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 = 0.0350688, antiderivative size = 65, normalized size of antiderivative = 1., 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 3657
Rule 12
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*}
Mathematica [A] time = 0.1956, 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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Maple [A] time = 0.003, size = 43, normalized size = 0.7 \begin{align*}{\frac{{a}^{4}}{d} \left ({\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{7}}{7}}+{\frac{3\, \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5}}+ \left ( \tan \left ( dx+c \right ) \right ) ^{3}+\tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.73162, size = 212, normalized size = 3.26 \begin{align*} 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.998153, size = 136, normalized size = 2.09 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.20402, size = 68, normalized size = 1.05 \begin{align*} \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}{\left (c \right )} + a\right )^{4} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.31117, size = 701, normalized size = 10.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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