Optimal. Leaf size=178 \[ -\frac {b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}-\frac {3 \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}+\frac {17 a b \cot (c+d x)}{12 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+2 a b x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.46, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2893, 3047, 3031, 3023, 2735, 3770} \[ -\frac {b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}-\frac {3 \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}+\frac {17 a b \cot (c+d x)}{12 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+2 a b x \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2735
Rule 2893
Rule 3023
Rule 3031
Rule 3047
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx &=\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}-\frac {\int \csc ^3(c+d x) (a+b \sin (c+d x))^2 \left (15 a^2+2 a b \sin (c+d x)-\left (12 a^2+b^2\right ) \sin ^2(c+d x)\right ) \, dx}{12 a^2}\\ &=\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}-\frac {\int \csc ^2(c+d x) (a+b \sin (c+d x)) \left (34 a^2 b-a \left (9 a^2-2 b^2\right ) \sin (c+d x)-b \left (39 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=\frac {17 a b \cot (c+d x)}{12 d}+\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac {\int \csc (c+d x) \left (9 a^2 \left (a^2-4 b^2\right )+48 a^3 b \sin (c+d x)+b^2 \left (39 a^2+2 b^2\right ) \sin ^2(c+d x)\right ) \, dx}{24 a^2}\\ &=-\frac {b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}+\frac {17 a b \cot (c+d x)}{12 d}+\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac {\int \csc (c+d x) \left (9 a^2 \left (a^2-4 b^2\right )+48 a^3 b \sin (c+d x)\right ) \, dx}{24 a^2}\\ &=2 a b x-\frac {b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}+\frac {17 a b \cot (c+d x)}{12 d}+\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}+\frac {1}{8} \left (3 \left (a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx\\ &=2 a b x-\frac {3 \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}-\frac {b^2 \left (39 a^2+2 b^2\right ) \cos (c+d x)}{24 a^2 d}+\frac {17 a b \cot (c+d x)}{12 d}+\frac {5 \cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{8 d}+\frac {b \cot (c+d x) \csc ^2(c+d x) (a+b \sin (c+d x))^3}{12 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^3}{4 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.74, size = 270, normalized size = 1.52 \[ \frac {-3 a^2 \csc ^4\left (\frac {1}{2} (c+d x)\right )+30 a^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+3 a^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )-30 a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )+72 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-72 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-256 a b \tan \left (\frac {1}{2} (c+d x)\right )+256 a b \cot \left (\frac {1}{2} (c+d x)\right )-8 a b \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+128 a b \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+384 a b c+384 a b d x-192 b^2 \cos (c+d x)-24 b^2 \csc ^2\left (\frac {1}{2} (c+d x)\right )+24 b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-288 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+288 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{192 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.71, size = 260, normalized size = 1.46 \[ \frac {96 \, a b d x \cos \left (d x + c\right )^{4} - 48 \, b^{2} \cos \left (d x + c\right )^{5} - 192 \, a b d x \cos \left (d x + c\right )^{2} + 96 \, a b d x - 30 \, {\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 18 \, {\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right ) - 9 \, {\left ({\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} - 4 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 9 \, {\left ({\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} - 4 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (4 \, a b \cos \left (d x + c\right )^{3} - 3 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.28, size = 244, normalized size = 1.37 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 384 \, {\left (d x + c\right )} a b - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, {\left (a^{2} - 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {384 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} - \frac {150 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 600 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 16 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.48, size = 223, normalized size = 1.25 \[ -\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}+\frac {3 a^{2} \cos \left (d x +c \right )}{8 d}+\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {2 a b \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {2 a b \cot \left (d x +c \right )}{d}+2 a b x +\frac {2 a b c}{d}-\frac {b^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d}-\frac {3 b^{2} \cos \left (d x +c \right )}{2 d}-\frac {3 b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.41, size = 166, normalized size = 0.93 \[ \frac {32 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a b - 3 \, a^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 12 \, b^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 10.89, size = 825, normalized size = 4.63 \[ \text {result too large to display} \]
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]
________________________________________________________________________________________