Optimal. Leaf size=194 \[ a^3 x-\frac {9}{2} a b^2 x+\frac {9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {b^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {9 a^2 b \cos (c+d x)}{2 d}+\frac {b^3 \cos (c+d x)}{d}+\frac {b^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{2 d}+\frac {3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.14, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {2801, 2672,
308, 212, 2671, 294, 327, 209, 3554, 8} \begin {gather*} -\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}+a^3 x-\frac {9 a^2 b \cos (c+d x)}{2 d}-\frac {3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}+\frac {9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {9 a b^2 \cot (c+d x)}{2 d}+\frac {3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {9}{2} a b^2 x+\frac {b^3 \cos ^3(c+d x)}{3 d}+\frac {b^3 \cos (c+d x)}{d}-\frac {b^3 \tanh ^{-1}(\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 209
Rule 212
Rule 294
Rule 308
Rule 327
Rule 2671
Rule 2672
Rule 2801
Rule 3554
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+b \sin (c+d x))^3 \, dx &=\int \left (b^3 \cos ^3(c+d x) \cot (c+d x)+3 a b^2 \cos ^2(c+d x) \cot ^2(c+d x)+3 a^2 b \cos (c+d x) \cot ^3(c+d x)+a^3 \cot ^4(c+d x)\right ) \, dx\\ &=a^3 \int \cot ^4(c+d x) \, dx+\left (3 a^2 b\right ) \int \cos (c+d x) \cot ^3(c+d x) \, dx+\left (3 a b^2\right ) \int \cos ^2(c+d x) \cot ^2(c+d x) \, dx+b^3 \int \cos ^3(c+d x) \cot (c+d x) \, dx\\ &=-\frac {a^3 \cot ^3(c+d x)}{3 d}-a^3 \int \cot ^2(c+d x) \, dx-\frac {\left (3 a^2 b\right ) \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^2} \, dx,x,\cot (c+d x)\right )}{d}-\frac {b^3 \text {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {a^3 \cot (c+d x)}{d}+\frac {3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+a^3 \int 1 \, dx+\frac {\left (9 a^2 b\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d}-\frac {\left (9 a b^2\right ) \text {Subst}\left (\int \frac {x^2}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}-\frac {b^3 \text {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=a^3 x-\frac {9 a^2 b \cos (c+d x)}{2 d}+\frac {b^3 \cos (c+d x)}{d}+\frac {b^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{2 d}+\frac {3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}+\frac {\left (9 a^2 b\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{2 d}+\frac {\left (9 a b^2\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (c+d x)\right )}{2 d}-\frac {b^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=a^3 x-\frac {9}{2} a b^2 x+\frac {9 a^2 b \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac {b^3 \tanh ^{-1}(\cos (c+d x))}{d}-\frac {9 a^2 b \cos (c+d x)}{2 d}+\frac {b^3 \cos (c+d x)}{d}+\frac {b^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cot (c+d x)}{d}-\frac {9 a b^2 \cot (c+d x)}{2 d}+\frac {3 a b^2 \cos ^2(c+d x) \cot (c+d x)}{2 d}-\frac {3 a^2 b \cos (c+d x) \cot ^2(c+d x)}{2 d}-\frac {a^3 \cot ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 6.18, size = 355, normalized size = 1.83 \begin {gather*} \frac {a \left (2 a^2-9 b^2\right ) (c+d x)}{2 d}+\frac {b \left (-12 a^2+5 b^2\right ) \cos (c+d x)}{4 d}+\frac {b^3 \cos (3 (c+d x))}{12 d}+\frac {\left (4 a^3 \cos \left (\frac {1}{2} (c+d x)\right )-9 a b^2 \cos \left (\frac {1}{2} (c+d x)\right )\right ) \csc \left (\frac {1}{2} (c+d x)\right )}{6 d}-\frac {3 a^2 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 \cot \left (\frac {1}{2} (c+d x)\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{24 d}+\frac {\left (9 a^2 b-2 b^3\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {\left (-9 a^2 b+2 b^3\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 a^2 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {\sec \left (\frac {1}{2} (c+d x)\right ) \left (-4 a^3 \sin \left (\frac {1}{2} (c+d x)\right )+9 a b^2 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{6 d}-\frac {3 a b^2 \sin (2 (c+d x))}{4 d}+\frac {a^3 \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 186, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a \,b^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(186\) |
default | \(\frac {a^{3} \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+3 a^{2} b \left (-\frac {\cos ^{5}\left (d x +c \right )}{2 \sin \left (d x +c \right )^{2}}-\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{2}-\frac {3 \cos \left (d x +c \right )}{2}-\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+3 a \,b^{2} \left (-\frac {\cos ^{5}\left (d x +c \right )}{\sin \left (d x +c \right )}-\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )-\frac {3 d x}{2}-\frac {3 c}{2}\right )+b^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )\right )}{3}+\cos \left (d x +c \right )+\ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right )}{d}\) | \(186\) |
