Optimal. Leaf size=374 \[ \frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}+\frac{\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left (61 a^2 b^2+4 a^4+40 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{a \left (4 a^2+167 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}+\frac{3 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{a+b \sin (c+d x)}} \]
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Rubi [A] time = 1.16537, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}+\frac{\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left (61 a^2 b^2+4 a^4+40 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^2 d \sqrt{a+b \sin (c+d x)}}-\frac{a \left (4 a^2+167 b^2\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{35 b^2 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}+\frac{3 a b \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{d \sqrt{a+b \sin (c+d x)}} \]
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) (a+b \sin (c+d x))^{3/2} \, dx &=-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}-\frac{2 \int \csc (c+d x) (a+b \sin (c+d x))^{3/2} \left (-\frac{21 b^2}{4}+\frac{9}{2} a b \sin (c+d x)+\frac{1}{4} \left (4 a^2+35 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{7 a b}\\ &=\frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}-\frac{4 \int \csc (c+d x) \sqrt{a+b \sin (c+d x)} \left (-\frac{105 a b^2}{8}+\frac{51}{4} a^2 b \sin (c+d x)+\frac{3}{8} a \left (4 a^2+65 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{35 a b}\\ &=\frac{\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}-\frac{8 \int \frac{\csc (c+d x) \left (-\frac{315}{16} a^2 b^2+\frac{3}{8} a b \left (53 a^2-20 b^2\right ) \sin (c+d x)+\frac{3}{16} a^2 \left (4 a^2+167 b^2\right ) \sin ^2(c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{105 a b}\\ &=\frac{\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}-\frac{1}{70} \left (a \left (167+\frac{4 a^2}{b^2}\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx+\frac{8 \int \frac{\csc (c+d x) \left (\frac{315 a^2 b^3}{16}+\frac{3}{16} a \left (4 a^4+61 a^2 b^2+40 b^4\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{105 a b^2}\\ &=\frac{\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}+\frac{1}{2} (3 a b) \int \frac{\csc (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx+\frac{\left (4 a^4+61 a^2 b^2+40 b^4\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{70 b^2}-\frac{\left (a \left (167+\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}{70 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}\\ &=\frac{\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}-\frac{a \left (167+\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)}}{35 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (3 a 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 (\left (4 a^4+61 a^2 b^2+40 b^4\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}{70 b^2 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{\left (4 a^2+65 b^2\right ) \cos (c+d x) \sqrt{a+b \sin (c+d x)}}{35 b d}+\frac{\left (4 a^2+35 b^2\right ) \cos (c+d x) (a+b \sin (c+d x))^{3/2}}{35 a b d}-\frac{2 \cos (c+d x) (a+b \sin (c+d x))^{5/2}}{7 b d}-\frac{\cot (c+d x) (a+b \sin (c+d x))^{5/2}}{a d}-\frac{a \left (167+\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)}}{35 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (4 a^4+61 a^2 b^2+40 b^4\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}}}{35 b^2 d \sqrt{a+b \sin (c+d x)}}+\frac{3 a 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 = 3.93311, size = 452, normalized size = 1.21 \[ \frac{\frac{8 \left (53 a^2-20 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{\sqrt{a+b \sin (c+d x)}}+\frac{2 a \left (4 a^2-43 b^2\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )}{b \sqrt{a+b \sin (c+d x)}}-\frac{2 \sqrt{a+b \sin (c+d x)} \left (\left (4 a^2-55 b^2\right ) \cos (c+d x)+b (16 a \sin (2 (c+d x))+70 a \cot (c+d x)-5 b \cos (3 (c+d x)))\right )}{b}+\frac{2 i \left (4 a^2+167 b^2\right ) \sec (c+d x) \sqrt{-\frac{b (\sin (c+d x)-1)}{a+b}} \sqrt{\frac{b (\sin (c+d x)+1)}{b-a}} \left (b \left (b \Pi \left (\frac{a+b}{a};i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt{-\frac{1}{a+b}} \sqrt{a+b \sin (c+d x)}\right )|\frac{a+b}{a-b}\right )\right )}{b^3 \sqrt{-\frac{1}{a+b}}}}{140 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.58, size = 726, normalized size = 1.9 \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{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} \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(-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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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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