Optimal. Leaf size=233 \[ \frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1155 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{4096 \sqrt{2} a^{5/2} d}-\frac{1155 \cos (c+d x)}{4096 a d (a \sin (c+d x)+a)^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}-\frac{77 \sec (c+d x)}{512 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.363838, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2681, 2687, 2650, 2649, 206} \[ \frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a \sin (c+d x)+a}}-\frac{1155 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{4096 \sqrt{2} a^{5/2} d}-\frac{1155 \cos (c+d x)}{4096 a d (a \sin (c+d x)+a)^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a \sin (c+d x)+a)^{3/2}}-\frac{\sec ^3(c+d x)}{8 d (a \sin (c+d x)+a)^{5/2}}-\frac{77 \sec (c+d x)}{512 a d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2681
Rule 2687
Rule 2650
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}+\frac{11 \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{16 a}\\ &=-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac{33 \int \frac{\sec ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{64 a^2}\\ &=-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{77 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{128 a}\\ &=-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac{77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{385 \int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{1024 a^2}\\ &=-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac{77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{1155 \int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx}{2048 a}\\ &=-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac{1155 \cos (c+d x)}{4096 a d (a+a \sin (c+d x))^{3/2}}-\frac{77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{1155 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{8192 a^2}\\ &=-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac{1155 \cos (c+d x)}{4096 a d (a+a \sin (c+d x))^{3/2}}-\frac{77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}-\frac{1155 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{4096 a^2 d}\\ &=-\frac{1155 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{4096 \sqrt{2} a^{5/2} d}-\frac{\sec ^3(c+d x)}{8 d (a+a \sin (c+d x))^{5/2}}-\frac{1155 \cos (c+d x)}{4096 a d (a+a \sin (c+d x))^{3/2}}-\frac{77 \sec (c+d x)}{512 a d (a+a \sin (c+d x))^{3/2}}-\frac{11 \sec ^3(c+d x)}{96 a d (a+a \sin (c+d x))^{3/2}}+\frac{385 \sec (c+d x)}{1024 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{11 \sec ^3(c+d x)}{64 a^2 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.515932, size = 394, normalized size = 1.69 \[ \frac{\frac{1920 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{256 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}-1545 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4+3090 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3-1036 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2+2072 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{1472 \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-\frac{384}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{768 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}+(3465+3465 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )-736}{12288 d (a (\sin (c+d x)+1))^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.192, size = 355, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( 24576\,\sin \left ( dx+c \right ) -24576 \right ) \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) d} \left ( 6930\,{a}^{11/2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}-924\, \left ( 16\,{a}^{11/2}+15\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{2}{a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) + \left ( -5632\,{a}^{11/2}+27720\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{2}{a}^{4} \right ) \sin \left ( dx+c \right ) + \left ( 16170\,{a}^{11/2}+3465\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{2}{a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}-1320\, \left ( 8\,{a}^{11/2}+21\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{2}{a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2560\,{a}^{11/2}+27720\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \left ( a-a\sin \left ( dx+c \right ) \right ) ^{3/2}\sqrt{2}{a}^{4} \right ){a}^{-{\frac{15}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 2.61488, size = 833, normalized size = 3.58 \begin{align*} \frac{3465 \, \sqrt{2}{\left (3 \, \cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3} +{\left (\cos \left (d x + c\right )^{5} - 4 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{a} \log \left (-\frac{a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )} + 3 \, a \cos \left (d x + c\right ) -{\left (a \cos \left (d x + c\right ) - 2 \, a\right )} \sin \left (d x + c\right ) + 2 \, a}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right ) + 4 \,{\left (8085 \, \cos \left (d x + c\right )^{4} - 5280 \, \cos \left (d x + c\right )^{2} + 11 \,{\left (315 \, \cos \left (d x + c\right )^{4} - 672 \, \cos \left (d x + c\right )^{2} - 256\right )} \sin \left (d x + c\right ) - 1280\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{49152 \,{\left (3 \, a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3} +{\left (a^{3} d \cos \left (d x + c\right )^{5} - 4 \, a^{3} d \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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