Optimal. Leaf size=119 \[ -\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc (c+d x)}{d}-\frac {a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2836, 12, 88} \[ -\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \csc ^6(c+d x)}{6 d}+\frac {3 a \csc ^5(c+d x)}{5 d}+\frac {3 a \csc ^4(c+d x)}{4 d}-\frac {a \csc ^3(c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}+\frac {a \csc (c+d x)}{d}-\frac {a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cot ^7(c+d x) \csc (c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^8 (a-x)^3 (a+x)^4}{x^8} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {a \operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^8} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \operatorname {Subst}\left (\int \left (\frac {a^7}{x^8}+\frac {a^6}{x^7}-\frac {3 a^5}{x^6}-\frac {3 a^4}{x^5}+\frac {3 a^3}{x^4}+\frac {3 a^2}{x^3}-\frac {a}{x^2}-\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a \csc (c+d x)}{d}-\frac {3 a \csc ^2(c+d x)}{2 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {3 a \csc ^4(c+d x)}{4 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^6(c+d x)}{6 d}-\frac {a \csc ^7(c+d x)}{7 d}-\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 115, normalized size = 0.97 \[ -\frac {a \csc ^7(c+d x)}{7 d}+\frac {3 a \csc ^5(c+d x)}{5 d}-\frac {a \csc ^3(c+d x)}{d}+\frac {a \csc (c+d x)}{d}-\frac {a \left (2 \cot ^6(c+d x)-3 \cot ^4(c+d x)+6 \cot ^2(c+d x)+12 \log (\tan (c+d x))+12 \log (\cos (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 172, normalized size = 1.45 \[ \frac {420 \, a \cos \left (d x + c\right )^{6} - 840 \, a \cos \left (d x + c\right )^{4} + 672 \, a \cos \left (d x + c\right )^{2} - 420 \, {\left (a \cos \left (d x + c\right )^{6} - 3 \, a \cos \left (d x + c\right )^{4} + 3 \, a \cos \left (d x + c\right )^{2} - a\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 35 \, {\left (18 \, a \cos \left (d x + c\right )^{4} - 27 \, a \cos \left (d x + c\right )^{2} + 11 \, a\right )} \sin \left (d x + c\right ) - 192 \, a}{420 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 106, normalized size = 0.89 \[ -\frac {420 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {1089 \, a \sin \left (d x + c\right )^{7} + 420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 217, normalized size = 1.82 \[ -\frac {a \left (\cot ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )^{7}}+\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{5}}-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{35 d \sin \left (d x +c \right )^{3}}+\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{7 d \sin \left (d x +c \right )}+\frac {16 a \sin \left (d x +c \right )}{35 d}+\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{7 d}+\frac {6 a \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{35 d}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{35 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 94, normalized size = 0.79 \[ -\frac {420 \, a \log \left (\sin \left (d x + c\right )\right ) - \frac {420 \, a \sin \left (d x + c\right )^{6} - 630 \, a \sin \left (d x + c\right )^{5} - 420 \, a \sin \left (d x + c\right )^{4} + 315 \, a \sin \left (d x + c\right )^{3} + 252 \, a \sin \left (d x + c\right )^{2} - 70 \, a \sin \left (d x + c\right ) - 60 \, a}{\sin \left (d x + c\right )^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.32, size = 270, normalized size = 2.27 \[ \frac {35\,a\,\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {35\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128\,d}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {29\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {7\,a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {29\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}-\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}+\frac {7\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{896\,d}-\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d} \]
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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