Optimal. Leaf size=159 \[ -\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac {164 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{105 d}+\frac {8 a \cos (c+d x)}{15 d \sqrt {a \sin (c+d x)+a}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d} \]
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Rubi [A] time = 0.49, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {2881, 2770, 2759, 2751, 2646, 3046, 2981, 2773, 206} \[ -\frac {2 a \sin ^3(c+d x) \cos (c+d x)}{7 d \sqrt {a \sin (c+d x)+a}}-\frac {12 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 a d}+\frac {164 \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{105 d}+\frac {8 a \cos (c+d x)}{15 d \sqrt {a \sin (c+d x)+a}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2646
Rule 2751
Rule 2759
Rule 2770
Rule 2773
Rule 2881
Rule 2981
Rule 3046
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \cot (c+d x) \sqrt {a+a \sin (c+d x)} \, dx &=\int \sin ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {6}{7} \int \sin ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx+\frac {2 \int \csc (c+d x) \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{3 a}\\ &=\frac {4 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {4 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {12 \int \left (\frac {3 a}{2}-a \sin (c+d x)\right ) \sqrt {a+a \sin (c+d x)} \, dx}{35 a}+\int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {4 a \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {164 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}+\frac {2}{5} \int \sqrt {a+a \sin (c+d x)} \, dx-\frac {(2 a) \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 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}+\frac {8 a \cos (c+d x)}{15 d \sqrt {a+a \sin (c+d x)}}-\frac {2 a \cos (c+d x) \sin ^3(c+d x)}{7 d \sqrt {a+a \sin (c+d x)}}+\frac {164 \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{105 d}-\frac {12 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 a d}\\ \end {align*}
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Mathematica [A] time = 0.33, size = 195, normalized size = 1.23 \[ \frac {\sqrt {a (\sin (c+d x)+1)} \left (-525 \sin \left (\frac {1}{2} (c+d x)\right )+175 \sin \left (\frac {3}{2} (c+d x)\right )-21 \sin \left (\frac {5}{2} (c+d x)\right )+15 \sin \left (\frac {7}{2} (c+d x)\right )+525 \cos \left (\frac {1}{2} (c+d x)\right )+175 \cos \left (\frac {3}{2} (c+d x)\right )+21 \cos \left (\frac {5}{2} (c+d x)\right )+15 \cos \left (\frac {7}{2} (c+d x)\right )-420 \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+420 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{420 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 294, normalized size = 1.85 \[ \frac {105 \, \sqrt {a} {\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (15 \, \cos \left (d x + c\right )^{4} + 18 \, \cos \left (d x + c\right )^{3} + 34 \, \cos \left (d x + c\right )^{2} + {\left (15 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 31 \, \cos \left (d x + c\right ) - 43\right )} \sin \left (d x + c\right ) + 74 \, \cos \left (d x + c\right ) + 43\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{210 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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 = 1.33, size = 141, normalized size = 0.89 \[ -\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (105 a^{\frac {7}{2}} \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )-15 \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}}+63 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a -35 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{2}-105 a^{3} \sqrt {a -a \sin \left (d x +c \right )}\right )}{105 a^{3} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
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 \[ \int \sqrt {a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,\sqrt {a+a\,\sin \left (c+d\,x\right )}}{\sin \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \cos ^{4}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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