Optimal. Leaf size=78 \[ -\frac {a^2 \cot (c+d x)}{d}+a^2 (-x)+\frac {2 a b \cos (c+d x)}{d}-\frac {2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac {b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {b^2 x}{2} \]
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Rubi [A] time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2722, 2635, 8, 2592, 321, 206, 3473} \[ -\frac {a^2 \cot (c+d x)}{d}+a^2 (-x)+\frac {2 a b \cos (c+d x)}{d}-\frac {2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac {b^2 \sin (c+d x) \cos (c+d x)}{2 d}+\frac {b^2 x}{2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 206
Rule 321
Rule 2592
Rule 2635
Rule 2722
Rule 3473
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+b \sin (c+d x))^2 \, dx &=\int \left (b^2 \cos ^2(c+d x)+2 a b \cos (c+d x) \cot (c+d x)+a^2 \cot ^2(c+d x)\right ) \, dx\\ &=a^2 \int \cot ^2(c+d x) \, dx+(2 a b) \int \cos (c+d x) \cot (c+d x) \, dx+b^2 \int \cos ^2(c+d x) \, dx\\ &=-\frac {a^2 \cot (c+d x)}{d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 d}-a^2 \int 1 \, dx+\frac {1}{2} b^2 \int 1 \, dx-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a^2 x+\frac {b^2 x}{2}+\frac {2 a b \cos (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {(2 a b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-a^2 x+\frac {b^2 x}{2}-\frac {2 a b \tanh ^{-1}(\cos (c+d x))}{d}+\frac {2 a b \cos (c+d x)}{d}-\frac {a^2 \cot (c+d x)}{d}+\frac {b^2 \cos (c+d x) \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 116, normalized size = 1.49 \[ \frac {2 a^2 \tan \left (\frac {1}{2} (c+d x)\right )-2 a^2 \cot \left (\frac {1}{2} (c+d x)\right )-4 a^2 c-4 a^2 d x+8 a b \cos (c+d x)+8 a b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-8 a b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+b^2 \sin (2 (c+d x))+2 b^2 c+2 b^2 d x}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 118, normalized size = 1.51 \[ -\frac {b^{2} \cos \left (d x + c\right )^{3} + 2 \, a b \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 2 \, a b \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right ) + {\left ({\left (2 \, a^{2} - b^{2}\right )} d x - 4 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 148, normalized size = 1.90 \[ \frac {4 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - {\left (2 \, a^{2} - b^{2}\right )} {\left (d x + c\right )} - \frac {4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 4 \, a b\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 102, normalized size = 1.31 \[ -a^{2} x -\frac {a^{2} \cot \left (d x +c \right )}{d}-\frac {a^{2} c}{d}+\frac {2 a b \cos \left (d x +c \right )}{d}+\frac {2 a b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d}+\frac {b^{2} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {b^{2} x}{2}+\frac {b^{2} c}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 79, normalized size = 1.01 \[ -\frac {4 \, {\left (d x + c + \frac {1}{\tan \left (d x + c\right )}\right )} a^{2} - {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} b^{2} - 4 \, a b {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.27, size = 277, normalized size = 3.55 \[ \frac {b^2\,\mathrm {atan}\left (\frac {-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )-2\,a^2\,\mathrm {atan}\left (\frac {-2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}{2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a^2+4\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,a\,b-\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,b^2}\right )+2\,a\,b\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a^2\,\cos \left (c+d\,x\right )-\frac {b^2\,\cos \left (c+d\,x\right )}{8}+\frac {b^2\,\cos \left (3\,c+3\,d\,x\right )}{8}-a\,b\,\sin \left (2\,c+2\,d\,x\right )}{d\,\sin \left (c+d\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{2} \cot ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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