Optimal. Leaf size=139 \[ -\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{4 \sqrt {2} d}+\frac {a^2 \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{5/2}}{5 d} \]
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Rubi [A]
time = 0.14, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2754, 2728,
212} \begin {gather*} -\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a \sin (c+d x)+a}}\right )}{4 \sqrt {2} d}+\frac {a^2 \sec (c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}+\frac {\sec ^5(c+d x) (a \sin (c+d x)+a)^{5/2}}{5 d}+\frac {a \sec ^3(c+d x) (a \sin (c+d x)+a)^{3/2}}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2754
Rubi steps
\begin {align*} \int \sec ^6(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{5/2}}{5 d}+\frac {1}{2} a \int \sec ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=\frac {a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{5/2}}{5 d}+\frac {1}{4} a^2 \int \sec ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {a^2 \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{5/2}}{5 d}+\frac {1}{8} a^3 \int \frac {1}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=\frac {a^2 \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{5/2}}{5 d}-\frac {a^3 \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 )}{4 d}\\ &=-\frac {a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {2} \sqrt {a+a \sin (c+d x)}}\right )}{4 \sqrt {2} d}+\frac {a^2 \sec (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {a \sec ^3(c+d x) (a+a \sin (c+d x))^{3/2}}{6 d}+\frac {\sec ^5(c+d x) (a+a \sin (c+d x))^{5/2}}{5 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.21, size = 129, normalized size = 0.93 \begin {gather*} \frac {\left ((15+15 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 )+\frac {89-15 \cos (2 (c+d x))-80 \sin (c+d x)}{2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}\right ) (a (1+\sin (c+d x)))^{5/2}}{60 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 = 0.46, size = 120, normalized size = 0.86
method | result | size |
default | \(-\frac {\left (1+\sin \left (d x +c \right )\right ) \left (30 a^{\frac {11}{2}} \left (\cos ^{2}\left (d x +c \right )\right )+80 a^{\frac {11}{2}} \sin \left (d x +c \right )-104 a^{\frac {11}{2}}+15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}\right )}{120 a^{\frac {5}{2}} \left (\sin \left (d x +c \right )-1\right )^{2} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 263 vs.
\(2 (116) = 232\).
time = 0.38, size = 263, normalized size = 1.89 \begin {gather*} \frac {15 \, {\left (\sqrt {2} a^{2} \cos \left (d x + c\right )^{3} + 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, \sqrt {2} a^{2} \cos \left (d x + c\right )\right )} \sqrt {a} \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 \, a^{2} \cos \left (d x + c\right )^{2} + 40 \, a^{2} \sin \left (d x + c\right ) - 52 \, a^{2}\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{240 \, {\left (d \cos \left (d x + c\right )^{3} + 2 \, d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, d \cos \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.98, size = 111, normalized size = 0.80 \begin {gather*} -\frac {\sqrt {2} a^{\frac {5}{2}} {\left (\frac {2 \, {\left (15 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} - 15 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + 15 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )\right )} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{240 \, 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 {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\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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