Optimal. Leaf size=65 \[ \frac{(5 A+4 C) \tan ^3(c+d x)}{15 d}+\frac{(5 A+4 C) \tan (c+d x)}{5 d}+\frac{C \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
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Rubi [A] time = 0.0439876, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4046, 3767} \[ \frac{(5 A+4 C) \tan ^3(c+d x)}{15 d}+\frac{(5 A+4 C) \tan (c+d x)}{5 d}+\frac{C \tan (c+d x) \sec ^4(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 4046
Rule 3767
Rubi steps
\begin{align*} \int \sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac{C \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{1}{5} (5 A+4 C) \int \sec ^4(c+d x) \, dx\\ &=\frac{C \sec ^4(c+d x) \tan (c+d x)}{5 d}-\frac{(5 A+4 C) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 d}\\ &=\frac{(5 A+4 C) \tan (c+d x)}{5 d}+\frac{C \sec ^4(c+d x) \tan (c+d x)}{5 d}+\frac{(5 A+4 C) \tan ^3(c+d x)}{15 d}\\ \end{align*}
Mathematica [A] time = 0.193881, size = 61, normalized size = 0.94 \[ \frac{A \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{C \left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 58, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ( -A \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \tan \left ( dx+c \right ) -C \left ( -{\frac{8}{15}}-{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \tan \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.927539, size = 58, normalized size = 0.89 \begin{align*} \frac{3 \, C \tan \left (d x + c\right )^{5} + 5 \,{\left (A + 2 \, C\right )} \tan \left (d x + c\right )^{3} + 15 \,{\left (A + C\right )} \tan \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.468252, size = 140, normalized size = 2.15 \begin{align*} \frac{{\left (2 \,{\left (5 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{4} +{\left (5 \, A + 4 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, C\right )} \sin \left (d x + c\right )}{15 \, d \cos \left (d x + c\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18374, size = 77, normalized size = 1.18 \begin{align*} \frac{3 \, C \tan \left (d x + c\right )^{5} + 5 \, A \tan \left (d x + c\right )^{3} + 10 \, C \tan \left (d x + c\right )^{3} + 15 \, A \tan \left (d x + c\right ) + 15 \, C \tan \left (d x + c\right )}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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