Optimal. Leaf size=38 \[ \frac {4 a^2 \sec ^{\frac {3}{4}}(c+d x) \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}} \]
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Rubi [A]
time = 0.04, 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 = {3899, 8}
\begin {gather*} \frac {4 a^2 \sin (c+d x) \sec ^{\frac {3}{4}}(c+d x)}{d \sqrt {a \sec (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3899
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{3/2}}{\sqrt [4]{\sec (c+d x)}} \, dx &=\frac {4 a^2 \sec ^{\frac {3}{4}}(c+d x) \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}}+(4 a) \int 0 \, dx\\ &=\frac {4 a^2 \sec ^{\frac {3}{4}}(c+d x) \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 45, normalized size = 1.18 \begin {gather*} \frac {4 \sec ^{\frac {3}{4}}(c+d x) (a (1+\sec (c+d x)))^{3/2} \sin (c+d x)}{d (1+\sec (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {\left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{\sec \left (d x +c \right )^{\frac {1}{4}}}\, dx\]
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 = 2.62, size = 50, normalized size = 1.32 \begin {gather*} \frac {4 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {1}{4}} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) + d} \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 = 0.75, size = 55, normalized size = 1.45 \begin {gather*} \frac {2\,a\,\sin \left (2\,c+2\,d\,x\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/4}\,\sqrt {\frac {a\,\left (\cos \left (c+d\,x\right )+1\right )}{\cos \left (c+d\,x\right )}}}{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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