Optimal. Leaf size=307 \[ -\frac{\left (8 a^2+b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{4 a b d \sqrt{a+b \sin (c+d x)}}+\frac{\left (8 a^2+3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{4 a^2 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left (4 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{4 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d} \]
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Rubi [A] time = 0.665193, antiderivative size = 307, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31, Rules used = {2893, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac{\left (8 a^2+b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{4 a b d \sqrt{a+b \sin (c+d x)}}+\frac{\left (8 a^2+3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{4 a^2 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{3 \left (4 a^2-b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{4 a^2 d \sqrt{a+b \sin (c+d x)}}+\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d} \]
Antiderivative was successfully verified.
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Rule 2893
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^3(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx &=\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d}-\frac{\int \frac{\csc (c+d x) \left (\frac{3}{4} \left (4 a^2-b^2\right )-\frac{1}{2} a b \sin (c+d x)-\frac{1}{4} \left (8 a^2+3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{2 a^2}\\ &=\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d}+\frac{\int \frac{\csc (c+d x) \left (-\frac{3}{4} b \left (4 a^2-b^2\right )-\frac{1}{4} a \left (8 a^2+b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{2 a^2 b}-\frac{\left (-8 a^2-3 b^2\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{8 a^2 b}\\ &=\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d}-\frac{1}{8} \left (\frac{8 a}{b}+\frac{b}{a}\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{1}{8} \left (3 \left (4-\frac{b^2}{a^2}\right )\right ) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{\left (\left (-8 a^2-3 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}{8 a^2 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d}+\frac{\left (8 a^2+3 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)}}{4 a^2 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (\left (\frac{8 a}{b}+\frac{b}{a}\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}{8 \sqrt{a+b \sin (c+d x)}}-\frac{\left (3 \left (4-\frac{b^2}{a^2}\right ) \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}{8 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{3 b \cot (c+d x) \sqrt{a+b \sin (c+d x)}}{4 a^2 d}-\frac{\cot (c+d x) \csc (c+d x) \sqrt{a+b \sin (c+d x)}}{2 a d}+\frac{\left (8 a^2+3 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)}}{4 a^2 b d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (\frac{8 a}{b}+\frac{b}{a}\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}}}{4 d \sqrt{a+b \sin (c+d x)}}-\frac{3 \left (4-\frac{b^2}{a^2}\right ) \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}}}{4 d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 3.36653, size = 443, normalized size = 1.44 \[ \frac{\frac{2 \left (16 a^2-9 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{a^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 i \left (8 a^2+3 b^2\right ) \cos (2 (c+d x)) \csc ^2(c+d x) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )\right )}{a^3 b^2 \sqrt{-\frac{1}{a+b}} \left (\csc ^2(c+d x)-2\right )}-\frac{4 \cot (c+d x) \sqrt{a+b \sin (c+d x)} (2 a \csc (c+d x)-3 b)}{a^2}-\frac{8 b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{a \sqrt{a+b \sin (c+d x)}}}{16 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.119, size = 913, normalized size = 3. \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (d x + c\right ) \cot \left (d x + c\right )^{3}}{\sqrt{b \sin \left (d x + c\right ) + a}}, 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{\cos{\left (c + d x \right )} \cot ^{3}{\left (c + d x \right )}}{\sqrt{a + b \sin{\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*} \int \frac{\cos \left (d x + c\right ) \cot \left (d x + c\right )^{3}}{\sqrt{b \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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