Optimal. Leaf size=44 \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.035826, antiderivative size = 44, 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} \[ \frac{a \tan ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 3767
Rubi steps
\begin{align*} \int \sec ^4(c+d x) (a+b \sin (c+d x)) \, dx &=\frac{b \sec ^3(c+d x)}{3 d}+a \int \sec ^4(c+d x) \, dx\\ &=\frac{b \sec ^3(c+d x)}{3 d}-\frac{a \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{b \sec ^3(c+d x)}{3 d}+\frac{a \tan (c+d x)}{d}+\frac{a \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.0793873, size = 41, normalized size = 0.93 \[ \frac{a \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{b \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 38, 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 ) +{\frac{b}{3\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967181, size = 47, normalized size = 1.07 \begin{align*} \frac{{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a + \frac{b}{\cos \left (d x + c\right )^{3}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98761, size = 92, normalized size = 2.09 \begin{align*} \frac{{\left (2 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + b}{3 \, d \cos \left (d x + c\right )^{3}} \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 [A] time = 1.0921, size = 103, normalized size = 2.34 \begin{align*} -\frac{2 \,{\left (3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b\right )}}{3 \,{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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