Optimal. Leaf size=55 \[ -\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)}{3 d}+\frac {\cot (c+d x) (2 a \sec (c+d x)+3 a)}{3 d}+a x \]
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Rubi [A] time = 0.05, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3882, 8} \[ -\frac {\cot ^3(c+d x) (a \sec (c+d x)+a)}{3 d}+\frac {\cot (c+d x) (2 a \sec (c+d x)+3 a)}{3 d}+a x \]
Antiderivative was successfully verified.
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Rule 8
Rule 3882
Rubi steps
\begin {align*} \int \cot ^4(c+d x) (a+a \sec (c+d x)) \, dx &=-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))}{3 d}+\frac {1}{3} \int \cot ^2(c+d x) (-3 a-2 a \sec (c+d x)) \, dx\\ &=-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))}{3 d}+\frac {\cot (c+d x) (3 a+2 a \sec (c+d x))}{3 d}+\frac {1}{3} \int 3 a \, dx\\ &=a x-\frac {\cot ^3(c+d x) (a+a \sec (c+d x))}{3 d}+\frac {\cot (c+d x) (3 a+2 a \sec (c+d x))}{3 d}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 62, normalized size = 1.13 \[ -\frac {a \cot ^3(c+d x) \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};-\tan ^2(c+d x)\right )}{3 d}-\frac {a \csc ^3(c+d x)}{3 d}+\frac {a \csc (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 72, normalized size = 1.31 \[ \frac {4 \, a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) + 3 \, {\left (a d x \cos \left (d x + c\right ) - a d x\right )} \sin \left (d x + c\right ) - 2 \, a}{3 \, {\left (d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.52, size = 56, normalized size = 1.02 \[ \frac {12 \, {\left (d x + c\right )} a - 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \frac {12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 86, normalized size = 1.56 \[ \frac {a \left (-\frac {\left (\cot ^{3}\left (d x +c \right )\right )}{3}+\cot \left (d x +c \right )+d x +c \right )+a \left (-\frac {\cos ^{4}\left (d x +c \right )}{3 \sin \left (d x +c \right )^{3}}+\frac {\cos ^{4}\left (d x +c \right )}{3 \sin \left (d x +c \right )}+\frac {\left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 59, normalized size = 1.07 \[ \frac {{\left (3 \, d x + 3 \, c + \frac {3 \, \tan \left (d x + c\right )^{2} - 1}{\tan \left (d x + c\right )^{3}}\right )} a + \frac {{\left (3 \, \sin \left (d x + c\right )^{2} - 1\right )} a}{\sin \left (d x + c\right )^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 53, normalized size = 0.96 \[ a\,x-\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4\,d}-\frac {\frac {a}{12}-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3} \]
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 \[ a \left (\int \cot ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \cot ^{4}{\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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