Optimal. Leaf size=138 \[ \frac {a^3 \sec ^7(c+d x)}{7 d}+\frac {a^3 \sec ^6(c+d x)}{2 d}+\frac {a^3 \sec ^5(c+d x)}{5 d}-\frac {5 a^3 \sec ^4(c+d x)}{4 d}-\frac {5 a^3 \sec ^3(c+d x)}{3 d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 138, 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} \[ \frac {a^3 \sec ^7(c+d x)}{7 d}+\frac {a^3 \sec ^6(c+d x)}{2 d}+\frac {a^3 \sec ^5(c+d x)}{5 d}-\frac {5 a^3 \sec ^4(c+d x)}{4 d}-\frac {5 a^3 \sec ^3(c+d x)}{3 d}+\frac {a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 88
Rule 3879
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^3 \tan ^5(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^2 (a+a x)^5}{x^8} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^7}{x^8}+\frac {3 a^7}{x^7}+\frac {a^7}{x^6}-\frac {5 a^7}{x^5}-\frac {5 a^7}{x^4}+\frac {a^7}{x^3}+\frac {3 a^7}{x^2}+\frac {a^7}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d}-\frac {5 a^3 \sec ^3(c+d x)}{3 d}-\frac {5 a^3 \sec ^4(c+d x)}{4 d}+\frac {a^3 \sec ^5(c+d x)}{5 d}+\frac {a^3 \sec ^6(c+d x)}{2 d}+\frac {a^3 \sec ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 140, normalized size = 1.01 \[ -\frac {a^3 \sec ^7(c+d x) (-4522 \cos (2 (c+d x))+1050 \cos (3 (c+d x))-2380 \cos (4 (c+d x))-210 \cos (5 (c+d x))-630 \cos (6 (c+d x))+2205 \cos (3 (c+d x)) \log (\cos (c+d x))+735 \cos (5 (c+d x)) \log (\cos (c+d x))+105 \cos (7 (c+d x)) \log (\cos (c+d x))+105 \cos (c+d x) (35 \log (\cos (c+d x))+8)-3732)}{6720 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 117, normalized size = 0.85 \[ -\frac {420 \, a^{3} \cos \left (d x + c\right )^{7} \log \left (-\cos \left (d x + c\right )\right ) - 1260 \, a^{3} \cos \left (d x + c\right )^{6} - 210 \, a^{3} \cos \left (d x + c\right )^{5} + 700 \, a^{3} \cos \left (d x + c\right )^{4} + 525 \, a^{3} \cos \left (d x + c\right )^{3} - 84 \, a^{3} \cos \left (d x + c\right )^{2} - 210 \, a^{3} \cos \left (d x + c\right ) - 60 \, a^{3}}{420 \, d \cos \left (d x + c\right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 4.00, size = 267, normalized size = 1.93 \[ \frac {420 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 420 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {2497 \, a^{3} + \frac {18319 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {58317 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {69475 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {56035 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {28749 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {8463 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1089 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 227, normalized size = 1.64 \[ \frac {a^{3} \left (\tan ^{4}\left (d x +c \right )\right )}{4 d}-\frac {a^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {22 a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )^{5}}-\frac {22 a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{105 d \cos \left (d x +c \right )^{3}}+\frac {22 a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{35 d \cos \left (d x +c \right )}+\frac {176 a^{3} \cos \left (d x +c \right )}{105 d}+\frac {22 a^{3} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}+\frac {88 a^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{105 d}+\frac {a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{6}}+\frac {a^{3} \left (\sin ^{6}\left (d x +c \right )\right )}{7 d \cos \left (d x +c \right )^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 110, normalized size = 0.80 \[ -\frac {420 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {1260 \, a^{3} \cos \left (d x + c\right )^{6} + 210 \, a^{3} \cos \left (d x + c\right )^{5} - 700 \, a^{3} \cos \left (d x + c\right )^{4} - 525 \, a^{3} \cos \left (d x + c\right )^{3} + 84 \, a^{3} \cos \left (d x + c\right )^{2} + 210 \, a^{3} \cos \left (d x + c\right ) + 60 \, a^{3}}{\cos \left (d x + c\right )^{7}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.48, size = 221, normalized size = 1.60 \[ \frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-14\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {128\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{3}-\frac {224\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {422\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{5}-\frac {382\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{15}+\frac {352\,a^3}{105}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 9.01, size = 255, normalized size = 1.85 \[ \begin {cases} \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{7 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 a^{3} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{5 d} + \frac {a^{3} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {4 a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{35 d} - \frac {a^{3} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{2 d} - \frac {4 a^{3} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{5 d} - \frac {a^{3} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {8 a^{3} \sec ^{3}{\left (c + d x \right )}}{105 d} + \frac {a^{3} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac {8 a^{3} \sec {\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{3} \tan ^{5}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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