Optimal. Leaf size=73 \[ \frac {\cot ^3(c+d x)}{3 a d}+\frac {2 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^7(c+d x)}{7 a d}-\frac {\csc ^7(c+d x)}{7 a d} \]
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Rubi [A]
time = 0.11, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3957, 2918,
2686, 30, 2687, 276} \begin {gather*} \frac {\cot ^7(c+d x)}{7 a d}+\frac {2 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^3(c+d x)}{3 a d}-\frac {\csc ^7(c+d x)}{7 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 276
Rule 2686
Rule 2687
Rule 2918
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc ^6(c+d x)}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cot (c+d x) \csc ^5(c+d x)}{-a-a \cos (c+d x)} \, dx\\ &=-\frac {\int \cot ^2(c+d x) \csc ^6(c+d x) \, dx}{a}+\frac {\int \cot (c+d x) \csc ^7(c+d x) \, dx}{a}\\ &=-\frac {\text {Subst}\left (\int x^6 \, dx,x,\csc (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int x^2 \left (1+x^2\right )^2 \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=-\frac {\csc ^7(c+d x)}{7 a d}-\frac {\text {Subst}\left (\int \left (x^2+2 x^4+x^6\right ) \, dx,x,-\cot (c+d x)\right )}{a d}\\ &=\frac {\cot ^3(c+d x)}{3 a d}+\frac {2 \cot ^5(c+d x)}{5 a d}+\frac {\cot ^7(c+d x)}{7 a d}-\frac {\csc ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(158\) vs. \(2(73)=146\).
time = 0.42, size = 158, normalized size = 2.16 \begin {gather*} \frac {\csc (c) \csc ^5(c+d x) \sec (c+d x) (-8960 \sin (c)+2560 \sin (d x)+1500 \sin (c+d x)+375 \sin (2 (c+d x))-750 \sin (3 (c+d x))-300 \sin (4 (c+d x))+150 \sin (5 (c+d x))+75 \sin (6 (c+d x))+640 \sin (c+2 d x)-1280 \sin (2 c+3 d x)-512 \sin (3 c+4 d x)+256 \sin (4 c+5 d x)+128 \sin (5 c+6 d x))}{53760 a d (1+\sec (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 88, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d a}\) | \(88\) |
default | \(\frac {-\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}-\frac {4 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {5}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {1}{5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {4}{3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}}{64 d a}\) | \(88\) |
risch | \(-\frac {16 i \left (70 \,{\mathrm e}^{6 i \left (d x +c \right )}+20 \,{\mathrm e}^{5 i \left (d x +c \right )}+5 \,{\mathrm e}^{4 i \left (d x +c \right )}-10 \,{\mathrm e}^{3 i \left (d x +c \right )}-4 \,{\mathrm e}^{2 i \left (d x +c \right )}+2 \,{\mathrm e}^{i \left (d x +c \right )}+1\right )}{105 a d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{5}}\) | \(104\) |
norman | \(\frac {-\frac {1}{320 a d}-\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{48 a d}-\frac {5 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 a d}-\frac {5 \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{80 a d}-\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{448 a d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}\) | \(117\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (65) = 130\).
time = 0.28, size = 136, normalized size = 1.86 \begin {gather*} -\frac {\frac {\frac {175 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {84 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a} + \frac {7 \, {\left (\frac {20 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 3\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{5}}{a \sin \left (d x + c\right )^{5}}}{6720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 131 vs.
\(2 (65) = 130\).
time = 3.67, size = 131, normalized size = 1.79 \begin {gather*} -\frac {8 \, \cos \left (d x + c\right )^{6} + 8 \, \cos \left (d x + c\right )^{5} - 20 \, \cos \left (d x + c\right )^{4} - 20 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )^{2} + 15 \, \cos \left (d x + c\right ) + 15}{105 \, {\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4} - 2 \, a d \cos \left (d x + c\right )^{3} - 2 \, a d \cos \left (d x + c\right )^{2} + a d \cos \left (d x + c\right ) + a d\right )} \sin \left (d x + c\right )} \end {gather*}
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 \begin {gather*} \frac {\int \frac {\csc ^{6}{\left (c + d x \right )}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 103, normalized size = 1.41 \begin {gather*} -\frac {\frac {7 \, {\left (75 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 20 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3\right )}}{a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}} + \frac {15 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 175 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}{a^{7}}}{6720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.16, size = 153, normalized size = 2.10 \begin {gather*} -\frac {21\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+140\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+525\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+175\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{6720\,a\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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