Optimal. Leaf size=131 \[ \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{4 d}+\frac {13 a^2 \cos (c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}-\frac {3 a \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
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Rubi [A] time = 0.52, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2874, 2975, 2981, 2773, 206} \[ \frac {13 a^2 \cos (c+d x)}{4 d \sqrt {a \sin (c+d x)+a}}+\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{4 d}-\frac {3 a \cot (c+d x) \sqrt {a \sin (c+d x)+a}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a \sin (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2773
Rule 2874
Rule 2975
Rule 2981
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\int \csc ^3(c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{5/2} \, dx}{a^2}\\ &=-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {\int \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \left (\frac {3 a^2}{2}-\frac {5}{2} a^2 \sin (c+d x)\right ) \, dx}{2 a^2}\\ &=-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (-\frac {a^3}{4}-\frac {13}{4} a^3 \sin (c+d x)\right ) \, dx}{2 a^2}\\ &=\frac {13 a^2 \cos (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}-\frac {1}{8} a \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {13 a^2 \cos (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}\\ &=\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{4 d}+\frac {13 a^2 \cos (c+d x)}{4 d \sqrt {a+a \sin (c+d x)}}-\frac {3 a \cot (c+d x) \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {\cot (c+d x) \csc (c+d x) (a+a \sin (c+d x))^{3/2}}{2 d}\\ \end {align*}
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Mathematica [B] time = 0.68, size = 271, normalized size = 2.07 \[ -\frac {a \csc ^7\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sin (c+d x)+1)} \left (22 \sin \left (\frac {1}{2} (c+d x)\right )+22 \sin \left (\frac {3}{2} (c+d x)\right )-8 \sin \left (\frac {5}{2} (c+d x)\right )-22 \cos \left (\frac {1}{2} (c+d x)\right )+22 \cos \left (\frac {3}{2} (c+d x)\right )+8 \cos \left (\frac {5}{2} (c+d x)\right )+\cos (2 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-\log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-\cos (2 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+\log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{4 d \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right ) \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 359, normalized size = 2.74 \[ \frac {{\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} - a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (8 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )^{2} - 6 \, a \cos \left (d x + c\right ) - {\left (8 \, a \cos \left (d x + c\right )^{2} - 7 \, a \cos \left (d x + c\right ) - 13 \, a\right )} \sin \left (d x + c\right ) - 13 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{16 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) + {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right ) - d\right )}} \]
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 = 1.20, size = 151, normalized size = 1.15 \[ \frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (8 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sin ^{2}\left (d x +c \right )\right ) a^{\frac {3}{2}}+\arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2}+7 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a}-9 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {3}{2}}\right )}{4 \sin \left (d x +c \right )^{2} \sqrt {a}\, \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \csc \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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