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 = 0.88, antiderivative size = 323, normalized size of antiderivative = 1.00, 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 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2805
Rule 2807
Rule 2894
Rule 3002
Rule 3049
Rule 3059
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*}
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Mathematica [C] time = 3.65, 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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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.90, size = 656, normalized size = 2.03 \[ -\frac {-6 a \,b^{4} \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+\left (2 a^{3} b^{2}+21 a \,b^{4}\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+\sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, \sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, \left (15 \EllipticPi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}-15 \EllipticPi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}+4 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b +42 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}+11 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}-57 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}-4 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}-53 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}+57 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}\right ) \sin \left (d x +c \right )-8 a^{2} b^{3} \left (\cos ^{4}\left (d x +c \right )\right )+23 a^{2} b^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{15 a \,b^{3} \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^2\,{\mathrm {cot}\left (c+d\,x\right )}^2\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a + b \sin {\left (c + d x \right )}} \cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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