Optimal. Leaf size=95 \[ -\frac{3 a}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} d}+\frac{\sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{2 d} \]
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Rubi [A] time = 0.123447, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2675, 2667, 51, 63, 206} \[ -\frac{3 a}{4 d \sqrt{a \sin (c+d x)+a}}+\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a \sin (c+d x)+a}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} d}+\frac{\sec ^2(c+d x) \sqrt{a \sin (c+d x)+a}}{2 d} \]
Antiderivative was successfully verified.
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Rule 2675
Rule 2667
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \sec ^3(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\frac{\sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}+\frac{1}{4} (3 a) \int \frac{\sec (c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=\frac{\sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}+\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{4 d}\\ &=-\frac{3 a}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=-\frac{3 a}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}+\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+a \sin (c+d x)}\right )}{4 d}\\ &=\frac{3 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} d}-\frac{3 a}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\sec ^2(c+d x) \sqrt{a+a \sin (c+d x)}}{2 d}\\ \end{align*}
Mathematica [C] time = 0.348932, size = 271, normalized size = 2.85 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (\frac{\left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )}+\frac{2 \sin \left (\frac{d x}{2}\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )}{\left (\cos \left (\frac{c}{2}\right )-\sin \left (\frac{c}{2}\right )\right ) \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^2}+(-3+3 i) \sqrt [4]{-1} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \sec \left (\frac{d x}{4}\right ) \left (\sin \left (\frac{1}{4} (2 c+d x)\right )+\cos \left (\frac{1}{4} (2 c+d x)\right )\right )\right )-2\right )}{4 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.145, size = 90, normalized size = 1. \begin{align*} 2\,{\frac{{a}^{3}}{d} \left ( -1/4\,{\frac{1}{{a}^{2}} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }}{a\sin \left ( dx+c \right ) -a}}-3/4\,{\frac{\sqrt{2}}{\sqrt{a}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a+a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) } \right ) }-1/4\,{\frac{1}{{a}^{2}\sqrt{a+a\sin \left ( dx+c \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71906, size = 274, normalized size = 2.88 \begin{align*} \frac{3 \, \sqrt{2} \sqrt{a} \cos \left (d x + c\right )^{2} \log \left (-\frac{a \sin \left (d x + c\right ) + 2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) + 4 \, \sqrt{a \sin \left (d x + c\right ) + a}{\left (3 \, \sin \left (d x + c\right ) - 1\right )}}{16 \, d \cos \left (d x + c\right )^{2}} \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 \sqrt{a \left (\sin{\left (c + d x \right )} + 1\right )} \sec ^{3}{\left (c + d x \right )}\, 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*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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