Optimal. Leaf size=89 \[ \frac {4 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d}+\frac {7 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^3(c+d x)}{a^3 d} \]
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Rubi [A] time = 0.37, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3872, 2875, 2873, 2607, 30, 2606, 270, 14} \[ \frac {4 \cot ^7(c+d x)}{7 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d}+\frac {7 \csc ^5(c+d x)}{5 a^3 d}-\frac {\csc ^3(c+d x)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 270
Rule 2606
Rule 2607
Rule 2873
Rule 2875
Rule 3872
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\int \frac {\cos (c+d x) \cot ^2(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 ^5(c+d x) \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \cot ^6(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^5(c+d x) \csc ^3(c+d x)-3 a^3 \cot ^4(c+d x) \csc ^4(c+d x)+a^3 \cot ^3(c+d x) \csc ^5(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \cot ^6(c+d x) \csc ^2(c+d x) \, dx}{a^3}+\frac {\int \cot ^3(c+d x) \csc ^5(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^4(c+d x) \csc ^4(c+d x) \, dx}{a^3}\\ &=-\frac {\operatorname {Subst}\left (\int x^6 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {\operatorname {Subst}\left (\int x^4 \left (-1+x^2\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^2 \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 ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {\cot ^7(c+d x)}{7 a^3 d}-\frac {\operatorname {Subst}\left (\int \left (-x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\csc (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int \left (x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^7(c+d x)}{7 a^3 d}-\frac {\csc ^3(c+d x)}{a^3 d}+\frac {7 \csc ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 137, normalized size = 1.54 \[ \frac {\csc (c) (602 \sin (c+d x)+602 \sin (2 (c+d x))+258 \sin (3 (c+d x))+43 \sin (4 (c+d x))-560 \sin (2 c+d x)+168 \sin (c+2 d x)-280 \sin (3 c+2 d x)-48 \sin (2 c+3 d x)-8 \sin (3 c+4 d x)-840 \sin (c)+448 \sin (d x)) \csc (c+d x) \sec ^3(c+d x)}{2240 a^3 d (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 95, normalized size = 1.07 \[ \frac {\cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )^{2} - 18 \, \cos \left (d x + c\right ) - 6}{35 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 73, normalized size = 0.82 \[ -\frac {\frac {35}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {5 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 14 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 70 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{21}}}{560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.73, size = 60, normalized size = 0.67 \[ \frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {2 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{16 d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 90, normalized size = 1.01 \[ -\frac {\frac {\frac {70 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} + \frac {35 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} \sin \left (d x + c\right )}}{560 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 84, normalized size = 0.94 \[ -\frac {-16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+72\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-34\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5}{560\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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 \[ \frac {\int \frac {\csc ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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