Optimal. Leaf size=92 \[ \frac{2 a^6 \cos (c+d x)}{15 d \left (a^3-a^3 \sin (c+d x)\right )}+\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2} \]
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Rubi [A] time = 0.0934226, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2670, 2650, 2648} \[ \frac{2 a^6 \cos (c+d x)}{15 d \left (a^3-a^3 \sin (c+d x)\right )}+\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 2670
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \sec ^6(c+d x) (a+a \sin (c+d x))^3 \, dx &=a^6 \int \frac{1}{(a-a \sin (c+d x))^3} \, dx\\ &=\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{1}{5} \left (2 a^5\right ) \int \frac{1}{(a-a \sin (c+d x))^2} \, dx\\ &=\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac{1}{15} \left (2 a^4\right ) \int \frac{1}{a-a \sin (c+d x)} \, dx\\ &=\frac{a^6 \cos (c+d x)}{5 d (a-a \sin (c+d x))^3}+\frac{2 a^5 \cos (c+d x)}{15 d (a-a \sin (c+d x))^2}+\frac{2 a^4 \cos (c+d x)}{15 d (a-a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0196873, size = 110, normalized size = 1.2 \[ \frac{2 a^3 \tan ^5(c+d x)}{15 d}+\frac{7 a^3 \sec ^5(c+d x)}{15 d}+\frac{a^3 \tan ^2(c+d x) \sec ^3(c+d x)}{3 d}-\frac{a^3 \tan ^3(c+d x) \sec ^2(c+d x)}{3 d}+\frac{a^3 \tan (c+d x) \sec ^4(c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 171, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ({\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{15\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{15\,\cos \left ( dx+c \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \right ) \cos \left ( dx+c \right ) }{15}} \right ) +3\,{a}^{3} \left ( 1/5\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}+2/15\,{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{3}}} \right ) +{\frac{3\,{a}^{3}}{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}}-{a}^{3} \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 = 0.971747, size = 139, normalized size = 1.51 \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^{3} + 3 \,{\left (3 \, \tan \left (d x + c\right )^{5} + 5 \, \tan \left (d x + c\right )^{3}\right )} a^{3} - \frac{{\left (5 \, \cos \left (d x + c\right )^{2} - 3\right )} a^{3}}{\cos \left (d x + c\right )^{5}} + \frac{9 \, a^{3}}{\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 = 1.60439, size = 367, normalized size = 3.99 \begin{align*} \frac{2 \, a^{3} \cos \left (d x + c\right )^{3} - 4 \, a^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3} +{\left (2 \, a^{3} \cos \left (d x + c\right )^{2} + 6 \, a^{3} \cos \left (d x + c\right ) - 3 \, a^{3}\right )} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{3} + 3 \, d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) -{\left (d \cos \left (d x + c\right )^{2} - 2 \, d \cos \left (d x + c\right ) - 4 \, d\right )} \sin \left (d x + c\right ) - 4 \, d\right )}} \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 = 1.15781, size = 116, normalized size = 1.26 \begin{align*} -\frac{2 \,{\left (15 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 30 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 20 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 7 \, a^{3}\right )}}{15 \, d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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