Optimal. Leaf size=124 \[ \frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]
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Rubi [A] time = 0.36, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2875, 2873, 2607, 30, 2611, 3768, 3770, 14} \[ \frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 2607
Rule 2611
Rule 2873
Rule 2875
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {\cos (c+d x) \cot ^7(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \cot ^2(c+d x) \csc ^5(c+d x) (a-a \sin (c+d x))^3 \, dx}{a^6}\\ &=\frac {\int \left (-a^3 \cot ^2(c+d x) \csc ^2(c+d x)+3 a^3 \cot ^2(c+d x) \csc ^3(c+d x)-3 a^3 \cot ^2(c+d x) \csc ^4(c+d x)+a^3 \cot ^2(c+d x) \csc ^5(c+d x)\right ) \, dx}{a^6}\\ &=-\frac {\int \cot ^2(c+d x) \csc ^2(c+d x) \, dx}{a^3}+\frac {\int \cot ^2(c+d x) \csc ^5(c+d x) \, dx}{a^3}+\frac {3 \int \cot ^2(c+d x) \csc ^3(c+d x) \, dx}{a^3}-\frac {3 \int \cot ^2(c+d x) \csc ^4(c+d x) \, dx}{a^3}\\ &=-\frac {3 \cot (c+d x) \csc ^3(c+d x)}{4 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {\int \csc ^5(c+d x) \, dx}{6 a^3}-\frac {3 \int \csc ^3(c+d x) \, dx}{4 a^3}-\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,-\cot (c+d x)\right )}{a^3 d}-\frac {3 \operatorname {Subst}\left (\int x^2 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{8 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {\int \csc ^3(c+d x) \, dx}{8 a^3}-\frac {3 \int \csc (c+d x) \, dx}{8 a^3}-\frac {3 \operatorname {Subst}\left (\int \left (x^2+x^4\right ) \, dx,x,-\cot (c+d x)\right )}{a^3 d}\\ &=\frac {3 \tanh ^{-1}(\cos (c+d x))}{8 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}-\frac {\int \csc (c+d x) \, dx}{16 a^3}\\ &=\frac {7 \tanh ^{-1}(\cos (c+d x))}{16 a^3 d}+\frac {4 \cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {7 \cot (c+d x) \csc (c+d x)}{16 a^3 d}-\frac {17 \cot (c+d x) \csc ^3(c+d x)}{24 a^3 d}-\frac {\cot (c+d x) \csc ^5(c+d x)}{6 a^3 d}\\ \end {align*}
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Mathematica [A] time = 1.01, size = 242, normalized size = 1.95 \[ \frac {\left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^6 \left (704 \tan \left (\frac {1}{2} (c+d x)\right )-704 \cot \left (\frac {1}{2} (c+d x)\right )+210 \csc ^2\left (\frac {1}{2} (c+d x)\right )+5 \sec ^6\left (\frac {1}{2} (c+d x)\right )+90 \sec ^4\left (\frac {1}{2} (c+d x)\right )-210 \sec ^2\left (\frac {1}{2} (c+d x)\right )-840 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+840 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+(18 \sin (c+d x)-5) \csc ^6\left (\frac {1}{2} (c+d x)\right )+(34 \sin (c+d x)-90) \csc ^4\left (\frac {1}{2} (c+d x)\right )-544 \sin ^4\left (\frac {1}{2} (c+d x)\right ) \csc ^3(c+d x)-36 \tan \left (\frac {1}{2} (c+d x)\right ) \sec ^4\left (\frac {1}{2} (c+d x)\right )\right )}{1920 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 196, normalized size = 1.58 \[ -\frac {210 \, \cos \left (d x + c\right )^{5} - 80 \, \cos \left (d x + c\right )^{3} - 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 105 \, {\left (\cos \left (d x + c\right )^{6} - 3 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 32 \, {\left (11 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right ) - 210 \, \cos \left (d x + c\right )}{480 \, {\left (a^{3} d \cos \left (d x + c\right )^{6} - 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 216, normalized size = 1.74 \[ -\frac {\frac {840 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} - \frac {2058 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 600 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 15 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 140 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6}} - \frac {5 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 36 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 140 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 600 \, a^{15} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{18}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.72, size = 246, normalized size = 1.98 \[ \frac {\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )}{384 d \,a^{3}}-\frac {3 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{160 d \,a^{3}}+\frac {7 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 d \,a^{3}}-\frac {7 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{96 d \,a^{3}}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a^{3} d}+\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 d \,a^{3}}-\frac {1}{384 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}-\frac {5}{16 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d \,a^{3}}+\frac {3}{160 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}+\frac {1}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {7}{128 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}+\frac {7}{96 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 274, normalized size = 2.21 \[ \frac {\frac {\frac {600 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {15 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {36 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{a^{3}} - \frac {840 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} + \frac {{\left (\frac {36 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {105 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {140 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {600 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 5\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{6}}{a^{3} \sin \left (d x + c\right )^{6}}}{1920 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.18, size = 339, normalized size = 2.73 \[ -\frac {5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+36\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}-36\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+600\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-15\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+105\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+840\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{1920\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6} \]
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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