Optimal. Leaf size=123 \[ -\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}+\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
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Rubi [A] time = 0.46, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2874, 2976, 2981, 2773, 206} \[ -\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}+\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}+\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2773
Rule 2874
Rule 2976
Rule 2981
Rubi steps
\begin {align*} \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\int \csc (c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{5/2} \, dx}{a^2}\\ &=\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {2 \int \csc (c+d x) (a+a \sin (c+d x))^{3/2} \left (\frac {5 a^2}{2}-\frac {3}{2} a^2 \sin (c+d x)\right ) \, dx}{5 a^2}\\ &=\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {4 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {15 a^3}{4}+\frac {3}{4} a^3 \sin (c+d x)\right ) \, dx}{15 a^2}\\ &=-\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+a \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=-\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\left (2 a^2\right ) \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 )}{d}\\ &=-\frac {2 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 145, normalized size = 1.18 \[ \frac {(a (\sin (c+d x)+1))^{3/2} \left (5 \sin \left (\frac {3}{2} (c+d x)\right )+\sin \left (\frac {5}{2} (c+d x)\right )+5 \cos \left (\frac {3}{2} (c+d x)\right )-\cos \left (\frac {5}{2} (c+d x)\right )-10 \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+10 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{10 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 282, normalized size = 2.29 \[ \frac {5 \, {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{10 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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 = 1.08, size = 121, normalized size = 0.98 \[ \frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (-5 a^{\frac {5}{2}} \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )+\left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}-5 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a +5 a^{2} \sqrt {a -a \sin \left (d x +c \right )}\right )}{5 a \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \csc \left (d x + c\right )\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{\sin \left (c+d\,x\right )} \,d x \]
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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