Optimal. Leaf size=73 \[ -\frac {\csc ^4(c+d x)}{4 a d}+\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d} \]
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Rubi [A]
time = 0.08, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2915, 12, 76}
\begin {gather*} -\frac {\csc ^7(c+d x)}{7 a d}+\frac {\csc ^6(c+d x)}{6 a d}+\frac {\csc ^5(c+d x)}{5 a d}-\frac {\csc ^4(c+d x)}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 76
Rule 2915
Rubi steps
\begin {align*} \int \frac {\cot ^5(c+d x) \csc ^3(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {a^8 (a-x)^2 (a+x)}{x^8} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a^3 \text {Subst}\left (\int \frac {(a-x)^2 (a+x)}{x^8} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \text {Subst}\left (\int \left (\frac {a^3}{x^8}-\frac {a^2}{x^7}-\frac {a}{x^6}+\frac {1}{x^5}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^4(c+d x)}{4 a d}+\frac {\csc ^5(c+d x)}{5 a d}+\frac {\csc ^6(c+d x)}{6 a d}-\frac {\csc ^7(c+d x)}{7 a d}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 48, normalized size = 0.66 \begin {gather*} \frac {\csc ^4(c+d x) \left (-105+84 \csc (c+d x)+70 \csc ^2(c+d x)-60 \csc ^3(c+d x)\right )}{420 a d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 49, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {-\frac {1}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}}{d a}\) | \(49\) |
default | \(\frac {-\frac {1}{7 \sin \left (d x +c \right )^{7}}+\frac {1}{6 \sin \left (d x +c \right )^{6}}+\frac {1}{5 \sin \left (d x +c \right )^{5}}-\frac {1}{4 \sin \left (d x +c \right )^{4}}}{d a}\) | \(49\) |
risch | \(-\frac {4 \left (-168 i {\mathrm e}^{9 i \left (d x +c \right )}+105 \,{\mathrm e}^{10 i \left (d x +c \right )}-144 i {\mathrm e}^{7 i \left (d x +c \right )}-35 \,{\mathrm e}^{8 i \left (d x +c \right )}-168 i {\mathrm e}^{5 i \left (d x +c \right )}+35 \,{\mathrm e}^{6 i \left (d x +c \right )}-105 \,{\mathrm e}^{4 i \left (d x +c \right )}\right )}{105 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{7}}\) | \(103\) |
norman | \(\frac {-\frac {1}{896 a d}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{672 a d}+\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{960 a d}-\frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{640 a d}+\frac {\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}-\frac {\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}-\frac {\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )}{64 a d}+\frac {\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )}{128 a d}-\frac {\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )}{640 a d}+\frac {\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )}{960 d a}+\frac {\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )}{672 d a}-\frac {\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )}{896 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}\) | \(242\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 46, normalized size = 0.63 \begin {gather*} -\frac {105 \, \sin \left (d x + c\right )^{3} - 84 \, \sin \left (d x + c\right )^{2} - 70 \, \sin \left (d x + c\right ) + 60}{420 \, a d \sin \left (d x + c\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 84, normalized size = 1.15 \begin {gather*} \frac {84 \, \cos \left (d x + c\right )^{2} - 35 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - 24}{420 \, {\left (a d \cos \left (d x + c\right )^{6} - 3 \, a d \cos \left (d x + c\right )^{4} + 3 \, a d \cos \left (d x + c\right )^{2} - 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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 46, normalized size = 0.63 \begin {gather*} -\frac {105 \, \sin \left (d x + c\right )^{3} - 84 \, \sin \left (d x + c\right )^{2} - 70 \, \sin \left (d x + c\right ) + 60}{420 \, a d \sin \left (d x + c\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.96, size = 46, normalized size = 0.63 \begin {gather*} \frac {-105\,{\sin \left (c+d\,x\right )}^3+84\,{\sin \left (c+d\,x\right )}^2+70\,\sin \left (c+d\,x\right )-60}{420\,a\,d\,{\sin \left (c+d\,x\right )}^7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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