Optimal. Leaf size=81 \[ -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d} \]
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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, 2607, 14, 2606, 270} \[ -\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d} \]
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 ^4(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx+a \int \cot ^5(c+d x) \csc ^4(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^2 \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^2-2 x^4+x^6\right ) \, dx,x,\csc (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 \cot ^6(c+d x)}{6 d}-\frac {a \cot ^8(c+d x)}{8 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 88, normalized size = 1.09 \[ -\frac {a \csc ^7(c+d x)}{7 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d}-\frac {a \left (3 \csc ^8(c+d x)-8 \csc ^6(c+d x)+6 \csc ^4(c+d x)\right )}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 109, normalized size = 1.35 \[ -\frac {210 \, a \cos \left (d x + c\right )^{4} - 140 \, a \cos \left (d x + c\right )^{2} + 8 \, {\left (35 \, a \cos \left (d x + c\right )^{4} - 28 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right ) + 35 \, a}{840 \, {\left (d \cos \left (d x + c\right )^{8} - 4 \, d \cos \left (d x + c\right )^{6} + 6 \, d \cos \left (d x + c\right )^{4} - 4 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
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 \[ -\frac {280 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, a \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, a \sin \left (d x + c\right ) + 105 \, a}{840 \, d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 148, normalized size = 1.83 \[ \frac {a \left (-\frac {\cos ^{6}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{8 \sin \left (d x +c \right )^{8}}-\frac {\cos ^{6}\left (d x +c \right )}{24 \sin \left (d x +c \right )^{6}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 70, normalized size = 0.86 \[ -\frac {280 \, a \sin \left (d x + c\right )^{5} + 210 \, a \sin \left (d x + c\right )^{4} - 336 \, a \sin \left (d x + c\right )^{3} - 280 \, a \sin \left (d x + c\right )^{2} + 120 \, a \sin \left (d x + c\right ) + 105 \, a}{840 \, d \sin \left (d x + c\right )^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.84, size = 70, normalized size = 0.86 \[ -\frac {280\,a\,{\sin \left (c+d\,x\right )}^5+210\,a\,{\sin \left (c+d\,x\right )}^4-336\,a\,{\sin \left (c+d\,x\right )}^3-280\,a\,{\sin \left (c+d\,x\right )}^2+120\,a\,\sin \left (c+d\,x\right )+105\,a}{840\,d\,{\sin \left (c+d\,x\right )}^8} \]
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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