+\frac {a^{2} d^{2} \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{4 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.63, size = 324, normalized size = 1.84 \[ \frac {32 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c d + 32 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} d^{2} - 3 \, a^{2} d^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (f x + e\right )^{3} - 5 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{4} - 2 \, \sin \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{2} c^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 48 \, a^{2} c d {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 12 \, a^{2} d^{2} {\left (\frac {2 \, \sin \left (f x + e\right )}{\sin \left (f x + e\right )^{2} - 1} - \log \left (\sin \left (f x + e\right ) + 1\right ) + \log \left (\sin \left (f x + e\right ) - 1\right )\right )} + 48 \, a^{2} c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 96 \, a^{2} c^{2} \tan \left (f x + e\right ) + 96 \, a^{2} c d \tan \left (f x + e\right )}{48 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.46, size = 237, normalized size = 1.35 \[ \frac {\left (-3\,a^2\,c^2-4\,a^2\,c\,d-\frac {7\,a^2\,d^2}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (11\,a^2\,c^2+\frac {44\,a^2\,c\,d}{3}+\frac {77\,a^2\,d^2}{12}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-13\,a^2\,c^2-\frac {68\,a^2\,c\,d}{3}-\frac {83\,a^2\,d^2}{12}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (5\,a^2\,c^2+12\,a^2\,c\,d+\frac {25\,a^2\,d^2}{4}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{2\,\left (6\,c^2+8\,c\,d+\frac {7\,d^2}{2}\right )}\right )\,\left (12\,c^2+16\,c\,d+7\,d^2\right )}{4\,f} \]
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 \[ a^{2} \left (\int c^{2} \sec {\left (e + f x \right )}\, dx + \int 2 c^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int c^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 2 d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{2}{\left (e + f x \right )}\, dx + \int 4 c d \sec ^{3}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{4}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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