Optimal. Leaf size=150 \[ \frac {7 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {5 a^2 c^4 \sec (e+f x) \tan (e+f x)}{16 f}-\frac {a^2 c^4 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac {a^2 c^4 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}-\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f} \]
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Rubi [A]
time = 0.18, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4043, 2691,
3855, 2687, 30, 3853} \begin {gather*} -\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac {7 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}+\frac {a^2 c^4 \tan ^3(e+f x) \sec ^3(e+f x)}{6 f}-\frac {a^2 c^4 \tan (e+f x) \sec ^3(e+f x)}{8 f}+\frac {a^2 c^4 \tan ^3(e+f x) \sec (e+f x)}{4 f}-\frac {5 a^2 c^4 \tan (e+f x) \sec (e+f x)}{16 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2687
Rule 2691
Rule 3853
Rule 3855
Rule 4043
Rubi steps
\begin {align*} \int \sec (e+f x) (a+a \sec (e+f x))^2 (c-c \sec (e+f x))^4 \, dx &=\left (a^2 c^2\right ) \int \left (c^2 \sec (e+f x) \tan ^4(e+f x)-2 c^2 \sec ^2(e+f x) \tan ^4(e+f x)+c^2 \sec ^3(e+f x) \tan ^4(e+f x)\right ) \, dx\\ &=\left (a^2 c^4\right ) \int \sec (e+f x) \tan ^4(e+f x) \, dx+\left (a^2 c^4\right ) \int \sec ^3(e+f x) \tan ^4(e+f x) \, dx-\left (2 a^2 c^4\right ) \int \sec ^2(e+f x) \tan ^4(e+f x) \, dx\\ &=\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac {a^2 c^4 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}-\frac {1}{2} \left (a^2 c^4\right ) \int \sec ^3(e+f x) \tan ^2(e+f x) \, dx-\frac {1}{4} \left (3 a^2 c^4\right ) \int \sec (e+f x) \tan ^2(e+f x) \, dx-\frac {\left (2 a^2 c^4\right ) \text {Subst}\left (\int x^4 \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {3 a^2 c^4 \sec (e+f x) \tan (e+f x)}{8 f}-\frac {a^2 c^4 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac {a^2 c^4 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}-\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac {1}{8} \left (a^2 c^4\right ) \int \sec ^3(e+f x) \, dx+\frac {1}{8} \left (3 a^2 c^4\right ) \int \sec (e+f x) \, dx\\ &=\frac {3 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{8 f}-\frac {5 a^2 c^4 \sec (e+f x) \tan (e+f x)}{16 f}-\frac {a^2 c^4 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac {a^2 c^4 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}-\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f}+\frac {1}{16} \left (a^2 c^4\right ) \int \sec (e+f x) \, dx\\ &=\frac {7 a^2 c^4 \tanh ^{-1}(\sin (e+f x))}{16 f}-\frac {5 a^2 c^4 \sec (e+f x) \tan (e+f x)}{16 f}-\frac {a^2 c^4 \sec ^3(e+f x) \tan (e+f x)}{8 f}+\frac {a^2 c^4 \sec (e+f x) \tan ^3(e+f x)}{4 f}+\frac {a^2 c^4 \sec ^3(e+f x) \tan ^3(e+f x)}{6 f}-\frac {2 a^2 c^4 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A]
time = 1.40, size = 91, normalized size = 0.61 \begin {gather*} \frac {a^2 c^4 \left (1680 \tanh ^{-1}(\sin (e+f x))+\sec ^6(e+f x) (330 \sin (e+f x)-240 \sin (2 (e+f x))-445 \sin (3 (e+f x))+192 \sin (4 (e+f x))-135 \sin (5 (e+f x))-48 \sin (6 (e+f x)))\right )}{3840 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 255, normalized size = 1.70
method | result | size |
norman | \(\frac {-\frac {7 a^{2} c^{4} \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {7 c^{4} a^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {119 c^{4} a^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {231 c^{4} a^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{20 f}+\frac {281 c^{4} a^{2} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{20 f}+\frac {119 c^{4} a^{2} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {7 c^{4} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{16 f}+\frac {7 c^{4} a^{2} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{16 f}\) | \(195\) |
risch | \(\frac {i a^{2} c^{4} \left (135 \,{\mathrm e}^{11 i \left (f x +e \right )}-480 \,{\mathrm e}^{10 i \left (f x +e \right )}+445 \,{\mathrm e}^{9 i \left (f x +e \right )}-480 \,{\mathrm e}^{8 i \left (f x +e \right )}-330 \,{\mathrm e}^{7 i \left (f x +e \right )}-960 \,{\mathrm e}^{6 i \left (f x +e \right )}+330 \,{\mathrm e}^{5 i \left (f x +e \right )}-960 \,{\mathrm e}^{4 i \left (f x +e \right )}-445 \,{\mathrm e}^{3 i \left (f x +e \right )}-96 \,{\mathrm e}^{2 i \left (f x +e \right )}-135 \,{\mathrm e}^{i \left (f x +e \right )}-96\right )}{120 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{6}}-\frac {7 c^{4} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}{16 f}+\frac {7 c^{4} a^{2} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}{16 f}\) | \(198\) |
