Optimal. Leaf size=62 \[ \frac{4 \tan ^3(c+d x)}{15 a d}+\frac{4 \tan (c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)} \]
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Rubi [A] time = 0.0593119, antiderivative size = 62, 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 = {2672, 3767} \[ \frac{4 \tan ^3(c+d x)}{15 a d}+\frac{4 \tan (c+d x)}{5 a d}-\frac{\sec ^3(c+d x)}{5 d (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 3767
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{a+a \sin (c+d x)} \, dx &=-\frac{\sec ^3(c+d x)}{5 d (a+a \sin (c+d x))}+\frac{4 \int \sec ^4(c+d x) \, dx}{5 a}\\ &=-\frac{\sec ^3(c+d x)}{5 d (a+a \sin (c+d x))}-\frac{4 \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{5 a d}\\ &=-\frac{\sec ^3(c+d x)}{5 d (a+a \sin (c+d x))}+\frac{4 \tan (c+d x)}{5 a d}+\frac{4 \tan ^3(c+d x)}{15 a d}\\ \end{align*}
Mathematica [A] time = 0.0969105, size = 66, normalized size = 1.06 \[ -\frac{\sec ^3(c+d x) (-2 (3 \sin (c+d x)+\sin (3 (c+d x)))+2 \cos (2 (c+d x))+\cos (4 (c+d x)))}{15 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 130, normalized size = 2.1 \begin{align*} 2\,{\frac{1}{da} \left ( -1/12\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-3}-1/8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{-2}-{\frac{5}{16\,\tan \left ( 1/2\,dx+c/2 \right ) -16}}-1/5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-5}+1/2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-4}-5/6\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-3}+3/4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{-2}-{\frac{11}{16\,\tan \left ( 1/2\,dx+c/2 \right ) +16}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.966609, size = 397, normalized size = 6.4 \begin{align*} \frac{2 \,{\left (\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{21 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{13 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{25 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{5 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{15 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 3\right )}}{15 \,{\left (a + \frac{2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac{6 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{6 \, a \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{2 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac{2 \, a \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.64399, size = 194, normalized size = 3.13 \begin{align*} -\frac{8 \, \cos \left (d x + c\right )^{4} - 4 \, \cos \left (d x + c\right )^{2} - 4 \,{\left (2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right ) - 1}{15 \,{\left (a d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{3}\right )}} \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*} \frac{\int \frac{\sec ^{4}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14159, size = 161, normalized size = 2.6 \begin{align*} -\frac{\frac{5 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 13\right )}}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1\right )}^{3}} + \frac{165 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 480 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 650 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 400 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 113}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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