Optimal. Leaf size=118 \[ -\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+a^2 x \]
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Rubi [A] time = 0.19, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2873, 3473, 8, 2611, 3770, 2607, 30} \[ -\frac {a^2 \cot ^5(c+d x)}{5 d}-\frac {a^2 \cot ^3(c+d x)}{3 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}+a^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 3473
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\int \left (a^2 \cot ^4(c+d x)+2 a^2 \cot ^4(c+d x) \csc (c+d x)+a^2 \cot ^4(c+d x) \csc ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^4(c+d x) \, dx+a^2 \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx+\left (2 a^2\right ) \int \cot ^4(c+d x) \csc (c+d x) \, dx\\ &=-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}-a^2 \int \cot ^2(c+d x) \, dx-\frac {1}{2} \left (3 a^2\right ) \int \cot ^2(c+d x) \csc (c+d x) \, dx+\frac {a^2 \operatorname {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}+\frac {1}{4} \left (3 a^2\right ) \int \csc (c+d x) \, dx+a^2 \int 1 \, dx\\ &=a^2 x-\frac {3 a^2 \tanh ^{-1}(\cos (c+d x))}{4 d}+\frac {a^2 \cot (c+d x)}{d}-\frac {a^2 \cot ^3(c+d x)}{3 d}-\frac {a^2 \cot ^5(c+d x)}{5 d}+\frac {3 a^2 \cot (c+d x) \csc (c+d x)}{4 d}-\frac {a^2 \cot ^3(c+d x) \csc (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.53, size = 200, normalized size = 1.69 \[ \frac {a^2 \left (-272 \tan \left (\frac {1}{2} (c+d x)\right )+272 \cot \left (\frac {1}{2} (c+d x)\right )+150 \csc ^2\left (\frac {1}{2} (c+d x)\right )+15 \sec ^4\left (\frac {1}{2} (c+d x)\right )-150 \sec ^2\left (\frac {1}{2} (c+d x)\right )+360 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-360 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-\frac {3}{2} \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+96 \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)+\frac {1}{2} (\sin (c+d x)-30) \csc ^4\left (\frac {1}{2} (c+d x)\right )-8 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+480 c+480 d x\right )}{480 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.52, size = 239, normalized size = 2.03 \[ \frac {136 \, a^{2} \cos \left (d x + c\right )^{5} - 280 \, a^{2} \cos \left (d x + c\right )^{3} + 120 \, a^{2} \cos \left (d x + c\right ) - 45 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 45 \, {\left (a^{2} \cos \left (d x + c\right )^{4} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 30 \, {\left (4 \, a^{2} d x \cos \left (d x + c\right )^{4} - 8 \, a^{2} d x \cos \left (d x + c\right )^{2} - 5 \, a^{2} \cos \left (d x + c\right )^{3} + 4 \, a^{2} d x + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 207, normalized size = 1.75 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 480 \, {\left (d x + c\right )} a^{2} + 360 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - 270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {822 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 270 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 170, normalized size = 1.44 \[ -\frac {a^{2} \left (\cot ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a^{2} \cot \left (d x +c \right )}{d}+a^{2} x +\frac {a^{2} c}{d}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{4}}+\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{2}}+\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a^{2} \cos \left (d x +c \right )}{4 d}+\frac {3 a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{4 d}-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 124, normalized size = 1.05 \[ \frac {40 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{2} - 15 \, a^{2} {\left (\frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {24 \, a^{2}}{\tan \left (d x + c\right )^{5}}}{120 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.16, size = 275, normalized size = 2.33 \[ \frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}-\frac {a^2\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}+\frac {2\,a^2\,\mathrm {atan}\left (\frac {4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+3\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )-4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {3\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {9\,a^2\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d} \]
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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