Optimal. Leaf size=111 \[ \frac{2 b \cos (c+d x)}{d \left (a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}+\frac{2 \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.0648819, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2664, 21, 2655, 2653} \[ \frac{2 b \cos (c+d x)}{d \left (a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}+\frac{2 \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2664
Rule 21
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{1}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac{2 b \cos (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sin (c+d x)}}-\frac{2 \int \frac{-\frac{a}{2}-\frac{1}{2} b \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{a^2-b^2}\\ &=\frac{2 b \cos (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sin (c+d x)}}+\frac{\int \sqrt{a+b \sin (c+d x)} \, dx}{a^2-b^2}\\ &=\frac{2 b \cos (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sin (c+d x)}}+\frac{\sqrt{a+b \sin (c+d x)} \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{\left (a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{2 b \cos (c+d x)}{\left (a^2-b^2\right ) d \sqrt{a+b \sin (c+d x)}}+\frac{2 E\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{a+b \sin (c+d x)}}{\left (a^2-b^2\right ) d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ \end{align*}
Mathematica [A] time = 0.289202, size = 87, normalized size = 0.78 \[ \frac{2 b \cos (c+d x)-2 (a+b) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{d (a-b) (a+b) \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.158, size = 443, normalized size = 4. \begin{align*} 2\,{\frac{1}{ \left ({a}^{2}-{b}^{2} \right ) b\cos \left ( dx+c \right ) \sqrt{a+b\sin \left ( dx+c \right ) }d} \left ({a}^{2}\sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}}\sqrt{-{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) b}{a+b}}}\sqrt{-{\frac{ \left ( 1+\sin \left ( dx+c \right ) \right ) b}{a-b}}}{\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ) -\sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}}\sqrt{-{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) b}{a+b}}}\sqrt{-{\frac{ \left ( 1+\sin \left ( dx+c \right ) \right ) b}{a-b}}}{\it EllipticF} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){b}^{2}-\sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}}\sqrt{-{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) b}{a+b}}}\sqrt{-{\frac{ \left ( 1+\sin \left ( dx+c \right ) \right ) b}{a-b}}}{\it EllipticE} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){a}^{2}+\sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}}\sqrt{-{\frac{ \left ( \sin \left ( dx+c \right ) -1 \right ) b}{a+b}}}\sqrt{-{\frac{ \left ( 1+\sin \left ( dx+c \right ) \right ) b}{a-b}}}{\it EllipticE} \left ( \sqrt{{\frac{a+b\sin \left ( dx+c \right ) }{a-b}}},\sqrt{{\frac{a-b}{a+b}}} \right ){b}^{2}- \left ( \sin \left ( dx+c \right ) \right ) ^{2}{b}^{2}+{b}^{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b \sin \left (d x + c\right ) + a}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b \sin{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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