derivativedivides | \(\frac {a^{2} c^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+2 c^{4} a^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-c^{4} a^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-4 c^{4} a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-c^{4} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{4} a^{2} \tan \left (f x +e \right )+c^{4} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}\) | \(255\) |
default | \(\frac {a^{2} c^{4} \left (-\left (-\frac {\left (\sec ^{5}\left (f x +e \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (f x +e \right )\right )}{24}-\frac {5 \sec \left (f x +e \right )}{16}\right ) \tan \left (f x +e \right )+\frac {5 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{16}\right )+2 c^{4} a^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-c^{4} a^{2} \left (-\left (-\frac {\left (\sec ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \sec \left (f x +e \right )}{8}\right ) \tan \left (f x +e \right )+\frac {3 \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{8}\right )-4 c^{4} a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-c^{4} a^{2} \left (\frac {\sec \left (f x +e \right ) \tan \left (f x +e \right )}{2}+\frac {\ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{2}\right )-2 c^{4} a^{2} \tan \left (f x +e \right )+c^{4} a^{2} \ln \left (\sec \left (f x +e \right )+\tan \left (f x +e \right )\right )}{f}\) | \(255\) |
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. 347 vs.
\(2 (148) = 296\).
time = 0.28, size = 347, normalized size = 2.31 \begin {gather*} -\frac {64 \, {\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^{2} c^{4} - 640 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{2} c^{4} + 5 \, a^{2} 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 )} - 30 \, a^{2} 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 )} - 120 \, a^{2} 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^{2} c^{4} \log \left (\sec \left (f x + e\right ) + \tan \left (f x + e\right )\right ) + 960 \, a^{2} 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 = 2.87, size = 172, normalized size = 1.15 \begin {gather*} \frac {105 \, a^{2} c^{4} \cos \left (f x + e\right )^{6} \log \left (\sin \left (f x + e\right ) + 1\right ) - 105 \, a^{2} c^{4} \cos \left (f x + e\right )^{6} \log \left (-\sin \left (f x + e\right ) + 1\right ) - 2 \, {\left (96 \, a^{2} c^{4} \cos \left (f x + e\right )^{5} + 135 \, a^{2} c^{4} \cos \left (f x + e\right )^{4} - 192 \, a^{2} c^{4} \cos \left (f x + e\right )^{3} + 10 \, a^{2} c^{4} \cos \left (f x + e\right )^{2} + 96 \, a^{2} c^{4} \cos \left (f x + e\right ) - 40 \, a^{2} 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^{2} c^{4} \left (\int \sec {\left (e + f x \right )}\, dx + \int \left (- 2 \sec ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- \sec ^{3}{\left (e + f x \right )}\right )\, dx + \int 4 \sec ^{4}{\left (e + f x \right )}\, dx + \int \left (- \sec ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 \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 = 0.61, size = 178, normalized size = 1.19 \begin {gather*} \frac {105 \, a^{2} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right ) - 105 \, a^{2} c^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right ) - \frac {2 \, {\left (105 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{11} - 595 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{9} - 1686 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 1386 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 595 \, a^{2} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 105 \, a^{2} 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 = 5.61, size = 219, normalized size = 1.46 \begin {gather*} \frac {-\frac {7\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}}{8}+\frac {119\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9}{24}+\frac {281\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{20}-\frac {231\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{20}+\frac {119\,a^2\,c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24}-\frac {7\,a^2\,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 {7\,a^2\,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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