Optimal. Leaf size=294 \[ \frac {\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a^2 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}} \]
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Rubi [A] time = 0.71, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {2890, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ \frac {\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\left (4 a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a b^2 d \sqrt {a+b \sin (c+d x)}}+\frac {\left (4 a^2-3 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a^2 b^2 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {3 b \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2805
Rule 2807
Rule 2890
Rule 3002
Rule 3059
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^{3/2}} \, dx &=\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}+\frac {2 \int \frac {\csc (c+d x) \left (-\frac {3 b^2}{4}-\frac {1}{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}{a^2 b}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}+\frac {1}{2} \left (-\frac {3}{a^2}+\frac {4}{b^2}\right ) \int \sqrt {a+b \sin (c+d x)} \, dx-\frac {2 \int \frac {\csc (c+d x) \left (\frac {3 b^3}{4}+\frac {1}{4} a \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{a^2 b^2}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}-\frac {(3 b) \int \frac {\csc (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{2 a^2}-\frac {\left (4 a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{2 a b^2}+\frac {\left (\left (-\frac {3}{a^2}+\frac {4}{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}{2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (\frac {3}{a^2}-\frac {4}{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)}}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (3 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^2 \sqrt {a+b \sin (c+d x)}}-\frac {\left (\left (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}{2 a b^2 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {\left (2 a^2-3 b^2\right ) \cos (c+d x)}{a^2 b d \sqrt {a+b \sin (c+d x)}}-\frac {\cot (c+d x)}{a d \sqrt {a+b \sin (c+d x)}}-\frac {\left (\frac {3}{a^2}-\frac {4}{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)}}{d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\left (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}}}{a b^2 d \sqrt {a+b \sin (c+d x)}}-\frac {3 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^2 d \sqrt {a+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 3.60, size = 433, normalized size = 1.47 \[ \frac {\frac {4 a \left (a^2-b^2\right ) \cos (c+d x)}{b \sqrt {a+b \sin (c+d x)}}-\frac {a \left (4 a^2-9 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} \Pi \left (2;\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{b \sqrt {a+b \sin (c+d x)}}+\frac {i \left (3 b^2-4 a^2\right ) \sec (c+d x) \sqrt {-\frac {b (\sin (c+d x)-1)}{a+b}} \sqrt {\frac {b (\sin (c+d x)+1)}{b-a}} \left (b \left (b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )}{b^3 \sqrt {-\frac {1}{a+b}}}+\frac {4 a^2 \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (c+d x)}}-2 a \cot (c+d x) \sqrt {a+b \sin (c+d x)}}{2 a^3 d} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos \left (d x + c\right )^{2} \cot \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.69, size = 618, normalized size = 2.10 \[ -\frac {\left (-2 a^{3} b^{2}+3 a \,b^{4}\right ) \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a +b}+\frac {b}{a +b}}\, \sqrt {-\frac {b \sin \left (d x +c \right )}{a -b}-\frac {b}{a -b}}\, \left (4 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{5}-7 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}+3 \EllipticE \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}-3 \EllipticPi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}+3 \EllipticPi \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right ) b^{5}-4 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{4} b +6 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{3} b^{2}+\EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a^{2} b^{3}-3 \EllipticF \left (\sqrt {\frac {b \sin \left (d x +c \right )}{a -b}+\frac {a}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a \,b^{4}\right ) \sin \left (d x +c \right )+a^{2} b^{3} \left (\cos ^{2}\left (d x +c \right )\right )}{b^{3} \sin \left (d x +c \right ) a^{3} \cos \left (d x +c \right ) \sqrt {a +b \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\mathrm {cot}\left (c+d\,x\right )}^2}{{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cos ^{2}{\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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