Optimal. Leaf size=22 \[ -\frac {2}{b d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2668, 32} \[ -\frac {2}{b d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2668
Rubi steps
\begin {align*} \int \frac {\cos (c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^{3/2}} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac {2}{b d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 1.00 \[ -\frac {2}{b d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 32, normalized size = 1.45 \[ -\frac {2 \, \sqrt {b \sin \left (d x + c\right ) + a}}{b^{2} d \sin \left (d x + c\right ) + a b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.90, size = 20, normalized size = 0.91 \[ -\frac {2}{\sqrt {b \sin \left (d x + c\right ) + a} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 0.95 \[ -\frac {2}{b d \sqrt {a +b \sin \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 20, normalized size = 0.91 \[ -\frac {2}{\sqrt {b \sin \left (d x + c\right ) + a} b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.14, size = 51, normalized size = 2.32 \[ -\frac {4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}}{b\,d\,\left (2\,a^2+4\,a\,b\,\sin \left (c+d\,x\right )+2\,b^2\,{\sin \left (c+d\,x\right )}^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.01, size = 56, normalized size = 2.55 \[ \begin {cases} \frac {x \cos {\relax (c )}}{a^{\frac {3}{2}}} & \text {for}\: b = 0 \wedge d = 0 \\\frac {\sin {\left (c + d x \right )}}{a^{\frac {3}{2}} d} & \text {for}\: b = 0 \\\frac {x \cos {\relax (c )}}{\left (a + b \sin {\relax (c )}\right )^{\frac {3}{2}}} & \text {for}\: d = 0 \\- \frac {2}{b d \sqrt {a + b \sin {\left (c + d x \right )}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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