Optimal. Leaf size=170 \[ \frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {\left (a^2+2 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {2 a b \cot ^3(c+d x)}{15 d}+\frac {2 a b \cot (c+d x)}{5 d}-\frac {a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d} \]
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Rubi [A] time = 0.40, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2889, 3048, 3031, 3021, 2748, 3767, 3768, 3770} \[ \frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}+\frac {\left (a^2+2 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {2 a b \cot ^3(c+d x)}{15 d}+\frac {2 a b \cot (c+d x)}{5 d}-\frac {a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d} \]
Antiderivative was successfully verified.
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Rule 2748
Rule 2889
Rule 3021
Rule 3031
Rule 3048
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \csc ^7(c+d x) (a+b \sin (c+d x))^2 \left (1-\sin ^2(c+d x)\right ) \, dx\\ &=-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}+\frac {1}{6} \int \csc ^6(c+d x) (a+b \sin (c+d x)) \left (2 b-a \sin (c+d x)-3 b \sin ^2(c+d x)\right ) \, dx\\ &=-\frac {a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}-\frac {1}{30} \int \csc ^5(c+d x) \left (5 \left (a^2-2 b^2\right )+12 a b \sin (c+d x)+15 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}-\frac {1}{120} \int \csc ^4(c+d x) \left (48 a b+15 \left (a^2+2 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}-\frac {1}{5} (2 a b) \int \csc ^4(c+d x) \, dx-\frac {1}{8} \left (a^2+2 b^2\right ) \int \csc ^3(c+d x) \, dx\\ &=\frac {\left (a^2+2 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}-\frac {1}{16} \left (a^2+2 b^2\right ) \int \csc (c+d x) \, dx+\frac {(2 a b) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{5 d}\\ &=\frac {\left (a^2+2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{16 d}+\frac {2 a b \cot (c+d x)}{5 d}+\frac {2 a b \cot ^3(c+d x)}{15 d}+\frac {\left (a^2+2 b^2\right ) \cot (c+d x) \csc (c+d x)}{16 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc ^3(c+d x)}{24 d}-\frac {a b \cot (c+d x) \csc ^4(c+d x)}{15 d}-\frac {\cot (c+d x) \csc ^5(c+d x) (a+b \sin (c+d x))^2}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.80, size = 296, normalized size = 1.74 \[ \frac {30 \left (a^2+2 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )+5 a^2 \sec ^6\left (\frac {1}{2} (c+d x)\right )-30 a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+120 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+\csc ^4\left (\frac {1}{2} (c+d x)\right ) \left (4 a b \sin (c+d x)-30 b^2\right )-256 a b \tan \left (\frac {1}{2} (c+d x)\right )+256 a b \cot \left (\frac {1}{2} (c+d x)\right )-a \csc ^6\left (\frac {1}{2} (c+d x)\right ) (5 a+12 b \sin (c+d x))+768 a b \sin ^6\left (\frac {1}{2} (c+d x)\right ) \csc ^5(c+d x)-64 a b \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+30 b^2 \sec ^4\left (\frac {1}{2} (c+d x)\right )-60 b^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-240 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+240 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{1920 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 283, normalized size = 1.66 \[ -\frac {30 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{5} - 80 \, a^{2} \cos \left (d x + c\right )^{3} - 30 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right ) - 15 \, {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 15 \, {\left ({\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{4} + 3 \, {\left (a^{2} + 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} - 2 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 64 \, {\left (2 \, a b \cos \left (d x + c\right )^{5} - 5 \, a b \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{480 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 276, normalized size = 1.62 \[ \frac {5 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 40 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 120 \, {\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {294 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 588 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 240 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 40 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 30 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 24 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 244, normalized size = 1.44 \[ -\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}}-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{4}}-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{16 d \sin \left (d x +c \right )^{2}}-\frac {a^{2} \cos \left (d x +c \right )}{16 d}-\frac {a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{16 d}-\frac {2 a b \left (\cos ^{3}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}-\frac {4 a b \left (\cos ^{3}\left (d x +c \right )\right )}{15 d \sin \left (d x +c \right )^{3}}-\frac {b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}-\frac {b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {b^{2} \cos \left (d x +c \right )}{8 d}-\frac {b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 186, normalized size = 1.09 \[ -\frac {5 \, a^{2} {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 8 \, \cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + 30 \, b^{2} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {64 \, {\left (5 \, \tan \left (d x + c\right )^{2} + 3\right )} a b}{\tan \left (d x + c\right )^{5}}}{480 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.44, size = 245, normalized size = 1.44 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {a^2}{16}+\frac {b^2}{8}\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^2}{6}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{2}+b^2\right )+\frac {4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}\right )}{64\,d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{128}+\frac {b^2}{64}\right )}{d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{48\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{80\,d}-\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d} \]
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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