Optimal. Leaf size=65 \[ -\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.12, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2834, 2606, 270, 2607, 30} \[ -\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 270
Rule 2606
Rule 2607
Rule 2834
Rubi steps
\begin {align*} \int \cot ^5(c+d x) \csc ^3(c+d x) (a+a \sin (c+d x)) \, dx &=a \int \cot ^5(c+d x) \csc ^2(c+d x) \, dx+a \int \cot ^5(c+d x) \csc ^3(c+d x) \, dx\\ &=-\frac {a \operatorname {Subst}\left (\int x^5 \, dx,x,-\cot (c+d x)\right )}{d}-\frac {a \operatorname {Subst}\left (\int x^2 \left (-1+x^2\right )^2 \, dx,x,\csc (c+d x)\right )}{d}\\ &=-\frac {a \cot ^6(c+d x)}{6 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 \cot ^6(c+d x)}{6 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 = 0.03, size = 65, normalized size = 1.00 \[ -\frac {a \cot ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}+\frac {2 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 106, normalized size = 1.63 \[ \frac {70 \, a \cos \left (d x + c\right )^{4} - 56 \, a \cos \left (d x + c\right )^{2} + 35 \, {\left (3 \, a \cos \left (d x + c\right )^{4} - 3 \, a \cos \left (d x + c\right )^{2} + a\right )} \sin \left (d x + c\right ) + 16 \, a}{210 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 70, normalized size = 1.08 \[ -\frac {105 \, a \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, a \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, a \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 128, normalized size = 1.97 \[ \frac {-\frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{6 \sin \left (d x +c \right )^{6}}+a \left (-\frac {\cos ^{6}\left (d x +c \right )}{7 \sin \left (d x +c \right )^{7}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )^{5}}+\frac {\cos ^{6}\left (d x +c \right )}{105 \sin \left (d x +c \right )^{3}}-\frac {\cos ^{6}\left (d x +c \right )}{35 \sin \left (d x +c \right )}-\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 70, normalized size = 1.08 \[ -\frac {105 \, a \sin \left (d x + c\right )^{5} + 70 \, a \sin \left (d x + c\right )^{4} - 105 \, a \sin \left (d x + c\right )^{3} - 84 \, a \sin \left (d x + c\right )^{2} + 35 \, a \sin \left (d x + c\right ) + 30 \, a}{210 \, d \sin \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.83, size = 70, normalized size = 1.08 \[ -\frac {105\,a\,{\sin \left (c+d\,x\right )}^5+70\,a\,{\sin \left (c+d\,x\right )}^4-105\,a\,{\sin \left (c+d\,x\right )}^3-84\,a\,{\sin \left (c+d\,x\right )}^2+35\,a\,\sin \left (c+d\,x\right )+30\,a}{210\,d\,{\sin \left (c+d\,x\right )}^7} \]
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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