Optimal. Leaf size=288 \[ \frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{48 f}+\frac {a \tan (e+f x) (a \sec (e+f x)+a)^2 \left (d \left (6 c^2+62 c d+31 d^2\right ) \sec (e+f x)+2 \left (4 c^3+74 c^2 d+66 c d^2+21 d^3\right )\right )}{120 f}+\frac {a \tan (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^3}{6 f}+\frac {a (3 c+8 d) \tan (e+f x) (a \sec (e+f x)+a)^2 (c+d \sec (e+f x))^2}{30 f} \]
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Rubi [A] time = 0.43, antiderivative size = 333, normalized size of antiderivative = 1.16, 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, 100, 147, 50, 63, 217, 203} \[ \frac {a^3 \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a^4 \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a (\sec (e+f x)+1)}}\right )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a \sec (e+f x)+a}}+\frac {\left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x) \left (a^3 \sec (e+f x)+a^3\right )}{48 f}+\frac {a \left (90 c^2 d+40 c^3+78 c d^2+23 d^3\right ) \tan (e+f x) (a \sec (e+f x)+a)^2}{120 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^3 \left (70 c^2+4 d (8 c+3 d) \sec (e+f x)+54 c d+19 d^2\right )}{120 f}+\frac {d \tan (e+f x) (a \sec (e+f x)+a)^3 (c+d \sec (e+f x))^2}{6 f} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 100
Rule 147
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))^3 \, dx &=-\frac {\left (a^2 \tan (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(a+a x)^{5/2} (c+d x)^3}{\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))^2 \tan (e+f x)}{6 f}+\frac {\tan (e+f x) \operatorname {Subst}\left (\int \frac {(a+a x)^{5/2} (c+d x) \left (-a^2 \left (6 c^2+3 c d+2 d^2\right )-a^2 d (8 c+3 d) x\right )}{\sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{6 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))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^2 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{40 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{24 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{16 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}-\frac {\left (a^5 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{16 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}+\frac {\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}+\frac {\left (a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\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 )}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\\ &=\frac {a^3 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan (e+f x)}{16 f}+\frac {a^4 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \tan ^{-1}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a+a \sec (e+f x)}}\right ) \tan (e+f x)}{8 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {a \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) (a+a \sec (e+f x))^2 \tan (e+f x)}{120 f}+\frac {\left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \left (a^3+a^3 \sec (e+f x)\right ) \tan (e+f x)}{48 f}+\frac {d (a+a \sec (e+f x))^3 (c+d \sec (e+f x))^2 \tan (e+f x)}{6 f}+\frac {d (a+a \sec (e+f x))^3 \left (70 c^2+54 c d+19 d^2+4 d (8 c+3 d) \sec (e+f x)\right ) \tan (e+f x)}{120 f}\\ \end {align*}
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Mathematica [A] time = 2.69, size = 380, normalized size = 1.32 \[ -\frac {a^3 (\cos (e+f x)+1)^3 \sec ^6\left (\frac {1}{2} (e+f x)\right ) \sec ^6(e+f x) \left (240 \left (40 c^3+90 c^2 d+78 c d^2+23 d^3\right ) \cos ^6(e+f x) \left (\log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )-2 \sin (e+f x) \left (2360 c^3 \cos (3 (e+f x))+360 c^3 \cos (4 (e+f x))+440 c^3 \cos (5 (e+f x))+1080 c^3+6840 c^2 d \cos (3 (e+f x))+1350 c^2 d \cos (4 (e+f x))+1080 c^2 d \cos (5 (e+f x))+4770 c^2 d+16 \left (305 c^3+945 c^2 d+984 c d^2+344 d^3\right ) \cos (e+f x)+20 \left (72 c^3+306 c^2 d+342 c d^2+115 d^3\right ) \cos (2 (e+f x))+6384 c d^2 \cos (3 (e+f x))+1170 c d^2 \cos (4 (e+f x))+912 c d^2 \cos (5 (e+f x))+5670 c d^2+1904 d^3 \cos (3 (e+f x))+345 d^3 \cos (4 (e+f x))+272 d^3 \cos (5 (e+f x))+2275 d^3\right )\right )}{30720 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 337, normalized size = 1.17 \[ \frac {15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 15 \, {\left (40 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) + 2 \, {\left (40 \, a^{3} d^{3} + 16 \, {\left (55 \, a^{3} c^{3} + 135 \, a^{3} c^{2} d + 114 \, a^{3} c d^{2} + 34 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{5} + 15 \, {\left (24 \, a^{3} c^{3} + 90 \, a^{3} c^{2} d + 78 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{4} + 16 \, {\left (5 \, a^{3} c^{3} + 45 \, a^{3} c^{2} d + 57 \, a^{3} c d^{2} + 17 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{3} + 10 \, {\left (18 \, a^{3} c^{2} d + 54 \, a^{3} c d^{2} + 23 \, a^{3} d^{3}\right )} \cos \left (f x + e\right )^{2} + 144 \, {\left (a^{3} c d^{2} + a^{3} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \]
