Optimal. Leaf size=68 \[ a^3 c^3 x-\frac {a^3 c^3 \tan (e+f x)}{f}+\frac {a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^3 \tan ^5(e+f x)}{5 f} \]
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Rubi [A]
time = 0.06, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3989, 3554, 8}
\begin {gather*} -\frac {a^3 c^3 \tan ^5(e+f x)}{5 f}+\frac {a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^3 \tan (e+f x)}{f}+a^3 c^3 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3554
Rule 3989
Rubi steps
\begin {align*} \int (a+a \sec (e+f x))^3 (c-c \sec (e+f x))^3 \, dx &=-\left (\left (a^3 c^3\right ) \int \tan ^6(e+f x) \, dx\right )\\ &=-\frac {a^3 c^3 \tan ^5(e+f x)}{5 f}+\left (a^3 c^3\right ) \int \tan ^4(e+f x) \, dx\\ &=\frac {a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^3 \tan ^5(e+f x)}{5 f}-\left (a^3 c^3\right ) \int \tan ^2(e+f x) \, dx\\ &=-\frac {a^3 c^3 \tan (e+f x)}{f}+\frac {a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^3 \tan ^5(e+f x)}{5 f}+\left (a^3 c^3\right ) \int 1 \, dx\\ &=a^3 c^3 x-\frac {a^3 c^3 \tan (e+f x)}{f}+\frac {a^3 c^3 \tan ^3(e+f x)}{3 f}-\frac {a^3 c^3 \tan ^5(e+f x)}{5 f}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 61, normalized size = 0.90 \begin {gather*} -a^3 c^3 \left (-\frac {\text {ArcTan}(\tan (e+f x))}{f}+\frac {\tan (e+f x)}{f}-\frac {\tan ^3(e+f x)}{3 f}+\frac {\tan ^5(e+f x)}{5 f}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 93, normalized size = 1.37
method | result | size |
risch | \(a^{3} c^{3} x -\frac {2 i c^{3} a^{3} \left (45 \,{\mathrm e}^{8 i \left (f x +e \right )}+90 \,{\mathrm e}^{6 i \left (f x +e \right )}+140 \,{\mathrm e}^{4 i \left (f x +e \right )}+70 \,{\mathrm e}^{2 i \left (f x +e \right )}+23\right )}{15 f \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )^{5}}\) | \(81\) |
derivativedivides | \(\frac {c^{3} a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \tan \left (f x +e \right )+c^{3} a^{3} \left (f x +e \right )}{f}\) | \(93\) |
default | \(\frac {c^{3} a^{3} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (f x +e \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (f x +e \right )\right )}{15}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (f x +e \right )\right )}{3}\right ) \tan \left (f x +e \right )-3 c^{3} a^{3} \tan \left (f x +e \right )+c^{3} a^{3} \left (f x +e \right )}{f}\) | \(93\) |
norman | \(\frac {a^{3} c^{3} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-a^{3} c^{3} x +5 a^{3} c^{3} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-10 a^{3} c^{3} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+10 a^{3} c^{3} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-5 a^{3} c^{3} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {2 c^{3} a^{3} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {32 c^{3} a^{3} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {356 c^{3} a^{3} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 f}-\frac {32 c^{3} a^{3} \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {2 c^{3} a^{3} \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}}{\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}\) | \(234\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 101, normalized size = 1.49 \begin {gather*} -\frac {{\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^{3} - 15 \, {\left (\tan \left (f x + e\right )^{3} + 3 \, \tan \left (f x + e\right )\right )} a^{3} c^{3} - 15 \, {\left (f x + e\right )} a^{3} c^{3} + 45 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.93, size = 86, normalized size = 1.26 \begin {gather*} \frac {15 \, a^{3} c^{3} f x \cos \left (f x + e\right )^{5} - {\left (23 \, a^{3} c^{3} \cos \left (f x + e\right )^{4} - 11 \, a^{3} c^{3} \cos \left (f x + e\right )^{2} + 3 \, a^{3} c^{3}\right )} \sin \left (f x + e\right )}{15 \, f \cos \left (f x + e\right )^{5}} \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^{3} \left (\int \left (-1\right )\, dx + \int 3 \sec ^{2}{\left (e + f x \right )}\, dx + \int \left (- 3 \sec ^{4}{\left (e + f x \right )}\right )\, dx + \int \sec ^{6}{\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 = 0.54, size = 65, normalized size = 0.96 \begin {gather*} -\frac {3 \, a^{3} c^{3} \tan \left (f x + e\right )^{5} - 5 \, a^{3} c^{3} \tan \left (f x + e\right )^{3} - 15 \, {\left (f x + e\right )} a^{3} c^{3} + 15 \, a^{3} c^{3} \tan \left (f x + e\right )}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.96, size = 122, normalized size = 1.79 \begin {gather*} a^3\,c^3\,x+\frac {2\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9-\frac {32\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7}{3}+\frac {356\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5}{15}-\frac {32\,a^3\,c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{3}+2\,a^3\,c^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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