risch | \(a^{3} x -\frac {9 a \,b^{2} x}{2}+\frac {3 i a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {3 b \,{\mathrm e}^{i \left (d x +c \right )} a^{2}}{2 d}+\frac {5 b^{3} {\mathrm e}^{i \left (d x +c \right )}}{8 d}-\frac {3 b \,{\mathrm e}^{-i \left (d x +c \right )} a^{2}}{2 d}+\frac {5 b^{3} {\mathrm e}^{-i \left (d x +c \right )}}{8 d}-\frac {3 i a \,b^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {a \left (12 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-18 i b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+9 a \,{\mathrm e}^{5 i \left (d x +c \right )} b -12 i a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+36 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+8 i a^{2}-18 i b^{2}-9 a \,{\mathrm e}^{i \left (d x +c \right )} b \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3}}-\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}+\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d}+\frac {9 a^{2} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}-\frac {b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d}+\frac {b^{3} \cos \left (3 d x +3 c \right )}{12 d}\) | \(338\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 187, normalized size = 0.96 \begin {gather*} \frac {4 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a^{3} - 18 \, {\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} + 2}{\tan \left (d x + c\right )^{3} + \tan \left (d x + c\right )}\right )} a b^{2} + 2 \, {\left (2 \, \cos \left (d x + c\right )^{3} + 6 \, \cos \left (d x + c\right ) - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} b^{3} + 9 \, a^{2} b {\left (\frac {2 \, \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1} - 4 \, \cos \left (d x + c\right ) + 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) - 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 293, normalized size = 1.51 \begin {gather*} \frac {18 \, a b^{2} \cos \left (d x + c\right )^{5} + 8 \, {\left (2 \, a^{3} - 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (9 \, a^{2} b - 2 \, b^{3} - {\left (9 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + 3 \, {\left (9 \, a^{2} b - 2 \, b^{3} - {\left (9 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) - 6 \, {\left (2 \, a^{3} - 9 \, a b^{2}\right )} \cos \left (d x + c\right ) + 2 \, {\left (2 \, b^{3} \cos \left (d x + c\right )^{5} + 3 \, {\left (2 \, a^{3} - 9 \, a b^{2}\right )} d x \cos \left (d x + c\right )^{2} - 2 \, {\left (9 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{3} - 9 \, a b^{2}\right )} d x + 3 \, {\left (9 \, a^{2} b - 2 \, b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\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 ^{4}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 421 vs.
\(2 (178) = 356\).
time = 6.46, size = 421, normalized size = 2.17 \begin {gather*} \frac {3 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 27 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 108 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, {\left (2 \, a^{3} - 9 \, a b^{2}\right )} {\left (d x + c\right )} - 36 \, {\left (9 \, a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + \frac {198 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 44 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 45 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 108 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 135 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 156 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 132 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 324 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 351 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 156 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 126 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 540 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 315 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 148 \, b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 36 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 108 \, a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 27 \, a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, a^{3}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}^{3}}}{72 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.81, size = 405, normalized size = 2.09 \begin {gather*} \frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {9\,a^2\,b}{2}-b^3\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )\,\left (\frac {a\,b^2\,9{}\mathrm {i}}{2}-a^3\,1{}\mathrm {i}\right )}{d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a\,b^2}{2}-\frac {5\,a^3}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (12\,a\,b^2-4\,a^3\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (5\,a^3+12\,a\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (60\,a\,b^2-14\,a^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (36\,a\,b^2-\frac {44\,a^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (51\,a^2\,b-32\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (57\,a^2\,b-\frac {64\,b^3}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (105\,a^2\,b-32\,b^3\right )+\frac {a^3}{3}+3\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+24\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {3\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )\,\left (2\,a^2-9\,b^2\right )\,1{}\mathrm {i}}{2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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