Optimal. Leaf size=89 \[ \frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-2 a b x+\frac {3 b^2 \cos (c+d x)}{2 d} \]
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Rubi [A] time = 0.31, antiderivative size = 89, 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 = {2889, 3048, 3031, 3023, 2735, 3770} \[ \frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-2 a b x+\frac {3 b^2 \cos (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2889
Rule 3023
Rule 3031
Rule 3048
Rule 3770
Rubi steps
\begin {align*} \int \cot ^2(c+d x) \csc (c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \csc ^3(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 (c+d x) (a+b \sin (c+d x))^2}{2 d}+\frac {1}{2} \int \csc ^2(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)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {1}{2} \int \csc (c+d x) \left (a^2-2 b^2+4 a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right ) \, dx\\ &=\frac {3 b^2 \cos (c+d x)}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {1}{2} \int \csc (c+d x) \left (a^2-2 b^2+4 a b \sin (c+d x)\right ) \, dx\\ &=-2 a b x+\frac {3 b^2 \cos (c+d x)}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}-\frac {1}{2} \left (a^2-2 b^2\right ) \int \csc (c+d x) \, dx\\ &=-2 a b x+\frac {\left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 d}+\frac {3 b^2 \cos (c+d x)}{2 d}-\frac {a b \cot (c+d x)}{d}-\frac {\cot (c+d x) \csc (c+d x) (a+b \sin (c+d x))^2}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 155, normalized size = 1.74 \[ \frac {a^2 \left (-\csc ^2\left (\frac {1}{2} (c+d x)\right )\right )+a^2 \sec ^2\left (\frac {1}{2} (c+d x)\right )-4 a^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+4 a^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+8 a b \tan \left (\frac {1}{2} (c+d x)\right )-8 a b \cot \left (\frac {1}{2} (c+d x)\right )-16 a b c-16 a b d x+8 b^2 \cos (c+d x)+8 b^2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 b^2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.65, size = 168, normalized size = 1.89 \[ -\frac {8 \, a b d x \cos \left (d x + c\right )^{2} - 4 \, b^{2} \cos \left (d x + c\right )^{3} - 8 \, a b d x - 8 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, {\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right ) - {\left ({\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 2 \, b^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left ({\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x + c\right )^{2} - a^{2} + 2 \, b^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 148, normalized size = 1.66 \[ \frac {a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, {\left (d x + c\right )} a b + 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, {\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {16 \, b^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + \frac {6 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.51, size = 126, normalized size = 1.42 \[ -\frac {a^{2} \left (\cos ^{3}\left (d x +c \right )\right )}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{2} \cos \left (d x +c \right )}{2 d}-\frac {a^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2 d}-2 a b x -\frac {2 a b \cot \left (d x +c \right )}{d}-\frac {2 a b c}{d}+\frac {b^{2} \cos \left (d x +c \right )}{d}+\frac {b^{2} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 103, normalized size = 1.16 \[ -\frac {8 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a b - a^{2} {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} + \log \left (\cos \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 2 \, b^{2} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.20, size = 397, normalized size = 4.46 \[ \frac {\cos \left (c+d\,x\right )\,\left (\frac {a^2}{2}-\frac {b^2}{4}\right )-\frac {b^2}{2}+\frac {a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4}-\frac {b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{2}+\frac {b^2\,\cos \left (2\,c+2\,d\,x\right )}{2}+\frac {b^2\,\cos \left (3\,c+3\,d\,x\right )}{4}-2\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )-\frac {a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{4}+\frac {b^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,\cos \left (2\,c+2\,d\,x\right )}{2}+a\,b\,\sin \left (2\,c+2\,d\,x\right )+2\,a\,b\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{-\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )\,\cos \left (2\,c+2\,d\,x\right )}{d\,\left ({\cos \left (c+d\,x\right )}^2-1\right )} \]
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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