Optimal. Leaf size=120 \[ -\frac{a^2}{24 d (a \sin (c+d x)+a)^3}+\frac{a}{32 d (a-a \sin (c+d x))^2}-\frac{3 a}{32 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{3}{16 d (a \sin (c+d x)+a)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d} \]
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Rubi [A] time = 0.106839, antiderivative size = 120, 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 = {2667, 44, 206} \[ -\frac{a^2}{24 d (a \sin (c+d x)+a)^3}+\frac{a}{32 d (a-a \sin (c+d x))^2}-\frac{3 a}{32 d (a \sin (c+d x)+a)^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{3}{16 d (a \sin (c+d x)+a)}+\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d} \]
Antiderivative was successfully verified.
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Rule 2667
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^5(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac{a^5 \operatorname{Subst}\left (\int \frac{1}{(a-x)^3 (a+x)^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^5 \operatorname{Subst}\left (\int \left (\frac{1}{16 a^4 (a-x)^3}+\frac{1}{8 a^5 (a-x)^2}+\frac{1}{8 a^3 (a+x)^4}+\frac{3}{16 a^4 (a+x)^3}+\frac{3}{16 a^5 (a+x)^2}+\frac{5}{16 a^5 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a}{32 d (a-a \sin (c+d x))^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{a^2}{24 d (a+a \sin (c+d x))^3}-\frac{3 a}{32 d (a+a \sin (c+d x))^2}-\frac{3}{16 d (a+a \sin (c+d x))}+\frac{5 \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{16 d}\\ &=\frac{5 \tanh ^{-1}(\sin (c+d x))}{16 a d}+\frac{a}{32 d (a-a \sin (c+d x))^2}+\frac{1}{8 d (a-a \sin (c+d x))}-\frac{a^2}{24 d (a+a \sin (c+d x))^3}-\frac{3 a}{32 d (a+a \sin (c+d x))^2}-\frac{3}{16 d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.149027, size = 97, normalized size = 0.81 \[ \frac{\sec ^4(c+d x) \left (-15 \sin ^4(c+d x)-15 \sin ^3(c+d x)+25 \sin ^2(c+d x)+25 \sin (c+d x)+15 (\sin (c+d x)-1)^2 (\sin (c+d x)+1)^3 \tanh ^{-1}(\sin (c+d x))-8\right )}{48 a d (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 126, normalized size = 1.1 \begin{align*}{\frac{1}{32\,da \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2}}}-{\frac{1}{8\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{5\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{32\,da}}-{\frac{1}{24\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}}}-{\frac{3}{32\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}-{\frac{3}{16\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}+{\frac{5\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{32\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.947961, size = 176, normalized size = 1.47 \begin{align*} -\frac{\frac{2 \,{\left (15 \, \sin \left (d x + c\right )^{4} + 15 \, \sin \left (d x + c\right )^{3} - 25 \, \sin \left (d x + c\right )^{2} - 25 \, \sin \left (d x + c\right ) + 8\right )}}{a \sin \left (d x + c\right )^{5} + a \sin \left (d x + c\right )^{4} - 2 \, a \sin \left (d x + c\right )^{3} - 2 \, a \sin \left (d x + c\right )^{2} + a \sin \left (d x + c\right ) + a} - \frac{15 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a} + \frac{15 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67871, size = 400, normalized size = 3.33 \begin{align*} -\frac{30 \, \cos \left (d x + c\right )^{4} - 10 \, \cos \left (d x + c\right )^{2} - 15 \,{\left (\cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \,{\left (\cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 10 \,{\left (3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sin \left (d x + c\right ) - 4}{96 \,{\left (a d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{4}\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 ^{5}{\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 [A] time = 1.18127, size = 157, normalized size = 1.31 \begin{align*} \frac{\frac{30 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{30 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{3 \,{\left (15 \, \sin \left (d x + c\right )^{2} - 38 \, \sin \left (d x + c\right ) + 25\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}^{2}} - \frac{55 \, \sin \left (d x + c\right )^{3} + 201 \, \sin \left (d x + c\right )^{2} + 255 \, \sin \left (d x + c\right ) + 117}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{3}}}{192 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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