Optimal. Leaf size=133 \[ \frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \csc ^6(c+d x)}{6 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^4(c+d x)}{4 d}+\frac {5 a^3 \csc ^3(c+d x)}{3 d}+\frac {5 a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc (c+d x)}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.13, antiderivative size = 133, 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 = {2836, 12, 88} \[ \frac {a^3 \sin (c+d x)}{d}-\frac {a^3 \csc ^6(c+d x)}{6 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^4(c+d x)}{4 d}+\frac {5 a^3 \csc ^3(c+d x)}{3 d}+\frac {5 a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc (c+d x)}{d}+\frac {3 a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cot ^5(c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^7 (a-x)^2 (a+x)^5}{x^7} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^5}{x^7} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (1+\frac {a^7}{x^7}+\frac {3 a^6}{x^6}+\frac {a^5}{x^5}-\frac {5 a^4}{x^4}-\frac {5 a^3}{x^3}+\frac {a^2}{x^2}+\frac {3 a}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {a^3 \csc (c+d x)}{d}+\frac {5 a^3 \csc ^2(c+d x)}{2 d}+\frac {5 a^3 \csc ^3(c+d x)}{3 d}-\frac {a^3 \csc ^4(c+d x)}{4 d}-\frac {3 a^3 \csc ^5(c+d x)}{5 d}-\frac {a^3 \csc ^6(c+d x)}{6 d}+\frac {3 a^3 \log (\sin (c+d x))}{d}+\frac {a^3 \sin (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 113, normalized size = 0.85 \[ a^3 \left (\frac {\sin (c+d x)}{d}-\frac {\csc ^6(c+d x)}{6 d}-\frac {3 \csc ^5(c+d x)}{5 d}-\frac {\csc ^4(c+d x)}{4 d}+\frac {5 \csc ^3(c+d x)}{3 d}+\frac {5 \csc ^2(c+d x)}{2 d}-\frac {\csc (c+d x)}{d}+\frac {3 \log (\sin (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 180, normalized size = 1.35 \[ -\frac {150 \, a^{3} \cos \left (d x + c\right )^{4} - 285 \, a^{3} \cos \left (d x + c\right )^{2} + 125 \, a^{3} - 180 \, {\left (a^{3} \cos \left (d x + c\right )^{6} - 3 \, a^{3} \cos \left (d x + c\right )^{4} + 3 \, a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 4 \, {\left (15 \, a^{3} \cos \left (d x + c\right )^{6} - 30 \, a^{3} \cos \left (d x + c\right )^{4} + 40 \, a^{3} \cos \left (d x + c\right )^{2} - 16 \, a^{3}\right )} \sin \left (d x + c\right )}{60 \, {\left (d \cos \left (d x + c\right )^{6} - 3 \, d \cos \left (d x + c\right )^{4} + 3 \, d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 122, normalized size = 0.92 \[ \frac {180 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {441 \, a^{3} \sin \left (d x + c\right )^{6} + 60 \, a^{3} \sin \left (d x + c\right )^{5} - 150 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} + 15 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{\sin \left (d x + c\right )^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 203, normalized size = 1.53 \[ -\frac {2 a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{15 d \sin \left (d x +c \right )^{3}}+\frac {2 a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )}+\frac {16 a^{3} \sin \left (d x +c \right )}{15 d}+\frac {2 a^{3} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{5 d}+\frac {8 a^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{15 d}-\frac {3 a^{3} \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {3 a^{3} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {3 a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{5 d \sin \left (d x +c \right )^{5}}-\frac {a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{6 d \sin \left (d x +c \right )^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 108, normalized size = 0.81 \[ \frac {180 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) + 60 \, a^{3} \sin \left (d x + c\right ) - \frac {60 \, a^{3} \sin \left (d x + c\right )^{5} - 150 \, a^{3} \sin \left (d x + c\right )^{4} - 100 \, a^{3} \sin \left (d x + c\right )^{3} + 15 \, a^{3} \sin \left (d x + c\right )^{2} + 36 \, a^{3} \sin \left (d x + c\right ) + 10 \, a^{3}}{\sin \left (d x + c\right )^{6}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.84, size = 296, normalized size = 2.23 \[ \frac {67\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{128\,d}+\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{32\,d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{384\,d}+\frac {a^3\,\left (5760\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-5760\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\right )}{1920\,d}-\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16\,d}+\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {31\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {67\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{128}+\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{96}+\frac {63\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{128}+\frac {23\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{240}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{384}-\frac {3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{160}-\frac {a^3}{384}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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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