Optimal. Leaf size=323 \[ \frac{\left (4 a^2+15 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}+\frac{a \left (4 a^2+11 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2+57 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}+\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.875501, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.323, Rules used = {2894, 3049, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{\left (4 a^2+15 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}+\frac{a \left (4 a^2+11 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2+57 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{15 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}+\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2894
Rule 3049
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \cot ^2(c+d x) \sqrt{a+b \sin (c+d x)} \, dx &=-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}-\frac{2 \int \csc (c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{5 b^2}{4}+\frac{7}{2} a b \sin (c+d x)+\frac{1}{4} \left (4 a^2+15 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{5 a b}\\ &=\frac{\left (4 a^2+15 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}-\frac{4 \int \frac{\csc (c+d x) \left (-\frac{15 a b^2}{8}+\frac{23}{4} a^2 b \sin (c+d x)+\frac{1}{8} a \left (4 a^2+57 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{15 a b}\\ &=\frac{\left (4 a^2+15 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}-\frac{1}{30} \left (57+\frac{4 a^2}{b^2}\right ) \int \sqrt{a+b \sin (c+d x)} \, dx+\frac{4 \int \frac{\csc (c+d x) \left (\frac{15 a b^3}{8}+\frac{1}{8} a^2 \left (4 a^2+11 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{15 a b^2}\\ &=\frac{\left (4 a^2+15 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}+\frac{1}{30} \left (a \left (11+\frac{4 a^2}{b^2}\right )\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx+\frac{1}{2} b \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{\left (\left (57+\frac{4 a^2}{b^2}\right ) \sqrt{a+b \sin (c+d x)}\right ) \int \sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}} \, dx}{30 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{\left (4 a^2+15 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}-\frac{\left (57+\frac{4 a^2}{b^2}\right ) 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)}}{15 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (a \left (11+\frac{4 a^2}{b^2}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{1}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{30 \sqrt{a+b \sin (c+d x)}}+\frac{\left (b \sqrt{\frac{a+b \sin (c+d x)}{a+b}}\right ) \int \frac{\csc (c+d x)}{\sqrt{\frac{a}{a+b}+\frac{b \sin (c+d x)}{a+b}}} \, dx}{2 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{\left (4 a^2+15 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{15 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{5 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{3/2}}{a d}-\frac{\left (57+\frac{4 a^2}{b^2}\right ) 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)}}{15 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{a \left (11+\frac{4 a^2}{b^2}\right ) F\left (\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{15 d \sqrt{a+b \sin (c+d x)}}+\frac{b \Pi \left (2;\frac{1}{2} \left (c-\frac{\pi }{2}+d x\right )|\frac{2 b}{a+b}\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}{d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 3.49874, size = 422, normalized size = 1.31 \[ \frac{\frac{2 \left (4 a^2+27 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \sin (c+d x)}}+\frac{2 i \left (4 a^2+57 b^2\right ) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )}{a b^3 \sqrt{-\frac{1}{a+b}}}+\frac{184 a \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}-\frac{4 \sqrt{a+b \sin (c+d x)} (2 a \cos (c+d x)+3 b (\sin (2 (c+d x))+5 \cot (c+d x)))}{b}}{60 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.615, size = 657, normalized size = 2. \begin{align*} \text{result too large to display} \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 \sqrt{b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sin{\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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