Optimal. Leaf size=98 \[ -\frac {a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {b \cot ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.17, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2611, 3768, 3770, 2607, 30} \[ -\frac {a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}-\frac {b \cot ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rule 2611
Rule 2838
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x)) \, dx &=a \int \cot ^4(c+d x) \csc ^3(c+d x) \, dx+b \int \cot ^4(c+d x) \csc ^2(c+d x) \, dx\\ &=-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}-\frac {1}{2} a \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx+\frac {b \operatorname {Subst}\left (\int x^4 \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {b \cot ^5(c+d x)}{5 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{8} a \int \csc ^3(c+d x) \, dx\\ &=-\frac {b \cot ^5(c+d x)}{5 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}+\frac {1}{16} a \int \csc (c+d x) \, dx\\ &=-\frac {a \tanh ^{-1}(\cos (c+d x))}{16 d}-\frac {b \cot ^5(c+d x)}{5 d}-\frac {a \cot (c+d x) \csc (c+d x)}{16 d}+\frac {a \cot (c+d x) \csc ^3(c+d x)}{8 d}-\frac {a \cot ^3(c+d x) \csc ^3(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 175, normalized size = 1.79 \[ -\frac {a \csc ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}+\frac {a \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^6\left (\frac {1}{2} (c+d x)\right )}{384 d}-\frac {a \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{16 d}-\frac {b \cot ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 187, normalized size = 1.91 \[ \frac {96 \, b \cos \left (d x + c\right )^{5} \sin \left (d x + c\right ) + 30 \, a \cos \left (d x + c\right )^{5} + 80 \, a \cos \left (d x + c\right )^{3} - 30 \, a \cos \left (d x + c\right ) - 15 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 201, normalized size = 2.05 \[ \frac {5 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 12 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 60 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 120 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {294 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 120 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 60 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 12 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 138, normalized size = 1.41 \[ -\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}}-\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{24 d \sin \left (d x +c \right )^{4}}+\frac {a \left (\cos ^{5}\left (d x +c \right )\right )}{48 d \sin \left (d x +c \right )^{2}}+\frac {a \left (\cos ^{3}\left (d x +c \right )\right )}{48 d}+\frac {a \cos \left (d x +c \right )}{16 d}+\frac {a \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {b \left (\cos ^{5}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 106, normalized size = 1.08 \[ \frac {5 \, a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} + 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - \frac {96 \, b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.52, size = 205, normalized size = 2.09 \[ \frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32\,d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+\frac {2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+\frac {a}{6}\right )}{64\,d}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16\,d} \]
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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