Optimal. Leaf size=116 \[ -\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {b \left (3 a^2-b^2\right ) \sin (c+d x)}{d}-\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {3 a b^2 \sin ^2(c+d x)}{2 d}-\frac {b^3 \sin ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.09, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2721, 894} \[ -\frac {b \left (3 a^2-b^2\right ) \sin (c+d x)}{d}-\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {3 a b^2 \sin ^2(c+d x)}{2 d}-\frac {b^3 \sin ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 894
Rule 2721
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+x)^3 \left (b^2-x^2\right )}{x^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-3 a^2 \left (1-\frac {b^2}{3 a^2}\right )+\frac {a^3 b^2}{x^3}+\frac {3 a^2 b^2}{x^2}+\frac {-a^3+3 a b^2}{x}-3 a x-x^2\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {3 a^2 b \csc (c+d x)}{d}-\frac {a^3 \csc ^2(c+d x)}{2 d}-\frac {a \left (a^2-3 b^2\right ) \log (\sin (c+d x))}{d}-\frac {b \left (3 a^2-b^2\right ) \sin (c+d x)}{d}-\frac {3 a b^2 \sin ^2(c+d x)}{2 d}-\frac {b^3 \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 97, normalized size = 0.84 \[ -\frac {3 a^3 \csc ^2(c+d x)-6 b \left (b^2-3 a^2\right ) \sin (c+d x)+6 a \left (a^2-3 b^2\right ) \log (\sin (c+d x))+18 a^2 b \csc (c+d x)+9 a b^2 \sin ^2(c+d x)+2 b^3 \sin ^3(c+d x)}{6 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 153, normalized size = 1.32 \[ \frac {18 \, a b^{2} \cos \left (d x + c\right )^{4} - 27 \, a b^{2} \cos \left (d x + c\right )^{2} + 6 \, a^{3} + 9 \, a b^{2} + 12 \, {\left (a^{3} - 3 \, a b^{2} - {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 4 \, {\left (b^{3} \cos \left (d x + c\right )^{4} + 18 \, a^{2} b - 2 \, b^{3} - {\left (9 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 131, normalized size = 1.13 \[ -\frac {2 \, b^{3} \sin \left (d x + c\right )^{3} + 9 \, a b^{2} \sin \left (d x + c\right )^{2} + 18 \, a^{2} b \sin \left (d x + c\right ) - 6 \, b^{3} \sin \left (d x + c\right ) + 6 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {3 \, {\left (3 \, a^{3} \sin \left (d x + c\right )^{2} - 9 \, a b^{2} \sin \left (d x + c\right )^{2} - 6 \, a^{2} b \sin \left (d x + c\right ) - a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 165, normalized size = 1.42 \[ -\frac {a^{3} \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {3 a^{2} b \left (\cos ^{4}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {3 a^{2} b \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{d}-\frac {6 a^{2} b \sin \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a \,b^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {b^{3} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {2 b^{3} \sin \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.14, size = 98, normalized size = 0.84 \[ -\frac {2 \, b^{3} \sin \left (d x + c\right )^{3} + 9 \, a b^{2} \sin \left (d x + c\right )^{2} + 6 \, {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + 6 \, {\left (3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right ) + \frac {3 \, {\left (6 \, a^{2} b \sin \left (d x + c\right ) + a^{3}\right )}}{\sin \left (d x + c\right )^{2}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.95, size = 312, normalized size = 2.69 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (3\,a\,b^2-a^3\right )}{d}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (3\,a\,b^2-a^3\right )}{d}-\frac {\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^3}{2}+24\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3}{2}+24\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (30\,a^2\,b-8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (42\,a^2\,b-8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (66\,a^2\,b-\frac {16\,b^3}{3}\right )+\frac {a^3}{2}+6\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \cot ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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