Optimal. Leaf size=132 \[ a^3 c^4 x-\frac {5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^3 \left (16 c^4-5 c^4 \sec (e+f x)\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f} \]
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Rubi [A]
time = 0.11, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3966,
3855} \begin {gather*} -\frac {5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^3 \tan ^5(e+f x) \left (6 c^4-5 c^4 \sec (e+f x)\right )}{30 f}+\frac {a^3 \tan ^3(e+f x) \left (8 c^4-5 c^4 \sec (e+f x)\right )}{24 f}-\frac {a^3 \tan (e+f x) \left (16 c^4-5 c^4 \sec (e+f x)\right )}{16 f}+a^3 c^4 x \end {gather*}
Antiderivative was successfully verified.
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Rule 3855
Rule 3966
Rule 3989
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^4 \, dx &=-\left (\left (a^3 c^3\right ) \int (c-c \sec (e+f x)) \tan ^6(e+f x) \, dx\right )\\ &=-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}+\frac {1}{6} \left (a^3 c^3\right ) \int (6 c-5 c \sec (e+f x)) \tan ^4(e+f x) \, dx\\ &=\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}-\frac {1}{24} \left (a^3 c^3\right ) \int (24 c-15 c \sec (e+f x)) \tan ^2(e+f x) \, dx\\ &=-\frac {a^3 \left (16 c^4-5 c^4 \sec (e+f x)\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}+\frac {1}{48} \left (a^3 c^3\right ) \int (48 c-15 c \sec (e+f x)) \, dx\\ &=a^3 c^4 x-\frac {a^3 \left (16 c^4-5 c^4 \sec (e+f x)\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}-\frac {1}{16} \left (5 a^3 c^4\right ) \int \sec (e+f x) \, dx\\ &=a^3 c^4 x-\frac {5 a^3 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {a^3 \left (16 c^4-5 c^4 \sec (e+f x)\right ) \tan (e+f x)}{16 f}+\frac {a^3 \left (8 c^4-5 c^4 \sec (e+f x)\right ) \tan ^3(e+f x)}{24 f}-\frac {a^3 \left (6 c^4-5 c^4 \sec (e+f x)\right ) \tan ^5(e+f x)}{30 f}\\ \end {align*}
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Mathematica [A]
time = 1.97, size = 165, normalized size = 1.25 \begin {gather*} \frac {a^3 c^4 \sec ^6(e+f x) \left (1200 e+1200 f x-1200 \tanh ^{-1}(\sin (e+f x)) \cos ^6(e+f x)+1800 (e+f x) \cos (2 (e+f x))+720 e \cos (4 (e+f x))+720 f x \cos (4 (e+f x))+120 e \cos (6 (e+f x))+120 f x \cos (6 (e+f x))+450 \sin (e+f x)-600 \sin (2 (e+f x))-25 \sin (3 (e+f x))-384 \sin (4 (e+f x))+165 \sin (5 (e+f x))-184 \sin (6 (e+f x))\right )}{3840 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(266\) vs.
\(2(124)=248\).
time = 0.11, size = 267, normalized size = 2.02
method | result | size |
risch | \(a^{3} c^{4} x -\frac {i c^{4} a^{3} \left (165 \,{\mathrm e}^{11 i \left (f x +e \right )}+720 \,{\mathrm e}^{10 i \left (f x +e \right )}-25 \,{\mathrm e}^{9 i \left (f x +e \right )}+2160 \,{\mathrm e}^{8 i \left (f x +e \right )}+450 \,{\mathrm e}^{7 i \left (f x +e \right )}+3680 \,{\mathrm e}^{6 i \left (f x +e \right )}-450 \,{\mathrm e}^{5 i \left (f x +e \right )}+3360 \,{\mathrm e}^{4 i \left (f x +e \right )}+25 \,{\mathrm e}^{3 i \left (f x +e \right )}+1488 \,{\mathrm e}^{2 i \left (f x +e \right )}-165 \,{\mathrm e}^{i \left (f x +e \right )}+368\right )}{120 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6}}+\frac {5 c^{4} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{16 f}-\frac {5 c^{4} a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{16 f}\) | \(206\) |
derivativedivides | \(\frac {c^{4} a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+c^{4} a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-3 c^{4} a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-3 c^{4} a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )+3 c^{4} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 c^{4} a^{3} \tan \left (f x +e \right )-c^{4} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{4} a^{3} \left (f x +e \right )}{f}\) | \(267\) |
default | \(\frac {c^{4} a^{3} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+c^{4} a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-3 c^{4} a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-3 c^{4} a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )+3 c^{4} a^{3} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-3 c^{4} a^{3} \tan \left (f x +e \right )-c^{4} a^{3} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )+c^{4} a^{3} \left (f x +e \right )}{f}\) | \(267\) |
norman | \(\frac {a^{3} c^{4} x +a^{3} c^{4} x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-6 a^{3} c^{4} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+15 a^{3} c^{4} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-20 a^{3} c^{4} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+15 a^{3} c^{4} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-6 a^{3} c^{4} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {11 c^{4} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {73 c^{4} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {523 c^{4} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{20 f}+\frac {853 c^{4} a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{20 f}-\frac {389 c^{4} a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}+\frac {21 c^{4} a^{3} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}+\frac {5 c^{4} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}-\frac {5 c^{4} a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(322\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 361 vs.
\(2 (131) = 262\).
time = 0.28, size = 361, normalized size = 2.73 \begin {gather*} -\frac {32 \, {\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^{4} - 480 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{4} - 480 \, {\left (f x + e\right )} a^{3} c^{4} + 5 \, a^{3} c^{4} {\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^{4} {\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^{4} {\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^{4} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 1440 \, a^{3} c^{4} \tan \left (f x + e\right )}{480 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.35, size = 191, normalized size = 1.45 \begin {gather*} \frac {480 \, a^{3} c^{4} f x \cos \left (f x + e\right )^{6} - 75 \, a^{3} c^{4} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) + 75 \, a^{3} c^{4} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (368 \, a^{3} c^{4} \cos \left (f x + e\right )^{5} - 165 \, a^{3} c^{4} \cos \left (f x + e\right )^{4} - 176 \, a^{3} c^{4} \cos \left (f x + e\right )^{3} + 130 \, a^{3} c^{4} \cos \left (f x + e\right )^{2} + 48 \, a^{3} c^{4} \cos \left (f x + e\right ) - 40 \, a^{3} c^{4}\right )} \sin \left (f x + e\right )}{480 \, f \cos \left (f x + e\right )^{6}} \end {gather*}
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 \begin {gather*} a^{3} c^{4} \left (\int 1\, dx + \int \left (- \sec {\left (e + f x \right )}\right )\, dx + \int \left (- 3 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int 3 \sec ^{3}{\left (e + f x \right )}\, dx + \int 3 \sec ^{4}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- \sec ^{6}{\left (e + f x \right )}\right )\, dx + \int \sec ^{7}{\left (e + f x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.58, size = 191, normalized size = 1.45 \begin {gather*} \frac {240 \, {\left (f x + e\right )} a^{3} c^{4} - 75 \, a^{3} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) + 75 \, a^{3} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) + \frac {2 \, {\left (315 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 1945 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} + 5118 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 3138 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 1095 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 165 \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{6}}}{240 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.59, size = 227, normalized size = 1.72 \begin {gather*} a^3\,c^4\,x+\frac {\frac {21\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}-\frac {389\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}+\frac {853\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{20}-\frac {523\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20}+\frac {73\,a^3\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{8}-\frac {11\,a^3\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8}}{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 )}-\frac {5\,a^3\,c^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{8\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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