Optimal. Leaf size=60 \[ \frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.0400507, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2669, 3767} \[ \frac{a \tan ^5(c+d x)}{5 d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 3767
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+b \sin (c+d x)) \, dx &=\frac{b \sec ^5(c+d x)}{5 d}+a \int \sec ^6(c+d x) \, dx\\ &=\frac{b \sec ^5(c+d x)}{5 d}-\frac{a \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{b \sec ^5(c+d x)}{5 d}+\frac{a \tan (c+d x)}{d}+\frac{2 a \tan ^3(c+d x)}{3 d}+\frac{a \tan ^5(c+d x)}{5 d}\\ \end{align*}
Mathematica [A] time = 0.179222, size = 53, normalized size = 0.88 \[ \frac{a \left (\frac{1}{5} \tan ^5(c+d x)+\frac{2}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{b \sec ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 48, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -a \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 ) +{\frac{b}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951036, size = 65, normalized size = 1.08 \begin{align*} \frac{{\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 + \frac{3 \, b}{\cos \left (d x + c\right )^{5}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09528, size = 127, normalized size = 2.12 \begin{align*} \frac{{\left (8 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 3 \, a\right )} \sin \left (d x + c\right ) + 3 \, b}{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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.10191, size = 162, normalized size = 2.7 \begin{align*} -\frac{2 \,{\left (15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 15 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 20 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 58 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 30 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 20 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 15 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 3 \, b\right )}}{15 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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