Optimal. Leaf size=38 \[ \frac {4 a^2 \sin (c+d x)}{d \sqrt [4]{\cos (c+d x)} \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.06, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2762, 8} \[ \frac {4 a^2 \sin (c+d x)}{d \sqrt [4]{\cos (c+d x)} \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2762
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^{3/2}}{\cos ^{\frac {5}{4}}(c+d x)} \, dx &=\frac {4 a^2 \sin (c+d x)}{d \sqrt [4]{\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}-(4 a) \int 0 \, dx\\ &=\frac {4 a^2 \sin (c+d x)}{d \sqrt [4]{\cos (c+d x)} \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 51, normalized size = 1.34 \[ \frac {2 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a (\cos (c+d x)+1))^{3/2}}{d \sqrt [4]{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.10, size = 50, normalized size = 1.32 \[ \frac {4 \, \sqrt {a \cos \left (d x + c\right ) + a} a \cos \left (d x + c\right )^{\frac {3}{4}} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{\cos \left (d x +c \right )^{\frac {5}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.13, size = 121, normalized size = 3.18 \[ \frac {4 \, {\left (\frac {\sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {\sqrt {2} a^{\frac {3}{2}} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{4}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{4}} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.58, size = 42, normalized size = 1.11 \[ \frac {4\,a\,\sin \left (c+d\,x\right )\,\sqrt {a\,\left (\cos \left (c+d\,x\right )+1\right )}}{d\,{\cos \left (c+d\,x\right )}^{1/4}\,\left (\cos \left (c+d\,x\right )+1\right )} \]
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 \[ \int \frac {\left (a \left (\cos {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}{\cos ^{\frac {5}{4}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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