Optimal. Leaf size=201 \[ \frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {34 a^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {68 a^2 \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {272 a^2 \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {544 a^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}} \]
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Rubi [A]
time = 0.21, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3898, 21, 3890,
3889} \begin {gather*} \frac {68 a^2 \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {34 a^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {544 a^2 \sin (c+d x) \sqrt {\sec (c+d x)}}{315 d \sqrt {a \sec (c+d x)+a}}+\frac {272 a^2 \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 3889
Rule 3890
Rule 3898
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac {9}{2}}(c+d x)} \, dx &=\frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {1}{9} (2 a) \int \frac {\frac {17 a}{2}+\frac {17}{2} a \sec (c+d x)}{\sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}} \, dx\\ &=\frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {1}{9} (17 a) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {34 a^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {1}{21} (34 a) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {34 a^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {68 a^2 \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {1}{105} (136 a) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {34 a^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {68 a^2 \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {272 a^2 \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {1}{315} (272 a) \int \frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a^2 \sin (c+d x)}{9 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {34 a^2 \sin (c+d x)}{63 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {68 a^2 \sin (c+d x)}{105 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {272 a^2 \sin (c+d x)}{315 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {544 a^2 \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.55, size = 80, normalized size = 0.40 \begin {gather*} \frac {2 a^2 \left (35+85 \sec (c+d x)+102 \sec ^2(c+d x)+136 \sec ^3(c+d x)+272 \sec ^4(c+d x)\right ) \sin (c+d x)}{315 d \sec ^{\frac {7}{2}}(c+d x) \sqrt {a (1+\sec (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 103, normalized size = 0.51
method | result | size |
default | \(-\frac {2 \left (35 \left (\cos ^{5}\left (d x +c \right )\right )+50 \left (\cos ^{4}\left (d x +c \right )\right )+17 \left (\cos ^{3}\left (d x +c \right )\right )+34 \left (\cos ^{2}\left (d x +c \right )\right )+136 \cos \left (d x +c \right )-272\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {9}{2}} \left (\cos ^{5}\left (d x +c \right )\right ) a}{315 d \sin \left (d x +c \right )}\) | \(103\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 396 vs.
\(2 (171) = 342\).
time = 0.58, size = 396, normalized size = 1.97 \begin {gather*} \frac {\sqrt {2} {\left (3780 \, a \cos \left (\frac {8}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 1050 \, a \cos \left (\frac {2}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 378 \, a \cos \left (\frac {4}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, a \cos \left (\frac {2}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) - 3780 \, a \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \sin \left (\frac {8}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) - 1050 \, a \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \sin \left (\frac {2}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) - 378 \, a \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \sin \left (\frac {4}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) - 135 \, a \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \sin \left (\frac {2}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 70 \, a \sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ) + 135 \, a \sin \left (\frac {7}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 378 \, a \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 1050 \, a \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right ) + 3780 \, a \sin \left (\frac {1}{9} \, \arctan \left (\sin \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right ), \cos \left (\frac {9}{2} \, d x + \frac {9}{2} \, c\right )\right )\right )\right )} \sqrt {a}}{5040 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.27, size = 103, normalized size = 0.51 \begin {gather*} \frac {2 \, {\left (35 \, a \cos \left (d x + c\right )^{5} + 85 \, a \cos \left (d x + c\right )^{4} + 102 \, a \cos \left (d x + c\right )^{3} + 136 \, a \cos \left (d x + c\right )^{2} + 272 \, a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right ) + d\right )} \sqrt {\cos \left (d x + c\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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.02, size = 105, normalized size = 0.52 \begin {gather*} \frac {a\,\cos \left (c+d\,x\right )\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {a\,\left (\cos \left (c+d\,x\right )+1\right )}{\cos \left (c+d\,x\right )}}\,\left (4830\,\sin \left (c+d\,x\right )+1428\,\sin \left (2\,c+2\,d\,x\right )+513\,\sin \left (3\,c+3\,d\,x\right )+170\,\sin \left (4\,c+4\,d\,x\right )+35\,\sin \left (5\,c+5\,d\,x\right )\right )}{2520\,d\,\left (\cos \left (c+d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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