Optimal. Leaf size=285 \[ \frac{\left (4 a^2-7 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2+3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 a b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}-\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a d \sqrt{a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.672747, antiderivative size = 285, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.29, Rules used = {2894, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac{\left (4 a^2-7 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\left (4 a^2+3 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 a b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}-\frac{b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{a d \sqrt{a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2894
Rule 3059
Rule 2655
Rule 2653
Rule 3002
Rule 2663
Rule 2661
Rule 2807
Rule 2805
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx &=-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}-\frac{2 \int \frac{\csc (c+d x) \left (\frac{3 b^2}{4}+\frac{5}{2} a b \sin (c+d x)+\frac{1}{4} \left (4 a^2+3 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{3 a b}\\ &=-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}+\frac{2 \int \frac{\csc (c+d x) \left (-\frac{3 b^3}{4}+\frac{1}{4} a \left (4 a^2-7 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{3 a b^2}-\frac{\left (4 a^2+3 b^2\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{6 a b^2}\\ &=-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}+\frac{1}{6} \left (-7+\frac{4 a^2}{b^2}\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx-\frac{b \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{2 a}-\frac{\left (\left (4 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}{6 a b^2 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}-\frac{\left (4 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)}}{3 a b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (\left (-7+\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}{6 \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 a \sqrt{a+b \sin (c+d x)}}\\ &=-\frac{2 \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{3 b d}-\frac{\cot (c+d x) \sqrt{a+b \sin (c+d x)}}{a d}-\frac{\left (4 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)}}{3 a b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{\left (7-\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}}}{3 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}}}{a d \sqrt{a+b \sin (c+d x)}}\\ \end{align*}
Mathematica [C] time = 3.46644, size = 416, normalized size = 1.46 \[ \frac{\frac{2 \left (4 a^2+9 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{a b \sqrt{a+b \sin (c+d x)}}+\frac{2 i \left (4 a^2+3 b^2\right ) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{-\frac{b (\sin (c+d x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )}{a^2 b^3 \sqrt{-\frac{1}{a+b}}}-\frac{4 \cot (c+d x) (2 a \sin (c+d x)+3 b) \sqrt{a+b \sin (c+d x)}}{a b}+\frac{40 \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.826, size = 704, normalized size = 2.5 \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 \frac{\cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2}}{\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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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 \frac{\cos ^{2}{\left (c + d x \right )} \cot ^{2}{\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 )^{2} \cot \left (d x + c\right )^{2}}{\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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