Optimal. Leaf size=79 \[ \frac{2 \left (a^2-b^2\right )}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac{4 a \sqrt{a+b \sin (c+d x)}}{b^3 d} \]
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Rubi [A] time = 0.0935693, antiderivative size = 79, 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} \[ \frac{2 \left (a^2-b^2\right )}{b^3 d \sqrt{a+b \sin (c+d x)}}-\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac{4 a \sqrt{a+b \sin (c+d x)}}{b^3 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{b^2-x^2}{(a+x)^{3/2}} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{-a^2+b^2}{(a+x)^{3/2}}+\frac{2 a}{\sqrt{a+x}}-\sqrt{a+x}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac{2 \left (a^2-b^2\right )}{b^3 d \sqrt{a+b \sin (c+d x)}}+\frac{4 a \sqrt{a+b \sin (c+d x)}}{b^3 d}-\frac{2 (a+b \sin (c+d x))^{3/2}}{3 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.0670612, size = 57, normalized size = 0.72 \[ \frac{16 a^2+8 a b \sin (c+d x)+b^2 \cos (2 (c+d x))-7 b^2}{3 b^3 d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.24, size = 54, normalized size = 0.7 \begin{align*}{\frac{2\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}+8\,ab\sin \left ( dx+c \right ) +16\,{a}^{2}-8\,{b}^{2}}{3\,{b}^{3}d}{\frac{1}{\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 = 0.971005, size = 90, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (\frac{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b \sin \left (d x + c\right ) + a} a}{b^{2}} - \frac{3 \,{\left (a^{2} - b^{2}\right )}}{\sqrt{b \sin \left (d x + c\right ) + a} b^{2}}\right )}}{3 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.02746, size = 161, normalized size = 2.04 \begin{align*} \frac{2 \,{\left (b^{2} \cos \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) + 8 \, a^{2} - 4 \, b^{2}\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{3 \,{\left (b^{4} d \sin \left (d x + c\right ) + a b^{3} d\right )}} \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 = 1.09855, size = 80, normalized size = 1.01 \begin{align*} -\frac{2 \,{\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} - 6 \, \sqrt{b \sin \left (d x + c\right ) + a} a - \frac{3 \,{\left (a^{2} - b^{2}\right )}}{\sqrt{b \sin \left (d x + c\right ) + a}}\right )}}{3 \, b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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