Optimal. Leaf size=103 \[ \frac {4 \cot ^9(c+d x)}{9 a^3 d}+\frac {\cot ^7(c+d x)}{a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}+\frac {\csc ^7(c+d x)}{a^3 d}-\frac {3 \csc ^5(c+d x)}{5 a^3 d} \]
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Rubi [A] time = 0.38, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3872, 2875, 2873, 2607, 14, 2606, 270} \[ \frac {4 \cot ^9(c+d x)}{9 a^3 d}+\frac {\cot ^7(c+d x)}{a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}+\frac {\csc ^7(c+d x)}{a^3 d}-\frac {3 \csc ^5(c+d x)}{5 a^3 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^4(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cot ^3(c+d x) \csc (c+d x)}{(-a-a \cos (c+d x))^3} \, dx\\ &=-\frac {\int (-a+a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^7(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \cot ^6(c+d x) \csc ^4(c+d x)+3 a^3 \cot ^5(c+d x) \csc ^5(c+d x)-3 a^3 \cot ^4(c+d x) \csc ^6(c+d x)+a^3 \cot ^3(c+d x) \csc ^7(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \cot ^6(c+d x) \csc ^4(c+d x) \, dx}{a^3}+\frac {\int \cot ^3(c+d x) \csc ^7(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^5(c+d x) \csc ^5(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^4(c+d x) \csc ^6(c+d x) \, dx}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int x^6 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\operatorname {Subst}\left (\int x^6 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^4 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (-x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {\operatorname {Subst}\left (\int \left (x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (x^4+2 x^6+x^8\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {\cot ^7(c+d x)}{a^3 d}+\frac {4 \cot ^9(c+d x)}{9 a^3 d}-\frac {3 \csc ^5(c+d x)}{5 a^3 d}+\frac {\csc ^7(c+d x)}{a^3 d}-\frac {4 \csc ^9(c+d x)}{9 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.87, size = 175, normalized size = 1.70 \[ -\frac {\csc (c) (-1764 \sin (c+d x)-1323 \sin (2 (c+d x))+98 \sin (3 (c+d x))+588 \sin (4 (c+d x))+294 \sin (5 (c+d x))+49 \sin (6 (c+d x))+3456 \sin (2 c+d x)-1152 \sin (c+2 d x)+2880 \sin (3 c+2 d x)-128 \sin (2 c+3 d x)-768 \sin (3 c+4 d x)-384 \sin (4 c+5 d x)-64 \sin (5 c+6 d x)+5376 \sin (c)-1152 \sin (d x)) \csc ^3(2 (c+d x))}{5760 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 146, normalized size = 1.42 \[ \frac {2 \, \cos \left (d x + c\right )^{6} + 6 \, \cos \left (d x + c\right )^{5} + 3 \, \cos \left (d x + c\right )^{4} - 7 \, \cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 6 \, \cos \left (d x + c\right ) + 2}{45 \, {\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 2 \, a^{3} d \cos \left (d x + c\right )^{3} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} - 3 \, a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.95, size = 73, normalized size = 0.71 \[ -\frac {\frac {15}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} + \frac {5 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 27 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 135 \, a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{27}}}{2880 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.81, size = 60, normalized size = 0.58 \[ \frac {-\frac {\left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{9}+\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 92, normalized size = 0.89 \[ -\frac {\frac {\frac {135 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {27 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{a^{3}} + \frac {15 \, {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{a^{3} \sin \left (d x + c\right )^{3}}}{2880 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 105, normalized size = 1.02 \[ -\frac {15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+135\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-27\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{2880\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]
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 \[ \frac {\int \frac {\csc ^{4}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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