Optimal. Leaf size=107 \[ -\frac {17 a^3}{8 d (1-\cos (c+d x))}+\frac {7 a^3}{8 d (1-\cos (c+d x))^2}-\frac {a^3}{6 d (1-\cos (c+d x))^3}-\frac {15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {a^3 \log (\cos (c+d x)+1)}{16 d} \]
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Rubi [A] time = 0.08, antiderivative size = 107, 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 = {3879, 88} \[ -\frac {17 a^3}{8 d (1-\cos (c+d x))}+\frac {7 a^3}{8 d (1-\cos (c+d x))^2}-\frac {a^3}{6 d (1-\cos (c+d x))^3}-\frac {15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {a^3 \log (\cos (c+d x)+1)}{16 d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int \cot ^7(c+d x) (a+a \sec (c+d x))^3 \, dx &=-\frac {a^8 \operatorname {Subst}\left (\int \frac {x^4}{(a-a x)^4 (a+a x)} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^8 \operatorname {Subst}\left (\int \left (\frac {1}{2 a^5 (-1+x)^4}+\frac {7}{4 a^5 (-1+x)^3}+\frac {17}{8 a^5 (-1+x)^2}+\frac {15}{16 a^5 (-1+x)}+\frac {1}{16 a^5 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^3}{6 d (1-\cos (c+d x))^3}+\frac {7 a^3}{8 d (1-\cos (c+d x))^2}-\frac {17 a^3}{8 d (1-\cos (c+d x))}-\frac {15 a^3 \log (1-\cos (c+d x))}{16 d}-\frac {a^3 \log (1+\cos (c+d x))}{16 d}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 102, normalized size = 0.95 \[ -\frac {a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (2 \csc ^6\left (\frac {1}{2} (c+d x)\right )-21 \csc ^4\left (\frac {1}{2} (c+d x)\right )+102 \csc ^2\left (\frac {1}{2} (c+d x)\right )+12 \left (15 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )\right )}{768 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.17, size = 178, normalized size = 1.66 \[ \frac {102 \, a^{3} \cos \left (d x + c\right )^{2} - 162 \, a^{3} \cos \left (d x + c\right ) + 68 \, a^{3} - 3 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 45 \, {\left (a^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) - a^{3}\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{48 \, {\left (d \cos \left (d x + c\right )^{3} - 3 \, d \cos \left (d x + c\right )^{2} + 3 \, d \cos \left (d x + c\right ) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.48, size = 165, normalized size = 1.54 \[ -\frac {90 \, a^{3} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 96 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - \frac {{\left (2 \, a^{3} + \frac {15 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {66 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {165 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 104, normalized size = 0.97 \[ \frac {a^{3} \ln \left (\sec \left (d x +c \right )\right )}{d}-\frac {a^{3}}{6 d \left (-1+\sec \left (d x +c \right )\right )^{3}}+\frac {3 a^{3}}{8 d \left (-1+\sec \left (d x +c \right )\right )^{2}}-\frac {7 a^{3}}{8 d \left (-1+\sec \left (d x +c \right )\right )}-\frac {15 a^{3} \ln \left (-1+\sec \left (d x +c \right )\right )}{16 d}-\frac {a^{3} \ln \left (1+\sec \left (d x +c \right )\right )}{16 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 96, normalized size = 0.90 \[ -\frac {3 \, a^{3} \log \left (\cos \left (d x + c\right ) + 1\right ) + 45 \, a^{3} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (51 \, a^{3} \cos \left (d x + c\right )^{2} - 81 \, a^{3} \cos \left (d x + c\right ) + 34 \, a^{3}\right )}}{\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) - 1}}{48 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 94, normalized size = 0.88 \[ \frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d}-\frac {\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{4}+\frac {a^3}{6}}{8\,d\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}-\frac {15\,a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{8\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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