Optimal. Leaf size=220 \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{128 a^{3/2} d}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 a^2 d}-\frac {3 \cot (c+d x)}{128 a d \sqrt {a \sin (c+d x)+a}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x)}{64 a d \sqrt {a \sin (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.88, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2880, 2772, 2773, 206, 3044, 2980} \[ -\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{128 a^{3/2} d}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a \sin (c+d x)+a}}{5 a^2 d}-\frac {3 \cot (c+d x)}{128 a d \sqrt {a \sin (c+d x)+a}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt {a \sin (c+d x)+a}}-\frac {\cot (c+d x) \csc (c+d x)}{64 a d \sqrt {a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 2772
Rule 2773
Rule 2880
Rule 2980
Rule 3044
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\int \csc ^6(c+d x) \sqrt {a+a \sin (c+d x)} \left (1+\sin ^2(c+d x)\right ) \, dx}{a^2}-\frac {2 \int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{a^2}\\ &=\frac {\cot (c+d x) \csc ^3(c+d x)}{2 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}+\frac {\int \csc ^5(c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {a}{2}+\frac {17}{2} a \sin (c+d x)\right ) \, dx}{5 a^3}-\frac {7 \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{4 a^2}\\ &=\frac {7 \cot (c+d x) \csc ^2(c+d x)}{12 a d \sqrt {a+a \sin (c+d x)}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}-\frac {35 \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{24 a^2}+\frac {143 \int \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{80 a^2}\\ &=\frac {35 \cot (c+d x) \csc (c+d x)}{48 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}-\frac {35 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{32 a^2}+\frac {143 \int \csc ^3(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{96 a^2}\\ &=\frac {35 \cot (c+d x)}{32 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}-\frac {35 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{64 a^2}+\frac {143 \int \csc ^2(c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{128 a^2}\\ &=-\frac {3 \cot (c+d x)}{128 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}+\frac {143 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx}{256 a^2}+\frac {35 \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 )}{32 a d}\\ &=\frac {35 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{32 a^{3/2} d}-\frac {3 \cot (c+d x)}{128 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}-\frac {143 \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 )}{128 a d}\\ &=-\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{128 a^{3/2} d}-\frac {3 \cot (c+d x)}{128 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc (c+d x)}{64 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{80 a d \sqrt {a+a \sin (c+d x)}}+\frac {19 \cot (c+d x) \csc ^3(c+d x)}{40 a d \sqrt {a+a \sin (c+d x)}}-\frac {\cot (c+d x) \csc ^4(c+d x) \sqrt {a+a \sin (c+d x)}}{5 a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.38, size = 412, normalized size = 1.87 \[ -\frac {\csc ^{15}\left (\frac {1}{2} (c+d x)\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3 \left (-7100 \sin \left (\frac {1}{2} (c+d x)\right )-2880 \sin \left (\frac {3}{2} (c+d x)\right )+144 \sin \left (\frac {5}{2} (c+d x)\right )-10 \sin \left (\frac {7}{2} (c+d x)\right )-30 \sin \left (\frac {9}{2} (c+d x)\right )+7100 \cos \left (\frac {1}{2} (c+d x)\right )-2880 \cos \left (\frac {3}{2} (c+d x)\right )-144 \cos \left (\frac {5}{2} (c+d x)\right )-10 \cos \left (\frac {7}{2} (c+d x)\right )+30 \cos \left (\frac {9}{2} (c+d x)\right )+150 \sin (c+d x) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-150 \sin (c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-75 \sin (3 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+75 \sin (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )+15 \sin (5 (c+d x)) \log \left (-\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )+1\right )-15 \sin (5 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )-\cos \left (\frac {1}{2} (c+d x)\right )+1\right )\right )}{640 d (a (\sin (c+d x)+1))^{3/2} \left (\csc ^2\left (\frac {1}{4} (c+d x)\right )-\sec ^2\left (\frac {1}{4} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.49, size = 492, normalized size = 2.24 \[ \frac {15 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{5} + \cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right ) + 1\right )} \sin \left (d x + c\right ) - 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \, {\left (15 \, \cos \left (d x + c\right )^{5} + 5 \, \cos \left (d x + c\right )^{4} - 38 \, \cos \left (d x + c\right )^{3} - 194 \, \cos \left (d x + c\right )^{2} - {\left (15 \, \cos \left (d x + c\right )^{4} + 10 \, \cos \left (d x + c\right )^{3} - 28 \, \cos \left (d x + c\right )^{2} + 166 \, \cos \left (d x + c\right ) + 317\right )} \sin \left (d x + c\right ) + 151 \, \cos \left (d x + c\right ) + 317\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2560 \, {\left (a^{2} d \cos \left (d x + c\right )^{6} - 3 \, a^{2} d \cos \left (d x + c\right )^{4} + 3 \, a^{2} d \cos \left (d x + c\right )^{2} - a^{2} d - {\left (a^{2} d \cos \left (d x + c\right )^{5} + a^{2} d \cos \left (d x + c\right )^{4} - 2 \, a^{2} d \cos \left (d x + c\right )^{3} - 2 \, a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.16, size = 808, normalized size = 3.67 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 1.36, size = 180, normalized size = 0.82 \[ -\frac {\left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (15 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {9}{2}} a^{\frac {7}{2}}-70 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {7}{2}} a^{\frac {9}{2}}+128 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {11}{2}}+15 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (d x +c \right )-1\right )}}{\sqrt {a}}\right ) a^{8} \left (\sin ^{5}\left (d x +c \right )\right )+70 \left (-a \left (\sin \left (d x +c \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {13}{2}}-15 \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, a^{\frac {15}{2}}\right )}{640 a^{\frac {19}{2}} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^4}{{\sin \left (c+d\,x\right )}^6\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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]
________________________________________________________________________________________