Optimal. Leaf size=81 \[ -\frac{2 \left (a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}{b^3 d}-\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d} \]
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Rubi [A] time = 0.0848283, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2668, 697} \[ -\frac{2 \left (a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}{b^3 d}-\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{\sqrt{a+x}} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-a^2+b^2}{\sqrt{a+x}}+2 a \sqrt{a+x}-(a+x)^{3/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac{2 \left (a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}{b^3 d}+\frac{4 a (a+b \sin (c+d x))^{3/2}}{3 b^3 d}-\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.0756197, size = 58, normalized size = 0.72 \[ \frac{2 \sqrt{a+b \sin (c+d x)} \left (-8 a^2+4 a b \sin (c+d x)-3 b^2 \sin ^2(c+d x)+15 b^2\right )}{15 b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.169, size = 55, normalized size = 0.7 \begin{align*} -{\frac{-6\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-8\,ab\sin \left ( dx+c \right ) +16\,{a}^{2}-24\,{b}^{2}}{15\,{b}^{3}d}\sqrt{a+b\sin \left ( dx+c \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966539, size = 101, normalized size = 1.25 \begin{align*} \frac{2 \,{\left (15 \, \sqrt{b \sin \left (d x + c\right ) + a} - \frac{3 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{2}}{b^{2}}\right )}}{15 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98215, size = 135, normalized size = 1.67 \begin{align*} \frac{2 \,{\left (3 \, b^{2} \cos \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) - 8 \, a^{2} + 12 \, b^{2}\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{15 \, b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0996, size = 97, normalized size = 1.2 \begin{align*} -\frac{2 \,{\left (3 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 10 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 15 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{2} - 15 \, \sqrt{b \sin \left (d x + c\right ) + a} b^{2}\right )}}{15 \, b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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