Optimal. Leaf size=194 \[ \frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac {2 b^3 \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d}-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
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Rubi [A] time = 0.95, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2889, 3056, 3055, 3001, 3770, 2660, 618, 204} \[ \frac {2 b^3 \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {a \tan \left (\frac {1}{2} (c+d x)\right )+b}{\sqrt {a^2-b^2}}\right )}{a^5 d}-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (4 a^2 b^2+a^4-8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
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 ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\int \frac {\csc ^5(c+d x) \left (1-\sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx\\ &=-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\int \frac {\csc ^4(c+d x) \left (-4 b-a \sin (c+d x)+3 b \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{4 a}\\ &=\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\int \frac {\csc ^3(c+d x) \left (-3 \left (a^2-4 b^2\right )+a b \sin (c+d x)-8 b^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{12 a^2}\\ &=\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\int \frac {\csc ^2(c+d x) \left (8 b \left (a^2-3 b^2\right )-a \left (3 a^2+4 b^2\right ) \sin (c+d x)-3 b \left (a^2-4 b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^3}\\ &=-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\int \frac {\csc (c+d x) \left (-3 \left (a^4+4 a^2 b^2-8 b^4\right )-3 a b \left (a^2-4 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{24 a^4}\\ &=-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\left (b^3 \left (a^2-b^2\right )\right ) \int \frac {1}{a+b \sin (c+d x)} \, dx}{a^5}-\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \int \csc (c+d x) \, dx}{8 a^5}\\ &=\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\left (2 b^3 \left (a^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^5 d}\\ &=\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\left (4 b^3 \left (a^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^5 d}\\ &=\frac {2 b^3 \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}+\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \tanh ^{-1}(\cos (c+d x))}{8 a^5 d}-\frac {b \left (a^2-3 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac {\left (a^2-4 b^2\right ) \cot (c+d x) \csc (c+d x)}{8 a^3 d}+\frac {b \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}\\ \end {align*}
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Mathematica [B] time = 6.27, size = 430, normalized size = 2.22 \[ \frac {b \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}-\frac {b \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{24 a^2 d}+\frac {2 b^3 \sqrt {a^2-b^2} \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a \sin \left (\frac {1}{2} (c+d x)\right )+b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{a^5 d}+\frac {\csc \left (\frac {1}{2} (c+d x)\right ) \left (3 b^3 \cos \left (\frac {1}{2} (c+d x)\right )-a^2 b \cos \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (a^2 b \sin \left (\frac {1}{2} (c+d x)\right )-3 b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac {\left (a^2-4 b^2\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\left (4 b^2-a^2\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 a^3 d}+\frac {\left (-a^4-4 a^2 b^2+8 b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}+\frac {\left (a^4+4 a^2 b^2-8 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 a^5 d}-\frac {\csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d}+\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 a d} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.02, size = 808, normalized size = 4.16 \[ \left [-\frac {6 \, {\left (a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} - 24 \, {\left (b^{3} \cos \left (d x + c\right )^{4} - 2 \, b^{3} \cos \left (d x + c\right )^{2} + b^{3}\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 ) + 6 \, {\left (a^{4} + 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4} - 2 \, {\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4} - 2 \, {\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (3 \, a b^{3} \cos \left (d x + c\right ) + {\left (a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (a^{5} d \cos \left (d x + c\right )^{4} - 2 \, a^{5} d \cos \left (d x + c\right )^{2} + a^{5} d\right )}}, -\frac {6 \, {\left (a^{4} - 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 48 \, {\left (b^{3} \cos \left (d x + c\right )^{4} - 2 \, b^{3} \cos \left (d x + c\right )^{2} + b^{3}\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 ) + 6 \, {\left (a^{4} + 