Optimal. Leaf size=65 \[ \frac {a^4 \tan ^7(c+d x)}{7 d}+\frac {3 a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^3(c+d x)}{d}+\frac {a^4 \tan (c+d x)}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3657, 12, 3767} \[ \frac {a^4 \tan ^7(c+d x)}{7 d}+\frac {3 a^4 \tan ^5(c+d x)}{5 d}+\frac {a^4 \tan ^3(c+d x)}{d}+\frac {a^4 \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3657
Rule 3767
Rubi steps
\begin {align*} \int \left (a+a \tan ^2(c+d x)\right )^4 \, dx &=\int a^4 \sec ^8(c+d x) \, dx\\ &=a^4 \int \sec ^8(c+d x) \, dx\\ &=-\frac {a^4 \operatorname {Subst}\left (\int \left (1+3 x^2+3 x^4+x^6\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac {a^4 \tan (c+d x)}{d}+\frac {a^4 \tan ^3(c+d x)}{d}+\frac {3 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 = 0.20, size = 46, normalized size = 0.71 \[ \frac {a^4 \left (\frac {1}{7} \tan ^7(c+d x)+\frac {3}{5} \tan ^5(c+d x)+\tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 56, normalized size = 0.86 \[ \frac {5 \, a^{4} \tan \left (d x + c\right )^{7} + 21 \, a^{4} \tan \left (d x + c\right )^{5} + 35 \, a^{4} \tan \left (d x + c\right )^{3} + 35 \, a^{4} \tan \left (d x + c\right )}{35 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 20.74, size = 519, normalized size = 7.98 \[ -\frac {35 \, a^{4} \tan \left (d x\right )^{7} \tan \relax (c)^{6} + 35 \, a^{4} \tan \left (d x\right )^{6} \tan \relax (c)^{7} + 35 \, a^{4} \tan \left (d x\right )^{7} \tan \relax (c)^{4} - 105 \, a^{4} \tan \left (d x\right )^{6} \tan \relax (c)^{5} - 105 \, a^{4} \tan \left (d x\right )^{5} \tan \relax (c)^{6} + 35 \, a^{4} \tan \left (d x\right )^{4} \tan \relax (c)^{7} + 21 \, a^{4} \tan \left (d x\right )^{7} \tan \relax (c)^{2} - 35 \, a^{4} \tan \left (d x\right )^{6} \tan \relax (c)^{3} + 315 \, a^{4} \tan \left (d x\right )^{5} \tan \relax (c)^{4} + 315 \, a^{4} \tan \left (d x\right )^{4} \tan \relax (c)^{5} - 35 \, a^{4} \tan \left (d x\right )^{3} \tan \relax (c)^{6} + 21 \, a^{4} \tan \left (d x\right )^{2} \tan \relax (c)^{7} + 5 \, a^{4} \tan \left (d x\right )^{7} - 7 \, a^{4} \tan \left (d x\right )^{6} \tan \relax (c) + 105 \, a^{4} \tan \left (d x\right )^{5} \tan \relax (c)^{2} - 315 \, a^{4} \tan \left (d x\right )^{4} \tan \relax (c)^{3} - 315 \, a^{4} \tan \left (d x\right )^{3} \tan \relax (c)^{4} + 105 \, a^{4} \tan \left (d x\right )^{2} \tan \relax (c)^{5} - 7 \, a^{4} \tan \left (d x\right ) \tan \relax (c)^{6} + 5 \, a^{4} \tan \relax (c)^{7} + 21 \, a^{4} \tan \left (d x\right )^{5} - 35 \, a^{4} \tan \left (d x\right )^{4} \tan \relax (c) + 315 \, a^{4} \tan \left (d x\right )^{3} \tan \relax (c)^{2} + 315 \, a^{4} \tan \left (d x\right )^{2} \tan \relax (c)^{3} - 35 \, a^{4} \tan \left (d x\right ) \tan \relax (c)^{4} + 21 \, a^{4} \tan \relax (c)^{5} + 35 \, a^{4} \tan \left (d x\right )^{3} - 105 \, a^{4} \tan \left (d x\right )^{2} \tan \relax (c) - 105 \, a^{4} \tan \left (d x\right ) \tan \relax (c)^{2} + 35 \, a^{4} \tan \relax (c)^{3} + 35 \, a^{4} \tan \left (d x\right ) + 35 \, a^{4} \tan \relax (c)}{35 \, {\left (d \tan \left (d x\right )^{7} \tan \relax (c)^{7} - 7 \, d \tan \left (d x\right )^{6} \tan \relax (c)^{6} + 21 \, d \tan \left (d x\right )^{5} \tan \relax (c)^{5} - 35 \, d \tan \left (d x\right )^{4} \tan \relax (c)^{4} + 35 \, d \tan \left (d x\right )^{3} \tan \relax (c)^{3} - 21 \, d \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 7 \, d \tan \left (d x\right ) \tan \relax (c) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 43, normalized size = 0.66 \[ \frac {a^{4} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}+\frac {3 \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\tan ^{3}\left (d x +c \right )+\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.47, size = 157, normalized size = 2.42 \[ a^{4} x + \frac {{\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} a^{4}}{105 \, d} + \frac {4 \, {\left (3 \, \tan \left (d x + c\right )^{5} - 5 \, \tan \left (d x + c\right )^{3} - 15 \, d x - 15 \, c + 15 \, \tan \left (d x + c\right )\right )} a^{4}}{15 \, d} + \frac {2 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, d x + 3 \, c - 3 \, \tan \left (d x + c\right )\right )} a^{4}}{d} - \frac {4 \, {\left (d x + c - \tan \left (d x + c\right )\right )} a^{4}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.52, size = 53, normalized size = 0.82 \[ \frac {\frac {a^4\,{\mathrm {tan}\left (c+d\,x\right )}^7}{7}+\frac {3\,a^4\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+a^4\,{\mathrm {tan}\left (c+d\,x\right )}^3+a^4\,\mathrm {tan}\left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.06, size = 68, normalized size = 1.05 \[ \begin {cases} \frac {a^{4} \tan ^{7}{\left (c + d x \right )}}{7 d} + \frac {3 a^{4} \tan ^{5}{\left (c + d x \right )}}{5 d} + \frac {a^{4} \tan ^{3}{\left (c + d x \right )}}{d} + \frac {a^{4} \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \tan ^{2}{\relax (c )} + a\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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