Optimal. Leaf size=74 \[ \frac {a^4 \tan ^7(c+d x)}{7 d}-\frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^3(c+d x)}{3 d}-\frac {a^4 \tan (c+d x)}{d}+a^4 x \]
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Rubi [A] time = 0.05, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4120, 3473, 8} \[ \frac {a^4 \tan ^7(c+d x)}{7 d}-\frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^3(c+d x)}{3 d}-\frac {a^4 \tan (c+d x)}{d}+a^4 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rule 4120
Rubi steps
\begin {align*} \int \left (a-a \sec ^2(c+d x)\right )^4 \, dx &=a^4 \int \tan ^8(c+d x) \, dx\\ &=\frac {a^4 \tan ^7(c+d x)}{7 d}-a^4 \int \tan ^6(c+d x) \, dx\\ &=-\frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^7(c+d x)}{7 d}+a^4 \int \tan ^4(c+d x) \, dx\\ &=\frac {a^4 \tan ^3(c+d x)}{3 d}-\frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^7(c+d x)}{7 d}-a^4 \int \tan ^2(c+d x) \, dx\\ &=-\frac {a^4 \tan (c+d x)}{d}+\frac {a^4 \tan ^3(c+d x)}{3 d}-\frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^7(c+d x)}{7 d}+a^4 \int 1 \, dx\\ &=a^4 x-\frac {a^4 \tan (c+d x)}{d}+\frac {a^4 \tan ^3(c+d x)}{3 d}-\frac {a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 72, normalized size = 0.97 \[ a^4 \left (\frac {\tan ^{-1}(\tan (c+d x))}{d}+\frac {\tan ^7(c+d x)}{7 d}-\frac {\tan ^5(c+d x)}{5 d}+\frac {\tan ^3(c+d x)}{3 d}-\frac {\tan (c+d x)}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 82, normalized size = 1.11 \[ \frac {105 \, a^{4} d x \cos \left (d x + c\right )^{7} - {\left (176 \, a^{4} \cos \left (d x + c\right )^{6} - 122 \, a^{4} \cos \left (d x + c\right )^{4} + 66 \, a^{4} \cos \left (d x + c\right )^{2} - 15 \, a^{4}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 66, normalized size = 0.89 \[ \frac {15 \, a^{4} \tan \left (d x + c\right )^{7} - 21 \, a^{4} \tan \left (d x + c\right )^{5} + 35 \, a^{4} \tan \left (d x + c\right )^{3} + 105 \, {\left (d x + c\right )} a^{4} - 105 \, a^{4} \tan \left (d x + c\right )}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.49, size = 125, normalized size = 1.69 \[ \frac {-a^{4} \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )+4 a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-6 a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-4 a^{4} \tan \left (d x +c \right )+a^{4} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 129, normalized size = 1.74 \[ a^{4} x + \frac {{\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{4}}{35 \, d} - \frac {4 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} a^{4}}{15 \, d} + \frac {2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{4}}{d} - \frac {4 \, a^{4} \tan \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.62, size = 61, normalized size = 0.82 \[ \frac {\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}-\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}-a^4\,\mathrm {tan}\left (c+d\,x\right )+d\,x\,a^4}{d} \]
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 \[ a^{4} \left (\int 1\, dx + \int \left (- 4 \sec ^{2}{\left (c + d x \right )}\right )\, dx + \int 6 \sec ^{4}{\left (c + d x \right )}\, dx + \int \left (- 4 \sec ^{6}{\left (c + d x \right )}\right )\, dx + \int \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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