Optimal. Leaf size=182 \[ \frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {3 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 d \sqrt {a^2-b^2}}-\frac {\left (a^2+6 b^2\right ) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 0.47, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2890, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac {3 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^4 d \sqrt {a^2-b^2}}-\frac {\left (a^2+6 b^2\right ) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}+\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}+\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2890
Rule 3001
Rule 3055
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \cot ^2(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc (c+d x) \left (-6 b^2-2 a b \sin (c+d x)+\left (a^2+3 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 a^2 b}\\ &=\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2}-\frac {\left (a^2+6 b^2\right ) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-6 b^2 \left (a^2-b^2\right )-3 a b \left (a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^3 b \left (a^2-b^2\right )}\\ &=\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2}-\frac {\left (a^2+6 b^2\right ) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}-\frac {(3 b) \int \csc (c+d x) \, dx}{a^4}-\frac {\left (3 \left (a^2-2 b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^4}\\ &=\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2}-\frac {\left (a^2+6 b^2\right ) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}-\frac {\left (3 \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^4 d}\\ &=\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2}-\frac {\left (a^2+6 b^2\right ) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}+\frac {\left (6 \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^4 d}\\ &=-\frac {3 \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^4 \sqrt {a^2-b^2} d}+\frac {3 b \tanh ^{-1}(\cos (c+d x))}{a^4 d}+\frac {\left (a^2-3 b^2\right ) \cos (c+d x)}{2 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x)}{a d (a+b \sin (c+d x))^2}-\frac {\left (a^2+6 b^2\right ) \cos (c+d x)}{2 a^3 b d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 2.55, size = 184, normalized size = 1.01 \[ \frac {-\frac {6 \left (a^2-2 b^2\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}+\frac {a^2 \left (a^2-b^2\right ) \cos (c+d x)}{b (a+b \sin (c+d x))^2}-\frac {a \left (a^2+4 b^2\right ) \cos (c+d x)}{b (a+b \sin (c+d x))}+a \tan \left (\frac {1}{2} (c+d x)\right )-a \cot \left (\frac {1}{2} (c+d x)\right )-6 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 1064, normalized size = 5.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 273, normalized size = 1.50 \[ -\frac {\frac {6 \, b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac {\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} + \frac {6 \, {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )} {\left (a^{2} - 2 \, b^{2}\right )}}{\sqrt {a^{2} - b^{2}} a^{4}} - \frac {6 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} - \frac {2 \, {\left (a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 14 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5 \, a^{2} b\right )}}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a\right )}^{2} a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.79, size = 489, normalized size = 2.69 \[ \frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{3}}-\frac {1}{2 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{4}}+\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \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 )^{2}}-\frac {6 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{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 )^{2}}-\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) 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 )^{2}}-\frac {10 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{d \,a^{4} \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 )^{2}}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \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 )^{2}}-\frac {14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{2}}{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 )^{2}}-\frac {5 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 )^{2}}-\frac {3 \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^{2} \sqrt {a^{2}-b^{2}}}+\frac {6 \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^{4} \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 = 9.88, size = 956, normalized size = 5.25 \[ \text {result too large to display} \]
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 \[ \int \frac {\cos ^{4}{\left (c + d x \right )} \csc ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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