Optimal. Leaf size=221 \[ -\frac {231 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.24, antiderivative size = 221, normalized size of antiderivative = 1.00, 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 = {2766, 2760,
2729, 2728, 212} \begin {gather*} -\frac {231 a \cos (c+d x)}{512 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a \sin (c+d x)+a}}-\frac {11 a \sec ^3(c+d x)}{60 d (a \sin (c+d x)+a)^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a \sin (c+d x)+a}}-\frac {77 a \sec (c+d x)}{320 d (a \sin (c+d x)+a)^{3/2}}-\frac {231 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{512 \sqrt {2} \sqrt {a} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rule 2760
Rule 2766
Rubi steps
\begin {align*} \int \frac {\sec ^6(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{10} (11 a) \int \frac {\sec ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {33}{40} \int \frac {\sec ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{80} (77 a) \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {77}{128} \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {1}{256} (231 a) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {231 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx}{1024}\\ &=-\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}-\frac {231 \text {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 )}{512 d}\\ &=-\frac {231 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{512 \sqrt {2} \sqrt {a} d}-\frac {231 a \cos (c+d x)}{512 d (a+a \sin (c+d x))^{3/2}}-\frac {77 a \sec (c+d x)}{320 d (a+a \sin (c+d x))^{3/2}}-\frac {11 a \sec ^3(c+d x)}{60 d (a+a \sin (c+d x))^{3/2}}+\frac {77 \sec (c+d x)}{128 d \sqrt {a+a \sin (c+d x)}}+\frac {11 \sec ^3(c+d x)}{40 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.45, size = 140, normalized size = 0.63 \begin {gather*} \frac {(3465+3465 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (c+d x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {1}{16} \sec ^5(c+d x) (11090+11352 \cos (2 (c+d x))+2310 \cos (4 (c+d x))+36850 \sin (c+d x)+17787 \sin (3 (c+d x))+3465 \sin (5 (c+d x)))}{7680 d \sqrt {a (1+\sin (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.60, size = 308, normalized size = 1.39
method | result | size |
default | \(-\frac {-6930 a^{\frac {11}{2}} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (-3696 a^{\frac {11}{2}}-3465 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\left (-2816 a^{\frac {11}{2}}+13860 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}\right ) \sin \left (d x +c \right )-2310 a^{\frac {11}{2}} \left (\cos ^{4}\left (d x +c \right )\right )+\left (-528 a^{\frac {11}{2}}-10395 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}\right ) \left (\cos ^{2}\left (d x +c \right )\right )-256 a^{\frac {11}{2}}+13860 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}{15360 a^{\frac {11}{2}} \left (\sin \left (d x +c \right )-1\right )^{2} \left (1+\sin \left (d x +c \right )\right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(308\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 250, normalized size = 1.13 \begin {gather*} \frac {3465 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + \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 (1155 \, \cos \left (d x + c\right )^{4} + 264 \, \cos \left (d x + c\right )^{2} + 11 \, {\left (315 \, \cos \left (d x + c\right )^{4} + 168 \, \cos \left (d x + c\right )^{2} + 128\right )} \sin \left (d x + c\right ) + 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{30720 \, {\left (a d \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{6}{\left (c + d x \right )}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.30, size = 262, normalized size = 1.19 \begin {gather*} \frac {\frac {3465 \, \sqrt {2} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {3465 \, \sqrt {2} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{\sqrt {a} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {10 \, \sqrt {2} {\left (213 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 472 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 267 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} - \frac {32 \, \sqrt {2} {\left (150 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \sqrt {a} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, \sqrt {a}\right )}}{a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{30720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^6\,\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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