Optimal. Leaf size=257 \[ \frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \tanh ^{-1}(\sin (e+f x))}{8 f}+\frac {a^3 \left (2 c^2-15 c d+76 d^2\right ) \tan (e+f x) (c+d \sec (e+f x))^2}{60 d^2 f}+\frac {a^3 \left (4 c^3-30 c^2 d+146 c d^2+195 d^3\right ) \tan (e+f x) \sec (e+f x)}{120 d f}+\frac {a^3 \left (2 c^4-15 c^3 d+72 c^2 d^2+180 c d^3+76 d^4\right ) \tan (e+f x)}{30 d^2 f}-\frac {a^3 (2 c-11 d) \tan (e+f x) (c+d \sec (e+f x))^3}{20 d^2 f}+\frac {\tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right ) (c+d \sec (e+f x))^3}{5 d f} \]
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Rubi [A] time = 0.30, antiderivative size = 273, normalized size of antiderivative = 1.06, number of steps used = 9, 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} \[ \frac {a^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a^4 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{4 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{24 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x) (a \sec (e+f x)+a)^2}{60 f}+\frac {3 d (2 c+d) \tan (e+f x) (a \sec (e+f x)+a)^3}{20 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))}{5 f} \]
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))^3 (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)^{5/2} (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))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(a+a x)^{5/2} \left (-a^2 \left (5 c^2+3 c d+d^2\right )-3 a^2 d (2 c+d) x\right )}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{5 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^2 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{5/2}}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{20 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^3 \left (20 c^2+30 c d+13 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 {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^4 \left (20 c^2+30 c d+13 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^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}-\frac {\left (a^5 \left (20 c^2+30 c d+13 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^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}+\frac {\left (a^4 \left (20 c^2+30 c d+13 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^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}+\frac {\left (a^4 \left (20 c^2+30 c d+13 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^3 \left (20 c^2+30 c d+13 d^2\right ) \tan (e+f x)}{8 f}+\frac {a^4 \left (20 c^2+30 c d+13 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 {a \left (20 c^2+30 c d+13 d^2\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{60 f}+\frac {3 d (2 c+d) (a+a \sec (e+f x))^3 \tan (e+f x)}{20 f}+\frac {\left (20 c^2+30 c d+13 d^2\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{24 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x)) \tan (e+f x)}{5 f}\\ \end {align*}
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Mathematica [A] time = 2.69, size = 433, normalized size = 1.68 \[ -\frac {a^3 (\cos (e+f x)+1)^3 \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sec ^5(e+f x) \left (240 \left (20 c^2+30 c d+13 d^2\right ) \cos ^5(e+f x) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-\sec (e) \left (-240 \left (7 c^2+10 c d+3 d^2\right ) \sin (2 e+f x)+80 \left (34 c^2+60 c d+29 d^2\right ) \sin (f x)+360 c^2 \sin (e+2 f x)+360 c^2 \sin (3 e+2 f x)+1840 c^2 \sin (2 e+3 f x)-360 c^2 \sin (4 e+3 f x)+180 c^2 \sin (3 e+4 f x)+180 c^2 \sin (5 e+4 f x)+440 c^2 \sin (4 e+5 f x)+1140 c d \sin (e+2 f x)+1140 c d \sin (3 e+2 f x)+3360 c d \sin (2 e+3 f x)-240 c d \sin (4 e+3 f x)+450 c d \sin (3 e+4 f x)+450 c d \sin (5 e+4 f x)+720 c d \sin (4 e+5 f x)+750 d^2 \sin (e+2 f x)+750 d^2 \sin (3 e+2 f x)+1520 d^2 \sin (2 e+3 f x)+195 d^2 \sin (3 e+4 f x)+195 d^2 \sin (5 e+4 f x)+304 d^2 \sin (4 e+5 f x)\right )\right )}{15360 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 245, normalized size = 0.95 \[ \frac {15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{5} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (20 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{5} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (24 \, a^{3} d^{2} + 8 \, {\left (55 \, a^{3} c^{2} + 90 \, a^{3} c d + 38 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{4} + 15 \, {\left (12 \, a^{3} c^{2} + 30 \, a^{3} c d + 13 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{3} + 8 \, {\left (5 \, a^{3} c^{2} + 30 \, a^{3} c d + 19 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )^{2} + 30 \, {\left (2 \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f \cos \left (f x + e\right )^{5}} \]
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 = 1.72, size = 342, normalized size = 1.33 \[ \frac {5 a^{3} c^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}+\frac {6 a^{3} c d \tan \left (f x +e \right )}{f}+\frac {13 a^{3} d^{2} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}+\frac {13 a^{3} d^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8 f}+\frac {11 a^{3} c^{2} \tan \left (f x +e \right )}{3 f}+\frac {15 a^{3} c d \sec \left (f x +e \right ) \tan \left (f x +e \right )}{4 f}+\frac {15 a^{3} c d \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{4 f}+\frac {38 a^{3} d^{2} \tan \left (f x +e \right )}{15 f}+\frac {19 a^{3} d^{2} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{15 f}+\frac {3 a^{3} c^{2} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {2 a^{3} c d \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{f}+\frac {3 a^{3} d^{2} \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{4 f}+\frac {a^{3} c^{2} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f}+\frac {a^{3} c d \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{2 f}+\frac {a^{3} d^{2} \tan \left (f x +e \right ) \left (\sec ^{4}\left (f x +e \right )\right )}{5 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 459, normalized size = 1.79 \[ \frac {80 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{2} + 480 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c d + 16 \, {\left (3 \, \tan \left (f x + e\right )^{5} + 10 \, \tan \left (f x + e\right )^{3} + 15 \, \tan \left (f x + e\right )\right )} a^{3} d^{2} + 240 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} d^{2} - 30 \, a^{3} c d {\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 )} - 45 \, a^{3} 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 )} - 180 \, a^{3} 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 )} - 360 \, a^{3} 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 )} - 60 \, a^{3} 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 )} + 240 \, a^{3} c^{2} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 720 \, a^{3} c^{2} \tan \left (f x + e\right ) + 480 \, a^{3} c d \tan \left (f x + e\right )}{240 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.52, size = 287, normalized size = 1.12 \[ \frac {a^3\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{2\,\left (10\,c^2+15\,c\,d+\frac {13\,d^2}{2}\right )}\right )\,\left (20\,c^2+30\,c\,d+13\,d^2\right )}{4\,f}-\frac {\left (5\,a^3\,c^2+\frac {15\,a^3\,c\,d}{2}+\frac {13\,a^3\,d^2}{4}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+\left (-\frac {70\,a^3\,c^2}{3}-35\,a^3\,c\,d-\frac {91\,a^3\,d^2}{6}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (\frac {128\,a^3\,c^2}{3}+64\,a^3\,c\,d+\frac {416\,a^3\,d^2}{15}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (-\frac {106\,a^3\,c^2}{3}-61\,a^3\,c\,d-\frac {133\,a^3\,d^2}{6}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (11\,a^3\,c^2+\frac {49\,a^3\,c\,d}{2}+\frac {51\,a^3\,d^2}{4}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}-5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )} \]
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 \[ a^{3} \left (\int c^{2} \sec {\left (e + f x \right )}\, dx + \int 3 c^{2} \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 c^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int c^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int d^{2} \sec ^{6}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{2}{\left (e + f x \right )}\, dx + \int 6 c d \sec ^{3}{\left (e + f x \right )}\, dx + \int 6 c d \sec ^{4}{\left (e + f x \right )}\, dx + \int 2 c d \sec ^{5}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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