Optimal. Leaf size=244 \[ \frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}+\frac {2 b^2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]
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Rubi [A] time = 1.05, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2893, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {2 b^2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^6 d}-\frac {\left (-20 a^2 b^2+3 a^4+15 b^4\right ) \cot (c+d x)}{15 a^5 d}+\frac {b \left (-12 a^2 b^2+3 a^4+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 2660
Rule 2893
Rule 3001
Rule 3055
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x) \csc ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc ^4(c+d x) \left (4 \left (6 a^2-5 b^2\right )-a b \sin (c+d x)-5 \left (4 a^2-3 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{20 a^2}\\ &=\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc ^3(c+d x) \left (-15 b \left (5 a^2-4 b^2\right )-a \left (12 a^2-5 b^2\right ) \sin (c+d x)+8 b \left (6 a^2-5 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{60 a^3}\\ &=-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc ^2(c+d x) \left (-8 \left (3 a^4-20 a^2 b^2+15 b^4\right )+a b \left (21 a^2-20 b^2\right ) \sin (c+d x)-15 b^2 \left (5 a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^4}\\ &=-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\int \frac {\csc (c+d x) \left (15 b \left (3 a^4-12 a^2 b^2+8 b^4\right )-15 a b^2 \left (5 a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{120 a^5}\\ &=-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\left (b^2 \left (a^2-b^2\right )^2\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^6}-\frac {\left (b \left (3 a^4-12 a^2 b^2+8 b^4\right )\right ) \int \csc (c+d x) \, dx}{8 a^6}\\ &=\frac {b \left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}+\frac {\left (2 b^2 \left (a^2-b^2\right )^2\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 (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}-\frac {\left (4 b^2 \left (a^2-b^2\right )^2\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 {2 b^2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^6 d}+\frac {b \left (3 a^4-12 a^2 b^2+8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^6 d}-\frac {\left (3 a^4-20 a^2 b^2+15 b^4\right ) \cot (c+d x)}{15 a^5 d}-\frac {b \left (5 a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^4 d}+\frac {\left (6 a^2-5 b^2\right ) \cot (c+d x) \csc ^2(c+d x)}{15 a^3 d}+\frac {b \cot (c+d x) \csc ^3(c+d x)}{4 a^2 d}-\frac {\cot (c+d x) \csc ^4(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [B] time = 1.82, size = 507, normalized size = 2.08 \[ \frac {96 a^5 \tan \left (\frac {1}{2} (c+d x)\right )-3 a^5 \sin (c+d x) \csc ^6\left (\frac {1}{2} (c+d x)\right )+21 a^5 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )-336 a^5 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+6 a^5 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )+15 a^4 b \csc ^4\left (\frac {1}{2} (c+d x)\right )-150 a^4 b \csc ^2\left (\frac {1}{2} (c+d x)\right )-15 a^4 b \sec ^4\left (\frac {1}{2} (c+d x)\right )+150 a^4 b \sec ^2\left (\frac {1}{2} (c+d x)\right )-360 a^4 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+360 a^4 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-640 a^3 b^2 \tan \left (\frac {1}{2} (c+d x)\right )-20 a^3 b^2 \sin (c+d x) \csc ^4\left (\frac {1}{2} (c+d x)\right )+320 a^3 b^2 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)+120 a^2 b^3 \csc ^2\left (\frac {1}{2} (c+d x)\right )-120 a^2 b^3 \sec ^2\left (\frac {1}{2} (c+d x)\right )+1440 a^2 b^3 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-1440 a^2 b^3 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+1920 b^2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )-32 \left (3 a^5-20 a^3 b^2+15 a b^4\right ) \cot \left (\frac {1}{2} (c+d x)\right )+480 a b^4 \tan \left (\frac {1}{2} (c+d x)\right )-960 b^5 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+960 b^5 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{960 a^6 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.83, size = 1051, normalized size = 4.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 484, normalized size = 1.98 \[ \frac {\frac {6 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 15 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 30 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 40 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 120 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 60 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 480 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{5}} - \frac {120 \, {\left (3 \, a^{4} b - 12 \, a^{2} b^{3} + 8 \, b^{5}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{6}} + \frac {1920 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\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 {822 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3288 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 2192 \, b^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 600 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 480 \, a b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 120 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a^{2} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 40 \, a^{3} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 15 \, a^{4} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 6 \, a^{5}}{a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}}}{960 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.49, size = 583, normalized size = 2.39 \[ \frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{160 d a}-\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{64 d \,a^{2}}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{32 d a}+\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{24 d \,a^{3}}+\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b}{8 d \,a^{2}}-\frac {\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{8 d \,a^{4}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a d}-\frac {5 b^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}+\frac {b^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{5}}-\frac {1}{160 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{32 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b^{2}}{24 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {1}{16 a d \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {5 b^{2}}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b^{4}}{2 d \,a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{64 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {b}{8 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {b^{3}}{8 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {3 b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{2}}+\frac {3 b^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{4}}-\frac {b^{5} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{6}}+\frac {2 b^{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 {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^{4}}{d \,a^{4} \sqrt {a^{2}-b^{2}}}+\frac {2 b^{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 )}{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 = 12.18, size = 1082, normalized size = 4.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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