Optimal. Leaf size=25 \[ \frac {b x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.00, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 8} \[ \frac {b x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rubi steps
\begin {align*} \int \frac {(b \cos (c+d x))^{3/2}}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int 1 \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {b x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 24, normalized size = 0.96 \[ \frac {x (b \cos (c+d x))^{3/2}}{\cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 95, normalized size = 3.80 \[ \left [\frac {\sqrt {-b} b \log \left (2 \, b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right )}{2 \, d}, \frac {b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right )}{d}\right ] \]
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 \[ \int \frac {\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 28, normalized size = 1.12 \[ \frac {\left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (d x +c \right )}{d \cos \left (d x +c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 26, normalized size = 1.04 \[ \frac {2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 21, normalized size = 0.84 \[ \frac {b\,x\,\sqrt {b\,\cos \left (c+d\,x\right )}}{\sqrt {\cos \left (c+d\,x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 125.71, size = 5, normalized size = 0.20 \[ b^{\frac {3}{2}} x \]
Verification of antiderivative is not currently implemented for this CAS.
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