Integrand size = 21, antiderivative size = 43 \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {2 \sqrt {d \cos (a+b x)}}{b d}+\frac {2 (d \cos (a+b x))^{5/2}}{5 b d^3} \]
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Time = 0.03 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2645, 14} \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {2 (d \cos (a+b x))^{5/2}}{5 b d^3}-\frac {2 \sqrt {d \cos (a+b x)}}{b d} \]
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Rule 14
Rule 2645
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1-\frac {x^2}{d^2}}{\sqrt {x}} \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {\text {Subst}\left (\int \left (\frac {1}{\sqrt {x}}-\frac {x^{3/2}}{d^2}\right ) \, dx,x,d \cos (a+b x)\right )}{b d} \\ & = -\frac {2 \sqrt {d \cos (a+b x)}}{b d}+\frac {2 (d \cos (a+b x))^{5/2}}{5 b d^3} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33 \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {\cos (a+b x) (-9+\cos (2 (a+b x)))+8 \cos ^2(a+b x)^{3/4} \sec (a+b x)}{5 b \sqrt {d \cos (a+b x)}} \]
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Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}{5}-2 d^{2} \sqrt {d \cos \left (b x +a \right )}}{b \,d^{3}}\) | \(37\) |
default | \(\frac {\frac {2 \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}}{5}-2 d^{2} \sqrt {d \cos \left (b x +a \right )}}{b \,d^{3}}\) | \(37\) |
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Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {2 \, \sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right )^{2} - 5\right )}}{5 \, b d} \]
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Time = 0.78 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\begin {cases} - \frac {2 \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )}}{b \sqrt {d \cos {\left (a + b x \right )}}} - \frac {8 \cos ^{3}{\left (a + b x \right )}}{5 b \sqrt {d \cos {\left (a + b x \right )}}} & \text {for}\: b \neq 0 \\\frac {x \sin ^{3}{\left (a \right )}}{\sqrt {d \cos {\left (a \right )}}} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=-\frac {2 \, {\left (5 \, \sqrt {d \cos \left (b x + a\right )} - \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}{d^{2}}\right )}}{5 \, b d} \]
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Time = 0.34 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\frac {2 \, {\left (\sqrt {d \cos \left (b x + a\right )} d^{2} \cos \left (b x + a\right )^{2} - 5 \, \sqrt {d \cos \left (b x + a\right )} d^{2}\right )}}{5 \, b d^{3}} \]
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Timed out. \[ \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^3}{\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \]
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