Optimal. Leaf size=178 \[ -\frac {3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}+\frac {171 a^2 \cos (c+d x)}{35 d \sqrt {a \sin (c+d x)+a}}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac {4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}+\frac {69 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{35 d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.65, antiderivative size = 178, 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 = {2881, 2759, 2751, 2647, 2646, 3044, 2976, 2981, 2773, 206} \[ \frac {171 a^2 \cos (c+d x)}{35 d \sqrt {a \sin (c+d x)+a}}-\frac {3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{5/2}}{7 a d}+\frac {4 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{35 d}+\frac {69 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{35 d}-\frac {\cot (c+d x) (a \sin (c+d x)+a)^{3/2}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2646
Rule 2647
Rule 2751
Rule 2759
Rule 2773
Rule 2881
Rule 2976
Rule 2981
Rule 3044
Rubi steps
\begin {align*} \int \cos ^2(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\int \sin ^2(c+d x) (a+a \sin (c+d x))^{3/2} \, dx+\int \csc ^2(c+d x) (a+a \sin (c+d x))^{3/2} \left (1-2 \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}+\frac {2 \int \left (\frac {5 a}{2}-a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{7 a}+\frac {\int \csc (c+d x) \left (\frac {3 a}{2}-\frac {7}{2} a \sin (c+d x)\right ) (a+a \sin (c+d x))^{3/2} \, dx}{a}\\ &=\frac {7 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{3 d}+\frac {4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}+\frac {19}{35} \int (a+a \sin (c+d x))^{3/2} \, dx+\frac {2 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {9 a^2}{4}-\frac {19}{4} a^2 \sin (c+d x)\right ) \, dx}{3 a}\\ &=\frac {19 a^2 \cos (c+d x)}{3 d \sqrt {a+a \sin (c+d x)}}+\frac {69 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}+\frac {4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}+\frac {1}{105} (76 a) \int \sqrt {a+a \sin (c+d x)} \, dx+\frac {1}{2} (3 a) \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx\\ &=\frac {171 a^2 \cos (c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}+\frac {69 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}+\frac {4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}-\frac {\left (3 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 {3 a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}+\frac {171 a^2 \cos (c+d x)}{35 d \sqrt {a+a \sin (c+d x)}}+\frac {69 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{35 d}+\frac {4 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{35 d}-\frac {\cot (c+d x) (a+a \sin (c+d x))^{3/2}}{d}-\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{5/2}}{7 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.32, size = 283, normalized size = 1.59 \[ -\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sin (c+d x)+1)} \left (-840 \sin \left (\frac {1}{2} (c+d x)\right )-574 \sin \left (\frac {3}{2} (c+d x)\right )-30 \sin \left (\frac {5}{2} (c+d x)\right )-21 \sin \left (\frac {7}{2} (c+d x)\right )-5 \sin \left (\frac {9}{2} (c+d x)\right )+840 \cos \left (\frac {1}{2} (c+d x)\right )-574 \cos \left (\frac {3}{2} (c+d x)\right )+30 \cos \left (\frac {5}{2} (c+d x)\right )-21 \cos \left (\frac {7}{2} (c+d x)\right )+5 \cos \left (\frac {9}{2} (c+d x)\right )+420 \sin (c+d x) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-420 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{140 d \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )-\sec \left (\frac {1}{4} (c+d x)\right )\right ) \left (\csc \left (\frac {1}{4} (c+d x)\right )+\sec \left (\frac {1}{4} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 360, normalized size = 2.02 \[ \frac {105 \, {\left (a \cos \left (d x + c\right )^{2} - {\left (a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) - a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (10 \, a \cos \left (d x + c\right )^{5} - 16 \, a \cos \left (d x + c\right )^{4} - 8 \, a \cos \left (d x + c\right )^{3} - 120 \, a \cos \left (d x + c\right )^{2} + 33 \, a \cos \left (d x + c\right ) - {\left (10 \, a \cos \left (d x + c\right )^{4} + 26 \, a \cos \left (d x + c\right )^{3} + 18 \, a \cos \left (d x + c\right )^{2} + 138 \, a \cos \left (d x + c\right ) + 171 \, a\right )} \sin \left (d x + c\right ) + 171 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{140 \, {\left (d \cos \left (d x + c\right )^{2} - {\left (d \cos \left (d x + c\right ) + d\right )} \sin \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.28, size = 180, normalized size = 1.01 \[ \frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (\sin \left (d x +c \right ) \left (140 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {7}{2}}+70 \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}} a^{\frac {5}{2}}-56 \left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}} a^{\frac {3}{2}}+10 \sqrt {a}\, \left (a -a \sin \left (d x +c \right )\right )^{\frac {7}{2}}-105 \arctanh \left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right ) a^{4}\right )-35 \sqrt {a -a \sin \left (d x +c \right )}\, a^{\frac {7}{2}}\right )}{35 a^{\frac {5}{2}} \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{4} \csc \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{{\sin \left (c+d\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________