Optimal. Leaf size=123 \[ \frac {\left (a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}+\frac {2 a b \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d} \]
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Rubi [A] time = 0.36, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 8, 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, 8, 3770} \[ \frac {\left (a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}+\frac {2 a b \cot (c+d x)}{3 d}-\frac {a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2748
Rule 2889
Rule 3021
Rule 3031
Rule 3048
Rule 3767
Rule 3770
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \csc ^5(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 ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}+\frac {1}{4} \int \csc ^4(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 ^2(c+d x)}{6 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {1}{12} \int \csc ^3(c+d x) \left (3 \left (a^2-2 b^2\right )+8 a b \sin (c+d x)+9 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {1}{24} \int \csc ^2(c+d x) \left (16 a b+3 \left (a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {1}{3} (2 a b) \int \csc ^2(c+d x) \, dx-\frac {1}{8} \left (a^2+4 b^2\right ) \int \csc (c+d x) \, dx\\ &=\frac {\left (a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}+\frac {(2 a b) \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{3 d}\\ &=\frac {\left (a^2+4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{8 d}+\frac {2 a b \cot (c+d x)}{3 d}+\frac {\left (a^2-2 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 d}-\frac {a b \cot (c+d x) \csc ^2(c+d x)}{6 d}-\frac {\cot (c+d x) \csc ^3(c+d x) (a+b \sin (c+d x))^2}{4 d}\\ \end {align*}
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Mathematica [B] time = 6.17, size = 579, normalized size = 4.71 \[ \frac {\left (a^2-4 b^2\right ) \sin ^2(c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{32 d (a+b \sin (c+d x))^2}+\frac {\left (-a^2-4 b^2\right ) \sin ^2(c+d x) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{8 d (a+b \sin (c+d x))^2}+\frac {\left (4 b^2-a^2\right ) \sin ^2(c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{32 d (a+b \sin (c+d x))^2}+\frac {\left (a^2+4 b^2\right ) \sin ^2(c+d x) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right ) (a \csc (c+d x)+b)^2}{8 d (a+b \sin (c+d x))^2}-\frac {a^2 \sin ^2(c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{64 d (a+b \sin (c+d x))^2}+\frac {a^2 \sin ^2(c+d x) \sec ^4\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{64 d (a+b \sin (c+d x))^2}-\frac {a b \sin ^2(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{12 d (a+b \sin (c+d x))^2}+\frac {a b \sin ^2(c+d x) \cot \left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{3 d (a+b \sin (c+d x))^2}-\frac {a b \sin ^2(c+d x) \tan \left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{3 d (a+b \sin (c+d x))^2}+\frac {a b \sin ^2(c+d x) \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (a \csc (c+d x)+b)^2}{12 d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 200, normalized size = 1.63 \[ -\frac {32 \, a b \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + 6 \, {\left (a^{2} - 4 \, b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 4 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{2} + 4 \, b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 4 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 182, normalized size = 1.48 \[ \frac {3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 16 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 48 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, {\left (a^{2} + 4 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {50 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 200 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 48 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 173, normalized size = 1.41 \[ -\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}-\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}-\frac {a^{2} \cos \left (d x +c \right )}{8 d}-\frac {a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d}-\frac {2 a b \left (\cos ^{3}\left (d x +c \right )\right )}{3 d \sin \left (d x +c \right )^{3}}-\frac {b^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {b^{2} \cos \left (d x +c \right )}{2 d}-\frac {b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 129, normalized size = 1.05 \[ -\frac {3 \, a^{2} {\left (\frac {2 \, {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1} - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 12 \, b^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} + \frac {32 \, a b}{\tan \left (d x + c\right )^{3}}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.33, size = 165, normalized size = 1.34 \[ \frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {a^2}{8}+\frac {b^2}{2}\right )}{d}+\frac {b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^2}{4}-4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\frac {4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+2\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{16\,d}+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12\,d}-\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,d} \]
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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