Optimal. Leaf size=165 \[ \frac {b \left (6 a^2-b^2\right ) \csc (c+d x)}{d}+\frac {a \left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {a^3 \csc ^4(c+d x)}{4 d}+\frac {a \left (a^2-6 b^2\right ) \log (\sin (c+d x))}{d}+\frac {b \left (3 a^2-2 b^2\right ) \sin (c+d x)}{d}+\frac {3 a b^2 \sin ^2(c+d x)}{2 d}+\frac {b^3 \sin ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.10, antiderivative size = 165, 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 = {2800, 962}
\begin {gather*} -\frac {a^3 \csc ^4(c+d x)}{4 d}+\frac {b \left (3 a^2-2 b^2\right ) \sin (c+d x)}{d}+\frac {a \left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 d}+\frac {b \left (6 a^2-b^2\right ) \csc (c+d x)}{d}+\frac {a \left (a^2-6 b^2\right ) \log (\sin (c+d x))}{d}-\frac {a^2 b \csc ^3(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 {gather*}
Antiderivative was successfully verified.
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Rule 962
Rule 2800
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+b \sin (c+d x))^3 \, dx &=\frac {\text {Subst}\left (\int \frac {(a+x)^3 \left (b^2-x^2\right )^2}{x^5} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (3 a^2 \left (1-\frac {2 b^2}{3 a^2}\right )+\frac {a^3 b^4}{x^5}+\frac {3 a^2 b^4}{x^4}+\frac {-2 a^3 b^2+3 a b^4}{x^3}+\frac {-6 a^2 b^2+b^4}{x^2}+\frac {a^3-6 a b^2}{x}+3 a x+x^2\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b \left (6 a^2-b^2\right ) \csc (c+d x)}{d}+\frac {a \left (2 a^2-3 b^2\right ) \csc ^2(c+d x)}{2 d}-\frac {a^2 b \csc ^3(c+d x)}{d}-\frac {a^3 \csc ^4(c+d x)}{4 d}+\frac {a \left (a^2-6 b^2\right ) \log (\sin (c+d x))}{d}+\frac {b \left (3 a^2-2 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 = 0.70, size = 144, normalized size = 0.87 \begin {gather*} \frac {-12 b \left (-6 a^2+b^2\right ) \csc (c+d x)+6 a \left (2 a^2-3 b^2\right ) \csc ^2(c+d x)-12 a^2 b \csc ^3(c+d x)-3 a^3 \csc ^4(c+d x)+2 \left (6 a \left (a^2-6 b^2\right ) \log (\sin (c+d x))+6 b \left (3 a^2-2 b^2\right ) \sin (c+d x)+9 a b^2 \sin ^2(c+d x)+2 b^3 \sin ^3(c+d x)\right )}{12 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 212, normalized size = 1.28
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a \,b^{2} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(212\) |
default | \(\frac {a^{3} \left (-\frac {\left (\cot ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (\cot ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\sin \left (d x +c \right )\right )\right )+3 a^{2} b \left (-\frac {\cos ^{6}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}+\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )+3 a \,b^{2} \left (-\frac {\cos ^{6}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{2}-\left (\cos ^{2}\left (d x +c \right )\right )-2 \ln \left (\sin \left (d x +c \right )\right )\right )+b^{3} \left (-\frac {\cos ^{6}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )\right )}{d}\) | \(212\) |
risch | \(-\frac {7 i b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i b^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {3 i b \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 d}-\frac {3 a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i b^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {7 i b^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}+6 i a \,b^{2} x +\frac {3 i b \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 d}-\frac {3 a \,b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}-\frac {2 i \left (-2 i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}-6 a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+2 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}-6 i a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+14 a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-3 b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-2 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-14 a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+3 b^{3} {\mathrm e}^{3 i \left (d x +c \right )}+6 a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}-b^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+\frac {12 i a \,b^{2} c}{d}-i a^{3} x -\frac {2 i a^{3} c}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}-\frac {6 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) b^{2}}{d}\) | \(452\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.45, size = 142, normalized size = 0.86 \begin {gather*} \frac {4 \, b^{3} \sin \left (d x + c\right )^{3} + 18 \, a b^{2} \sin \left (d x + c\right )^{2} + 12 \, {\left (a^{3} - 6 \, a b^{2}\right )} \log \left (\sin \left (d x + c\right )\right ) + 12 \, {\left (3 \, a^{2} b - 2 \, b^{3}\right )} \sin \left (d x + c\right ) - \frac {3 \, {\left (4 \, a^{2} b \sin \left (d x + c\right ) - 4 \, {\left (6 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )^{3} + a^{3} - 2 \, {\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (d x + c\right )^{2}\right )}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 225, normalized size = 1.36 \begin {gather*} -\frac {18 \, a b^{2} \cos \left (d x + c\right )^{6} - 45 \, a b^{2} \cos \left (d x + c\right )^{4} - 9 \, a^{3} + 9 \, a b^{2} + 6 \, {\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{2} - 12 \, {\left ({\left (a^{3} - 6 \, a b^{2}\right )} \cos \left (d x + c\right )^{4} + a^{3} - 6 \, a b^{2} - 2 \, {\left (a^{3} - 6 \, 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 )^{6} - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{4} - 24 \, a^{2} b + 8 \, b^{3} + 12 \, {\left (3 \, 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 )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end {gather*}
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 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right )^{3} \cot ^{5}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.30, size = 185, normalized size = 1.12 \begin {gather*} \frac {4 \, b^{3} \sin \left (d x + c\right )^{3} + 18 \, a b^{2} \sin \left (d x + c\right )^{2} + 36 \, a^{2} b \sin \left (d x + c\right ) - 24 \, b^{3} \sin \left (d x + c\right ) + 12 \, {\left (a^{3} - 6 \, a b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {25 \, a^{3} \sin \left (d x + c\right )^{4} - 150 \, a b^{2} \sin \left (d x + c\right )^{4} - 72 \, a^{2} b \sin \left (d x + c\right )^{3} + 12 \, b^{3} \sin \left (d x + c\right )^{3} - 12 \, a^{3} \sin \left (d x + c\right )^{2} + 18 \, a b^{2} \sin \left (d x + c\right )^{2} + 12 \, a^{2} b \sin \left (d x + c\right ) + 3 \, a^{3}}{\sin \left (d x + c\right )^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.97, size = 424, normalized size = 2.57 \begin {gather*} \frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (6\,a\,b^2-a^3\right )}{d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{64\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a\,b^2}{8}-\frac {3\,a^3}{16}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (6\,a\,b^2-a^3\right )}{d}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (3\,a^3+90\,a\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (18\,a\,b^2-\frac {33\,a^3}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (6\,a\,b^2-\frac {9\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {35\,a^3}{4}+78\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (36\,a^2\,b-8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (138\,a^2\,b-72\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (216\,a^2\,b-88\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (316\,a^2\,b-\frac {328\,b^3}{3}\right )-\frac {a^3}{4}-2\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {21\,a^2\,b}{8}-\frac {b^3}{2}\right )}{d}-\frac {a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{8\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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