Optimal. Leaf size=289 \[ \frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]
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Rubi [A] time = 1.07, antiderivative size = 289, 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 = {2724, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {\left (-19 a^2 b^2+2 a^4+20 b^4\right ) \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2724
Rule 3001
Rule 3055
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\int \frac {\csc ^3(c+d x) \left (2 \left (3 a^2-10 b^2\right )-2 a b \sin (c+d x)-3 \left (a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx}{6 a^2 b}\\ &=\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^3(c+d x) \left (12 \left (a^4-6 a^2 b^2+5 b^4\right )-5 a b \left (a^2-b^2\right ) \sin (c+d x)-2 \left (3 a^4-23 a^2 b^2+20 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^3 b \left (a^2-b^2\right )}\\ &=-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc ^2(c+d x) \left (-2 b \left (17 a^4-77 a^2 b^2+60 b^4\right )+20 a b^2 \left (a^2-b^2\right ) \sin (c+d x)+12 b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^4 b \left (a^2-b^2\right )}\\ &=\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\int \frac {\csc (c+d x) \left (6 b^2 \left (9 a^4-29 a^2 b^2+20 b^4\right )+12 a b \left (a^4-6 a^2 b^2+5 b^4\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^5 b \left (a^2-b^2\right )}\\ &=\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (b \left (9 a^4-29 a^2 b^2+20 b^4\right )\right ) \int \csc (c+d x) \, dx}{2 a^6 \left (a^2-b^2\right )}+\frac {\left (2 a^6-21 a^4 b^2+39 a^2 b^4-20 b^6\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{2 a^6 \left (a^2-b^2\right )}\\ &=-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}+\frac {\left (2 a^4-19 a^2 b^2+20 b^4\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^6 d}\\ &=-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}-\frac {\left (2 \left (2 a^4-19 a^2 b^2+20 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^6 d}\\ &=\frac {\left (2 a^4-19 a^2 b^2+20 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^6 \sqrt {a^2-b^2} d}-\frac {b \left (9 a^2-20 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^6 d}+\frac {\left (17 a^2-60 b^2\right ) \cot (c+d x)}{6 a^5 d}-\frac {\left (a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{a^4 b d}+\frac {\left (3 a^2-5 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^2 b d (a+b \sin (c+d x))^2}-\frac {\cot (c+d x) \csc ^2(c+d x)}{3 a d (a+b \sin (c+d x))^2}+\frac {\left (3 a^2-20 b^2\right ) \cot (c+d x) \csc (c+d x)}{6 a^3 b d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 6.21, size = 459, normalized size = 1.59 \[ \frac {3 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {3 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 a^4 d}-\frac {\cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}+\frac {\tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^3 d}+\frac {\left (9 a^2 b-20 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {\left (20 b^3-9 a^2 b\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 a^6 d}+\frac {3 a^2 b \cos (c+d x)-8 b^3 \cos (c+d x)}{2 a^5 d (a+b \sin (c+d x))}+\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \left (2 a^2 \cos \left (\frac {1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (9 b^2 \sin \left (\frac {1}{2} (c+d x)\right )-2 a^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^5 d}+\frac {a^2 b \cos (c+d x)-b^3 \cos (c+d x)}{2 a^4 d (a+b \sin (c+d x))^2}+\frac {\left (2 a^4-19 a^2 b^2+20 b^4\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d \sqrt {a^2-b^2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.05, size = 2027, normalized size = 7.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.92, size = 451, normalized size = 1.56 \[ \frac {\frac {12 \, {\left (9 \, a^{2} b - 20 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac {24 \, {\left (2 \, a^{4} - 19 \, a^{2} b^{2} + 20 \, b^{4}\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 )}}{\sqrt {a^{2} - b^{2}} a^{6}} + \frac {24 \, {\left (5 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 10 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 18 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 11 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 26 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, a^{4} b - 9 \, a^{2} b^{3}\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^{6}} + \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}} - \frac {198 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 440 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 72 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 9 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{3}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.33, size = 780, normalized size = 2.70 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{3}}-\frac {3 b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{4}}-\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}+\frac {3 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{5}}-\frac {1}{24 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {5}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {3 b^{2}}{d \,a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3 b}{8 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {9 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{4}}-\frac {10 b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{6}}+\frac {5 \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 {10 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}}{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}}+\frac {4 \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 {\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 {18 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{5}}{d \,a^{6} \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 {11 \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 {26 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{4}}{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}}+\frac {4 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 {9 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 {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^{2} \sqrt {a^{2}-b^{2}}}-\frac {19 \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}}}+\frac {20 \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^{4}}{d \,a^{6} \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 = 7.31, size = 1261, normalized size = 4.36 \[ \text {result too large to display} \]
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 \[ \int \frac {\cot ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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