Optimal. Leaf size=197 \[ -\frac {63 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {2} d}-\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d} \]
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Rubi [A]
time = 0.21, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2754, 2766,
2760, 2729, 2728, 212} \begin {gather*} -\frac {63 a^2 \cos (c+d x)}{128 d (a \sin (c+d x)+a)^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^5(c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a \sin (c+d x)+a}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a \sin (c+d x)+a}}-\frac {63 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{128 \sqrt {2} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rule 2754
Rule 2760
Rule 2766
Rubi steps
\begin {align*} \int \sec ^6(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{10} (9 a) \int \frac {\sec ^4(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{20} \left (21 a^2\right ) \int \frac {\sec ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{32} (21 a) \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{64} \left (63 a^2\right ) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {1}{256} (63 a) \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}-\frac {(63 a) \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 )}{128 d}\\ &=-\frac {63 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{128 \sqrt {2} d}-\frac {63 a^2 \cos (c+d x)}{128 d (a+a \sin (c+d x))^{3/2}}-\frac {21 a^2 \sec (c+d x)}{80 d (a+a \sin (c+d x))^{3/2}}+\frac {21 a \sec (c+d x)}{32 d \sqrt {a+a \sin (c+d x)}}+\frac {3 a \sec ^3(c+d x)}{10 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^5(c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.41, size = 191, normalized size = 0.97 \begin {gather*} \frac {\sqrt {a (1+\sin (c+d x))} \left ((-2520-2520 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \sec \left (\frac {d x}{4}\right ) \left (\cos \left (\frac {1}{4} (2 c+d x)\right )-\sin \left (\frac {1}{4} (2 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 )^4+\frac {649+1092 \cos (2 (c+d x))+315 \cos (4 (c+d x))+1572 \sin (c+d x)+420 \sin (3 (c+d x))}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}\right )}{5120 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.52, size = 244, normalized size = 1.24
method | result | size |
default | \(-\frac {-420 a^{\frac {9}{2}} \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+\left (630 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}-288 a^{\frac {9}{2}}\right ) \sin \left (d x +c \right )-630 a^{\frac {9}{2}} \left (\cos ^{4}\left (d x +c \right )\right )+\left (-315 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+84 a^{\frac {9}{2}}\right ) \left (\cos ^{2}\left (d x +c \right )\right )+630 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{2}+32 a^{\frac {9}{2}}}{1280 a^{\frac {7}{2}} \left (\sin \left (d x +c \right )-1\right )^{2} \left (1+\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(244\) |
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 = 0.37, size = 210, normalized size = 1.07 \begin {gather*} \frac {315 \, \sqrt {2} \sqrt {a} \cos \left (d x + c\right )^{5} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (\sqrt {2} \cos \left (d x + c\right ) - \sqrt {2} \sin \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {a} + 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 (315 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 6 \, {\left (35 \, \cos \left (d x + c\right )^{2} + 24\right )} \sin \left (d x + c\right ) - 16\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2560 \, d \cos \left (d x + c\right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.26, size = 239, normalized size = 1.21 \begin {gather*} \frac {\sqrt {2} {\left (315 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 315 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - \frac {10 \, {\left (15 \, \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 )^{3} - 17 \, \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 )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2}} - \frac {16 \, {\left (30 \, \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 )^{4} + 5 \, \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 )^{2} + \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}\right )} \sqrt {a}}{2560 \, 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 = -0.01 \begin {gather*} \int \frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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