Optimal. Leaf size=38 \[ \frac {4 a^2 \sqrt [4]{\sec (c+d x)} \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {4307, 2841, 8}
\begin {gather*} \frac {4 a^2 \sin (c+d x) \sqrt [4]{\sec (c+d x)}}{d \sqrt {a \cos (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2841
Rule 4307
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{3/2} \sec ^{\frac {5}{4}}(c+d x) \, dx &=\left (\sqrt [4]{\cos (c+d x)} \sqrt [4]{\sec (c+d x)}\right ) \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {5}{4}}(c+d x)} \, dx\\ &=\frac {4 a^2 \sqrt [4]{\sec (c+d x)} \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}-\left (4 a \sqrt [4]{\cos (c+d x)} \sqrt [4]{\sec (c+d x)}\right ) \int 0 \, dx\\ &=\frac {4 a^2 \sqrt [4]{\sec (c+d x)} \sin (c+d x)}{d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 51, normalized size = 1.34 \begin {gather*} \frac {2 (a (1+\cos (c+d x)))^{3/2} \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt [4]{\sec (c+d x)} \tan \left (\frac {1}{2} (c+d x)\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.16, size = 0, normalized size = 0.00 \[\int \left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (\sec ^{\frac {5}{4}}\left (d x +c \right )\right )\, dx\]
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. 121 vs.
\(2 (34) = 68\).
time = 0.51, size = 121, normalized size = 3.18 \begin {gather*} \frac {4 \, {\left (\frac {\sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{4}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{4}} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {1}{4}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 41, normalized size = 1.08 \begin {gather*} \frac {4 \, \sqrt {a \cos \left (d x + c\right ) + a} a \sin \left (d x + c\right )}{{\left (d \cos \left (d x + c\right ) + d\right )} \cos \left (d x + c\right )^{\frac {1}{4}}} \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(-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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Mupad [B]
time = 0.74, size = 44, normalized size = 1.16 \begin {gather*} \frac {4\,a\,\sin \left (c+d\,x\right )\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{1/4}}{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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