Optimal. Leaf size=256 \[ -\frac{3003 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{8192 \sqrt{2} a^{3/2} d}-\frac{3003 \cos (c+d x)}{8192 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^5(c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a \sin (c+d x)+a}}-\frac{143 \sec ^3(c+d x)}{960 d (a \sin (c+d x)+a)^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a \sin (c+d x)+a}}-\frac{1001 \sec (c+d x)}{5120 d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.429009, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ -\frac{3003 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{8192 \sqrt{2} a^{3/2} d}-\frac{3003 \cos (c+d x)}{8192 d (a \sin (c+d x)+a)^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a \sin (c+d x)+a}}-\frac{\sec ^5(c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a \sin (c+d x)+a}}-\frac{143 \sec ^3(c+d x)}{960 d (a \sin (c+d x)+a)^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a \sin (c+d x)+a}}-\frac{1001 \sec (c+d x)}{5120 d (a \sin (c+d x)+a)^{3/2}} \]
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 ^6(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{13 \int \frac{\sec ^6(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{16 a}\\ &=-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{143}{160} \int \frac{\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{429 \int \frac{\sec ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{640 a}\\ &=-\frac{143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{1001 \int \frac{\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{1280}\\ &=-\frac{1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac{143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{1001 \int \frac{\sec ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx}{2048 a}\\ &=-\frac{1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac{143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a+a \sin (c+d x)}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{3003 \int \frac{1}{(a+a \sin (c+d x))^{3/2}} \, dx}{4096}\\ &=-\frac{3003 \cos (c+d x)}{8192 d (a+a \sin (c+d x))^{3/2}}-\frac{1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac{143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a+a \sin (c+d x)}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}+\frac{3003 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{16384 a}\\ &=-\frac{3003 \cos (c+d x)}{8192 d (a+a \sin (c+d x))^{3/2}}-\frac{1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac{143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a+a \sin (c+d x)}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}-\frac{3003 \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 )}{8192 a d}\\ &=-\frac{3003 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{8192 \sqrt{2} a^{3/2} d}-\frac{3003 \cos (c+d x)}{8192 d (a+a \sin (c+d x))^{3/2}}-\frac{1001 \sec (c+d x)}{5120 d (a+a \sin (c+d x))^{3/2}}-\frac{143 \sec ^3(c+d x)}{960 d (a+a \sin (c+d x))^{3/2}}-\frac{\sec ^5(c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac{1001 \sec (c+d x)}{2048 a d \sqrt{a+a \sin (c+d x)}}+\frac{143 \sec ^3(c+d x)}{640 a d \sqrt{a+a \sin (c+d x)}}+\frac{13 \sec ^5(c+d x)}{80 a d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.40622, size = 444, normalized size = 1.73 \[ \frac{\frac{28800 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}{\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )}+\frac{6400 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}+\frac{1536 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}{\left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^5}-16245 \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2+32490 \sin \left (\frac{1}{2} (c+d x)\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{17720 \sin \left (\frac{1}{2} (c+d x)\right )}{\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )}-\frac{4960}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2}+\frac{9920 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3}-\frac{1920}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^4}+\frac{3840 \sin \left (\frac{1}{2} (c+d x)\right )}{\left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5}+(45045+45045 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^3 \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )-8860}{122880 d (a (\sin (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 367, normalized size = 1.4 \begin{align*} -{\frac{1}{245760\, \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2} \left ( 1+\sin \left ( dx+c \right ) \right ) ^{3}\cos \left ( dx+c \right ) d} \left ( -120120\,{a}^{13/2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+ \left ( -54912\,{a}^{13/2}-180180\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) + \left ( -39936\,{a}^{13/2}+360360\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \sin \left ( dx+c \right ) +90090\,{a}^{13/2} \left ( \cos \left ( dx+c \right ) \right ) ^{6}+9009\, \left ( -8\,{a}^{13/2}+5\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+ \left ( -18304\,{a}^{13/2}-360360\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){a}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-9216\,{a}^{13/2}+360360\, \left ( a-a\sin \left ( dx+c \right ) \right ) ^{5/2}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ){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 = 2.89251, size = 805, normalized size = 3.14 \begin{align*} \frac{45045 \, \sqrt{2}{\left (\cos \left (d x + c\right )^{7} - 2 \, \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right )^{5}\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 (45045 \, \cos \left (d x + c\right )^{6} - 36036 \, \cos \left (d x + c\right )^{4} - 9152 \, \cos \left (d x + c\right )^{2} - 156 \,{\left (385 \, \cos \left (d x + c\right )^{4} + 176 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) - 4608\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{491520 \,{\left (a^{2} d \cos \left (d x + c\right )^{7} - 2 \, a^{2} d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) - 2 \, a^{2} d \cos \left (d x + c\right )^{5}\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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