Optimal. Leaf size=43 \[ \frac {2 (d \cos (a+b x))^{5/2}}{5 b d^3}-\frac {2 \sqrt {d \cos (a+b x)}}{b d} \]
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Rubi [A] time = 0.05, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2565, 14} \[ \frac {2 (d \cos (a+b x))^{5/2}}{5 b d^3}-\frac {2 \sqrt {d \cos (a+b x)}}{b d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2565
Rubi steps
\begin {align*} \int \frac {\sin ^3(a+b x)}{\sqrt {d \cos (a+b x)}} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1-\frac {x^2}{d^2}}{\sqrt {x}} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=-\frac {\operatorname {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*}
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Mathematica [A] time = 0.18, size = 57, normalized size = 1.33 \[ \frac {\cos (a+b x) (\cos (2 (a+b x))-9)+8 \cos ^2(a+b x)^{3/4} \sec (a+b x)}{5 b \sqrt {d \cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 28, normalized size = 0.65 \[ \frac {2 \, \sqrt {d \cos \left (b x + a\right )} {\left (\cos \left (b x + a\right )^{2} - 5\right )}}{5 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.03, size = 40, normalized size = 0.93 \[ \frac {2 \, {\left (\sqrt {d \cos \left (b x + a\right )} \cos \left (b x + a\right )^{2} - 5 \, \sqrt {d \cos \left (b x + a\right )}\right )}}{5 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 92, normalized size = 2.14 \[ \frac {8 \sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-8 \sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-8 \sqrt {-2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) d +d}}{5 d b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 36, normalized size = 0.84 \[ -\frac {2 \, {\left (5 \, \sqrt {d \cos \left (b x + a\right )} - \frac {\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}{d^{2}}\right )}}{5 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\sin \left (a+b\,x\right )}^3}{\sqrt {d\,\cos \left (a+b\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 6.17, size = 63, normalized size = 1.47 \[ \begin {cases} - \frac {2 \sin ^{2}{\left (a + b x \right )} \sqrt {\cos {\left (a + b x \right )}}}{b \sqrt {d}} - \frac {8 \cos ^{\frac {5}{2}}{\left (a + b x \right )}}{5 b \sqrt {d}} & \text {for}\: b \neq 0 \\\frac {x \sin ^{3}{\relax (a )}}{\sqrt {d \cos {\relax (a )}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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