4 \, a^{2} b^{2}\right )} \cos \left (d x + c\right ) - 3 \, {\left ({\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4} - 2 \, {\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4} - 2 \, {\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 16 \, {\left (3 \, a b^{3} \cos \left (d x + c\right ) + {\left (a^{3} b - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{48 \, {\left (a^{5} d \cos \left (d x + c\right )^{4} - 2 \, a^{5} d \cos \left (d x + c\right )^{2} + a^{5} d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 336, normalized size = 1.73 \[ \frac {\frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 8 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 24 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 96 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{4}} - \frac {24 \, {\left (a^{4} + 4 \, a^{2} b^{2} - 8 \, b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{5}} + \frac {384 \, {\left (a^{2} b^{3} - b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {50 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 200 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 400 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 24 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 96 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 24 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 8 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{4}}{a^{5} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{192 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 315, normalized size = 1.62 \[ \frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d a}-\frac {b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{2}}+\frac {b^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{8 d \,a^{2}}-\frac {b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \,a^{4}}-\frac {1}{64 d a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a d}-\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{2}}{2 d \,a^{3}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{4}}{d \,a^{5}}+\frac {b}{24 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}-\frac {b^{2}}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {b}{8 d \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b^{3}}{2 d \,a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {2 \sqrt {a^{2}-b^{2}}\, b^{3} \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{d \,a^{5}} \]
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 = 12.07, size = 873, normalized size = 4.50 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {b}{8\,a^2}-\frac {b^3}{2\,a^4}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^2\,b-8\,b^3\right )+\frac {a^3}{4}-\frac {2\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+2\,a\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{16\,a^4\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a^4+4\,a^2\,b^2-8\,b^4\right )}{8\,a^5\,d}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^2\,d}+\frac {b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}+\frac {b^3\,\mathrm {atan}\left (\frac {\frac {b^3\,\sqrt {b^2-a^2}\,\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^9+2\,a^7\,b^2-32\,a^5\,b^4+32\,a^3\,b^6\right )}{4\,a^7}-\frac {a^9\,b+12\,a^7\,b^3-16\,a^5\,b^5}{4\,a^8}+\frac {b^3\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {b^2-a^2}}{a^5}\right )\,1{}\mathrm {i}}{a^5}-\frac {b^3\,\sqrt {b^2-a^2}\,\left (\frac {a^9\,b+12\,a^7\,b^3-16\,a^5\,b^5}{4\,a^8}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^9+2\,a^7\,b^2-32\,a^5\,b^4+32\,a^3\,b^6\right )}{4\,a^7}+\frac {b^3\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {b^2-a^2}}{a^5}\right )\,1{}\mathrm {i}}{a^5}}{\frac {a^6\,b^3+3\,a^4\,b^5-12\,a^2\,b^7+8\,b^9}{2\,a^8}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^4\,b^4-10\,a^2\,b^6+8\,b^8\right )}{2\,a^7}+\frac {b^3\,\sqrt {b^2-a^2}\,\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^9+2\,a^7\,b^2-32\,a^5\,b^4+32\,a^3\,b^6\right )}{4\,a^7}-\frac {a^9\,b+12\,a^7\,b^3-16\,a^5\,b^5}{4\,a^8}+\frac {b^3\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {b^2-a^2}}{a^5}\right )}{a^5}+\frac {b^3\,\sqrt {b^2-a^2}\,\left (\frac {a^9\,b+12\,a^7\,b^3-16\,a^5\,b^5}{4\,a^8}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (a^9+2\,a^7\,b^2-32\,a^5\,b^4+32\,a^3\,b^6\right )}{4\,a^7}+\frac {b^3\,\left (2\,a^2\,b-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (24\,a^{10}-32\,a^8\,b^2\right )}{4\,a^7}\right )\,\sqrt {b^2-a^2}}{a^5}\right )}{a^5}}\right )\,\sqrt {b^2-a^2}\,2{}\mathrm {i}}{a^5\,d} \]
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 \[ \int \frac {\cos ^{2}{\left (c + d x \right )} \csc ^{5}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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