Optimal. Leaf size=102 \[ \frac {(8 a+3 b) \log (1-\cos (c+d x))}{16 d}+\frac {(8 a-3 b) \log (\cos (c+d x)+1)}{16 d}-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d} \]
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Rubi [A] time = 0.13, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3882, 3883, 2668, 633, 31} \[ \frac {(8 a+3 b) \log (1-\cos (c+d x))}{16 d}+\frac {(8 a-3 b) \log (\cos (c+d x)+1)}{16 d}-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 2668
Rule 3882
Rule 3883
Rubi steps
\begin {align*} \int \cot ^5(c+d x) (a+b \sec (c+d x)) \, dx &=-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {1}{4} \int \cot ^3(c+d x) (-4 a-3 b \sec (c+d x)) \, dx\\ &=-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}+\frac {1}{8} \int \cot (c+d x) (8 a+3 b \sec (c+d x)) \, dx\\ &=-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}+\frac {1}{8} \int (3 b+8 a \cos (c+d x)) \csc (c+d x) \, dx\\ &=-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}-\frac {a \operatorname {Subst}\left (\int \frac {3 b+x}{64 a^2-x^2} \, dx,x,8 a \cos (c+d x)\right )}{d}\\ &=-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}-\frac {(8 a-3 b) \operatorname {Subst}\left (\int \frac {1}{-8 a-x} \, dx,x,8 a \cos (c+d x)\right )}{16 d}-\frac {(8 a+3 b) \operatorname {Subst}\left (\int \frac {1}{8 a-x} \, dx,x,8 a \cos (c+d x)\right )}{16 d}\\ &=\frac {(8 a+3 b) \log (1-\cos (c+d x))}{16 d}+\frac {(8 a-3 b) \log (1+\cos (c+d x))}{16 d}-\frac {\cot ^4(c+d x) (a+b \sec (c+d x))}{4 d}+\frac {\cot ^2(c+d x) (4 a+3 b \sec (c+d x))}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 166, normalized size = 1.63 \[ \frac {a \left (-\cot ^4(c+d x)+2 \cot ^2(c+d x)+4 \log (\tan (c+d x))+4 \log (\cos (c+d x))\right )}{4 d}-\frac {b \csc ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}+\frac {5 b \csc ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {b \sec ^4\left (\frac {1}{2} (c+d x)\right )}{64 d}-\frac {5 b \sec ^2\left (\frac {1}{2} (c+d x)\right )}{32 d}+\frac {3 b \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}-\frac {3 b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 168, normalized size = 1.65 \[ -\frac {10 \, b \cos \left (d x + c\right )^{3} + 16 \, a \cos \left (d x + c\right )^{2} - 6 \, b \cos \left (d x + c\right ) - {\left ({\left (8 \, a - 3 \, b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (8 \, a - 3 \, b\right )} \cos \left (d x + c\right )^{2} + 8 \, a - 3 \, b\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - {\left ({\left (8 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (8 \, a + 3 \, b\right )} \cos \left (d x + c\right )^{2} + 8 \, a + 3 \, b\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 12 \, a}{16 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.97, size = 266, normalized size = 2.61 \[ \frac {4 \, {\left (8 \, a + 3 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 64 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (a + b + \frac {12 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {48 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {18 \, b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}} - \frac {12 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.54, size = 134, normalized size = 1.31 \[ -\frac {a \left (\cot ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {b \left (\cos ^{5}\left (d x +c \right )\right )}{4 d \sin \left (d x +c \right )^{4}}+\frac {b \left (\cos ^{5}\left (d x +c \right )\right )}{8 d \sin \left (d x +c \right )^{2}}+\frac {b \left (\cos ^{3}\left (d x +c \right )\right )}{8 d}+\frac {3 b \cos \left (d x +c \right )}{8 d}+\frac {3 b \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 99, normalized size = 0.97 \[ \frac {{\left (8 \, a - 3 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) + {\left (8 \, a + 3 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (5 \, b \cos \left (d x + c\right )^{3} + 8 \, a \cos \left (d x + c\right )^{2} - 3 \, b \cos \left (d x + c\right ) - 6 \, a\right )}}{\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 128, normalized size = 1.25 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {3\,a}{16}-\frac {b}{8}\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a}{64}-\frac {b}{64}\right )}{d}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\left (-3\,a-2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a}{4}+\frac {b}{4}\right )}{16\,d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (a+\frac {3\,b}{8}\right )}{d} \]
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 \[ \int \left (a + b \sec {\left (c + d x \right )}\right ) \cot ^{5}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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