Optimal. Leaf size=137 \[ -\frac {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{8 \sqrt {2} d}-\frac {5 a^2 \cos (c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {5 a \sec (c+d x)}{6 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d} \]
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Rubi [A]
time = 0.11, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2754, 2766,
2729, 2728, 212} \begin {gather*} -\frac {5 a^2 \cos (c+d x)}{8 d (a \sin (c+d x)+a)^{3/2}}+\frac {\sec ^3(c+d x) \sqrt {a \sin (c+d x)+a}}{3 d}+\frac {5 a \sec (c+d x)}{6 d \sqrt {a \sin (c+d x)+a}}-\frac {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{8 \sqrt {2} d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rule 2754
Rule 2766
Rubi steps
\begin {align*} \int \sec ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\frac {\sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {1}{6} (5 a) \int \frac {\sec ^2(c+d x)}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {5 a \sec (c+d x)}{6 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {1}{4} \left (5 a^2\right ) \int \frac {1}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac {5 a^2 \cos (c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {5 a \sec (c+d x)}{6 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {1}{16} (5 a) \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {5 a^2 \cos (c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {5 a \sec (c+d x)}{6 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {(5 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 )}{8 d}\\ &=-\frac {5 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{8 \sqrt {2} d}-\frac {5 a^2 \cos (c+d x)}{8 d (a+a \sin (c+d x))^{3/2}}+\frac {5 a \sec (c+d x)}{6 d \sqrt {a+a \sin (c+d x)}}+\frac {\sec ^3(c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 302, normalized size = 2.20 \begin {gather*} \frac {\left (\frac {6 \sin \left (\frac {d x}{2}\right )}{\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )}-\frac {3 \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )}-(15+15 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 )^2+\frac {4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {12 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right ) \sqrt {a (1+\sin (c+d x))}}{24 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 153, normalized size = 1.12
method | result | size |
default | \(\frac {\sin \left (d x +c \right ) \left (15 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a -20 a^{\frac {5}{2}}\right )-30 a^{\frac {5}{2}} \left (\cos ^{2}\left (d x +c \right )\right )+15 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a +4 a^{\frac {5}{2}}}{48 a^{\frac {3}{2}} \left (\sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(153\) |
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 = 188, normalized size = 1.37 \begin {gather*} \frac {15 \, \sqrt {2} \sqrt {a} \cos \left (d x + c\right )^{3} \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 (15 \, \cos \left (d x + c\right )^{2} + 10 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{96 \, d \cos \left (d x + c\right )^{3}} \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 \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \sec ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 8.19, size = 178, normalized size = 1.30 \begin {gather*} \frac {\sqrt {2} {\left (15 \, \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 ) - 15 \, \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 {6 \, \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 )}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {4 \, {\left (6 \, \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 )^{3}}\right )} \sqrt {a}}{96 \, 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 )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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