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 = 1.95, size = 523, normalized size = 1.82 \[ \frac {23 a^{3} d^{3} \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{24 f}+\frac {23 a^{3} d^{3} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{16 f}+\frac {3 a^{3} c^{3} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{2 f}+\frac {a^{3} d^{3} \tan \left (f x +e \right ) \left (\sec ^{5}\left (f x +e \right )\right )}{6 f}+\frac {45 a^{3} c^{2} d \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}+\frac {9 a^{3} c \,d^{2} \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{4 f}+\frac {34 a^{3} d^{3} \tan \left (f x +e \right )}{15 f}+\frac {17 a^{3} d^{3} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{15 f}+\frac {3 a^{3} d^{3} \tan \left (f x +e \right ) \left (\sec ^{4}\left (f x +e \right )\right )}{5 f}+\frac {a^{3} c^{3} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{3 f}+\frac {3 a^{3} c^{2} d \tan \left (f x +e \right ) \left (\sec ^{3}\left (f x +e \right )\right )}{4 f}+\frac {38 a^{3} c \,d^{2} \tan \left (f x +e \right )}{5 f}+\frac {19 a^{3} c \,d^{2} \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{5 f}+\frac {3 a^{3} c^{2} d \tan \left (f x +e \right ) \left (\sec ^{2}\left (f x +e \right )\right )}{f}+\frac {3 a^{3} c \,d^{2} \tan \left (f x +e \right ) \left (\sec ^{4}\left (f x +e \right )\right )}{5 f}+\frac {9 a^{3} c^{2} d \tan \left (f x +e \right )}{f}+\frac {39 a^{3} c \,d^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8 f}+\frac {45 a^{3} c^{2} d \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8 f}+\frac {39 a^{3} c \,d^{2} \sec \left (f x +e \right ) \tan \left (f x +e \right )}{8 f}+\frac {5 a^{3} c^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2 f}+\frac {11 a^{3} c^{3} \tan \left (f x +e \right )}{3 f}+\frac {23 a^{3} d^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 701, normalized size = 2.43 \[ \frac {160 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{3} + 1440 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{2} d + 96 \, {\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} c d^{2} + 1440 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c d^{2} + 96 \, {\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^{3} + 160 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} d^{3} - 5 \, a^{3} d^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (f x + e\right )^{5} - 40 \, \sin \left (f x + e\right )^{3} + 33 \, \sin \left (f x + e\right )\right )}}{\sin \left (f x + e\right )^{6} - 3 \, \sin \left (f x + e\right )^{4} + 3 \, \sin \left (f x + e\right )^{2} - 1} - 15 \, \log \left (\sin \left (f x + e\right ) + 1\right ) + 15 \, \log \left (\sin \left (f x + e\right ) - 1\right )\right )} - 90 \, a^{3} c^{2} 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 )} - 270 \, a^{3} c 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 )} - 90 \, a^{3} d^{3} {\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 )} - 360 \, a^{3} c^{3} {\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 )} - 1080 \, a^{3} c^{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 )} - 360 \, a^{3} c 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 )} + 480 \, a^{3} c^{3} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 1440 \, a^{3} c^{3} \tan \left (f x + e\right ) + 1440 \, a^{3} c^{2} d \tan \left (f x + e\right )}{480 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.24, size = 411, normalized size = 1.43 \[ \frac {a^3\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (40\,c^3+90\,c^2\,d+78\,c\,d^2+23\,d^3\right )}{4\,\left (10\,c^3+\frac {45\,c^2\,d}{2}+\frac {39\,c\,d^2}{2}+\frac {23\,d^3}{4}\right )}\right )\,\left (40\,c^3+90\,c^2\,d+78\,c\,d^2+23\,d^3\right )}{8\,f}-\frac {\left (5\,a^3\,c^3+\frac {45\,a^3\,c^2\,d}{4}+\frac {39\,a^3\,c\,d^2}{4}+\frac {23\,a^3\,d^3}{8}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+\left (-\frac {85\,a^3\,c^3}{3}-\frac {255\,a^3\,c^2\,d}{4}-\frac {221\,a^3\,c\,d^2}{4}-\frac {391\,a^3\,d^3}{24}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+\left (66\,a^3\,c^3+\frac {297\,a^3\,c^2\,d}{2}+\frac {1287\,a^3\,c\,d^2}{10}+\frac {759\,a^3\,d^3}{20}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7+\left (-78\,a^3\,c^3-\frac {375\,a^3\,c^2\,d}{2}-\frac {1497\,a^3\,c\,d^2}{10}-\frac {969\,a^3\,d^3}{20}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+\left (\frac {139\,a^3\,c^3}{3}+\frac {513\,a^3\,c^2\,d}{4}+\frac {419\,a^3\,c\,d^2}{4}+\frac {211\,a^3\,d^3}{8}\right )\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+\left (-11\,a^3\,c^3-\frac {147\,a^3\,c^2\,d}{4}-\frac {153\,a^3\,c\,d^2}{4}-\frac {105\,a^3\,d^3}{8}\right )\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8-20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4-6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]
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 \[ a^{3} \left (\int c^{3} \sec {\left (e + f x \right )}\, dx + \int 3 c^{3} \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 c^{3} \sec ^{3}{\left (e + f x \right )}\, dx + \int c^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 d^{3} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 d^{3} \sec ^{6}{\left (e + f x \right )}\, dx + \int d^{3} \sec ^{7}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{3}{\left (e + f x \right )}\, dx + \int 9 c d^{2} \sec ^{4}{\left (e + f x \right )}\, dx + \int 9 c d^{2} \sec ^{5}{\left (e + f x \right )}\, dx + \int 3 c d^{2} \sec ^{6}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{2}{\left (e + f x \right )}\, dx + \int 9 c^{2} d \sec ^{3}{\left (e + f x \right )}\, dx + \int 9 c^{2} d \sec ^{4}{\left (e + f x \right )}\, dx + \int 3 c^{2} d \sec ^{5}{\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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