Optimal. Leaf size=115 \[ \frac {2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 \cot (c+d x)}{a^2 d}-\frac {2 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \sqrt {a^2-b^2}}+\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.45, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2723, 3056, 3001, 3770, 2660, 618, 204} \[ -\frac {2 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^3 d \sqrt {a^2-b^2}}+\frac {2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 \cot (c+d x)}{a^2 d}+\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2723
Rule 3001
Rule 3056
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\int \frac {\csc ^2(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx\\ &=\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (2 \left (a^2-b^2\right )-\left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {2 \cot (c+d x)}{a^2 d}+\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-2 b \left (a^2-b^2\right )-a \left (a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {2 \cot (c+d x)}{a^2 d}+\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))}-\frac {(2 b) \int \csc (c+d x) \, dx}{a^3}-\frac {\left (a^2-2 b^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^3}\\ &=\frac {2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 \cot (c+d x)}{a^2 d}+\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))}-\frac {\left (2 \left (a^2-2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d}\\ &=\frac {2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 \cot (c+d x)}{a^2 d}+\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))}+\frac {\left (4 \left (a^2-2 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a^3 d}\\ &=-\frac {2 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^3 \sqrt {a^2-b^2} d}+\frac {2 b \tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac {2 \cot (c+d x)}{a^2 d}+\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 139, normalized size = 1.21 \[ -\frac {\frac {4 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {2 a b \cos (c+d x)}{a+b \sin (c+d x)}-a \tan \left (\frac {1}{2} (c+d x)\right )+a \cot \left (\frac {1}{2} (c+d x)\right )+4 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 768, normalized size = 6.68 \[ \left [\frac {4 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \, {\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) + 2 \, {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right ) - 2 \, {\left (a^{2} b^{2} - b^{4} - {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (a^{2} b^{2} - b^{4} - {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{2 \, {\left ({\left (a^{5} b - a^{3} b^{3}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{6} - a^{4} b^{2}\right )} d \sin \left (d x + c\right ) - {\left (a^{5} b - a^{3} b^{3}\right )} d\right )}}, \frac {2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{2} b - 2 \, b^{3} - {\left (a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} - 2 \, a b^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \sin \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) + {\left (a^{4} - a^{2} b^{2}\right )} \cos \left (d x + c\right ) - {\left (a^{2} b^{2} - b^{4} - {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{2} b^{2} - b^{4} - {\left (a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{{\left (a^{5} b - a^{3} b^{3}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{6} - a^{4} b^{2}\right )} d \sin \left (d x + c\right ) - {\left (a^{5} b - a^{3} b^{3}\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 218, normalized size = 1.90 \[ -\frac {\frac {12 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{2}} + \frac {12 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} {\left (a^{2} - 2 \, b^{2}\right )}}{\sqrt {a^{2} - b^{2}} a^{3}} - \frac {4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 \, b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 14 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{2}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} a^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.28, size = 245, normalized size = 2.13 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{2}}-\frac {1}{2 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}}-\frac {2 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{3} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}-\frac {2 b}{d \,a^{2} \left (\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right )}-\frac {2 \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d a \sqrt {a^{2}-b^{2}}}+\frac {4 \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right ) b^{2}}{d \,a^{3} \sqrt {a^{2}-b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.32, size = 1616, normalized size = 14.05 \[ \text {result too large to display} \]
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 \[ \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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