Optimal. Leaf size=81 \[ -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d} \]
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Rubi [A] time = 0.12, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2606, 270, 2607, 14} \[ -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^9(c+d x)}{9 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 270
Rule 2606
Rule 2607
Rule 2834
Rubi steps
\begin {align*} \int \cot ^5(c+d x) \csc ^5(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^4(c+d x) \, dx+a \int \cot ^5(c+d x) \csc ^5(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^4 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int x^5 \left (1+x^2\right ) \, dx,x,-\cot (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \left (x^5+x^7\right ) \, dx,x,-\cot (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^5(c+d x)}{5 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^9(c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 88, normalized size = 1.09 \[ -\frac {a \csc ^9(c+d x)}{9 d}+\frac {2 a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^5(c+d x)}{5 d}-\frac {a \left (3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 115, normalized size = 1.42 \[ -\frac {504 \, a \cos \left (d x + c\right )^{4} - 288 \, a \cos \left (d x + c\right )^{2} + 105 \, {\left (6 \, a \cos \left (d x + c\right )^{4} - 4 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 64 \, a}{2520 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 70, normalized size = 0.86 \[ -\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.38, size = 166, normalized size = 2.05 \[ \frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{6}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{6}}\right )+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{9 \sin \left (d x +c \right )^{9}}-\frac {\cos ^{6}\left (d x +c \right )}{21 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{315 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{105}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 70, normalized size = 0.86 \[ -\frac {630 \, a \sin \left (d x + c\right )^{5} + 504 \, a \sin \left (d x + c\right )^{4} - 840 \, a \sin \left (d x + c\right )^{3} - 720 \, a \sin \left (d x + c\right )^{2} + 315 \, a \sin \left (d x + c\right ) + 280 \, a}{2520 \, d \sin \left (d x + c\right )^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.88, size = 70, normalized size = 0.86 \[ -\frac {\frac {a\,{\sin \left (c+d\,x\right )}^5}{4}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{5}-\frac {a\,{\sin \left (c+d\,x\right )}^3}{3}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^2}{7}+\frac {a\,\sin \left (c+d\,x\right )}{8}+\frac {a}{9}}{d\,{\sin \left (c+d\,x\right )}^9} \]
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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