Optimal. Leaf size=269 \[ \frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 d \left (a^2-b^2\right )}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d \left (a^2-b^2\right )}+\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{3/2}}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 1.15, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2889, 3056, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {b \left (-19 a^2 b^2+6 a^4+12 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{3/2}}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 d \left (a^2-b^2\right )}+\frac {\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 d \left (a^2-b^2\right )}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2889
Rule 3001
Rule 3055
Rule 3056
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^2(c+d x) \csc (c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\int \frac {\csc ^3(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^3} \, dx\\ &=\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^3(c+d x) \left (4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (2 \left (5 a^4-11 a^2 b^2+6 b^4\right )-a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (3 a^2-4 b^2\right ) \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 b \left (11 a^4-23 a^2 b^2+12 b^4\right )-2 a \left (a^4-3 a^2 b^2+2 b^4\right ) \sin (c+d x)+2 b \left (5 a^4-11 a^2 b^2+6 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (-2 \left (a^2-12 b^2\right ) \left (a^2-b^2\right )^2+2 a b \left (5 a^4-11 a^2 b^2+6 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (a^2-12 b^2\right ) \int \csc (c+d x) \, dx}{2 a^5}+\frac {\left (b \left (6 a^4-19 a^2 b^2+12 b^4\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^5 \left (a^2-b^2\right )}\\ &=\frac {\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac {\left (b \left (6 a^4-19 a^2 b^2+12 b^4\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^5 \left (a^2-b^2\right ) d}\\ &=\frac {\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}-\frac {\left (2 b \left (6 a^4-19 a^2 b^2+12 b^4\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^5 \left (a^2-b^2\right ) d}\\ &=\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 \left (a^2-b^2\right )^{3/2} d}+\frac {\left (a^2-12 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^5 d}+\frac {b \left (11 a^2-12 b^2\right ) \cot (c+d x)}{2 a^4 \left (a^2-b^2\right ) d}-\frac {\left (5 a^2-6 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^3 \left (a^2-b^2\right ) d}+\frac {\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{2 a^2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.33, size = 330, normalized size = 1.23 \[ -\frac {3 b \tan \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}+\frac {3 b \cot \left (\frac {1}{2} (c+d x)\right )}{2 a^4 d}+\frac {b^2 \cos (c+d x)}{2 a^3 d (a+b \sin (c+d x))^2}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^3 d}+\frac {\left (12 b^2-a^2\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {\left (a^2-12 b^2\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^5 d}+\frac {5 a^2 b^2 \cos (c+d x)-6 b^4 \cos (c+d x)}{2 a^4 d (a-b) (a+b) (a+b \sin (c+d x))}+\frac {b \left (6 a^4-19 a^2 b^2+12 b^4\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d \left (a^2-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.62, size = 1922, normalized size = 7.14 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 526, normalized size = 1.96 \[ \frac {\frac {8 \, {\left (6 \, a^{4} b - 19 \, a^{2} b^{3} + 12 \, b^{5}\right )} {\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^{7} - a^{5} b^{2}\right )} \sqrt {a^{2} - b^{2}}} + \frac {2 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 26 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 24 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 20 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 32 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 53 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 64 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 16 \, b^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 28 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 60 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 112 \, a b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 68 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 76 \, a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a^{6} + a^{4} b^{2}}{{\left (a^{7} - a^{5} b^{2}\right )} {\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 )}^{2}} - \frac {4 \, {\left (a^{2} - 12 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 12 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{6}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.93, size = 803, normalized size = 2.99 \[ \frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}-\frac {3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{2 d \,a^{4}}-\frac {1}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}+\frac {6 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{d \,a^{5}}+\frac {3 b}{2 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {7 b^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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} \left (a^{2}-b^{2}\right )}-\frac {8 b^{5} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{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} \left (a^{2}-b^{2}\right )}+\frac {6 b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \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} a \left (a^{2}-b^{2}\right )}+\frac {5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}}{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} \left (a^{2}-b^{2}\right )}-\frac {14 b^{6} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{5} \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} \left (a^{2}-b^{2}\right )}+\frac {17 b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{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} \left (a^{2}-b^{2}\right )}-\frac {20 b^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{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} \left (a^{2}-b^{2}\right )}+\frac {6 b^{2}}{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} \left (a^{2}-b^{2}\right )}-\frac {7 b^{4}}{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} \left (a^{2}-b^{2}\right )}+\frac {6 b \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 \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}-\frac {19 b^{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^{3} \left (a^{2}-b^{2}\right )^{\frac {3}{2}}}+\frac {12 b^{5} \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^{5} \left (a^{2}-b^{2}\right )^{\frac {3}{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 = 11.01, size = 1906, normalized size = 7.09 \[ \text {result too large to display} \]
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 \[ \int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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