Optimal. Leaf size=65 \[ \frac {a^2 \sec ^4(c+d x)}{4 d}+\frac {2 a^2 \sec ^3(c+d x)}{3 d}-\frac {2 a^2 \sec (c+d x)}{d}+\frac {a^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 65, 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, 75} \[ \frac {a^2 \sec ^4(c+d x)}{4 d}+\frac {2 a^2 \sec ^3(c+d x)}{3 d}-\frac {2 a^2 \sec (c+d x)}{d}+\frac {a^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 75
Rule 3879
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^2 \tan ^3(c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a-a x) (a+a x)^3}{x^5} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^4}{x^5}+\frac {2 a^4}{x^4}-\frac {2 a^4}{x^2}-\frac {a^4}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {a^2 \log (\cos (c+d x))}{d}-\frac {2 a^2 \sec (c+d x)}{d}+\frac {2 a^2 \sec ^3(c+d x)}{3 d}+\frac {a^2 \sec ^4(c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 83, normalized size = 1.28 \[ \frac {a^2 \sec ^4(c+d x) (3 (-4 \cos (3 (c+d x))+4 \cos (2 (c+d x)) \log (\cos (c+d x))+\cos (4 (c+d x)) \log (\cos (c+d x))+3 \log (\cos (c+d x))+2)-20 \cos (c+d x))}{24 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 65, normalized size = 1.00 \[ \frac {12 \, a^{2} \cos \left (d x + c\right )^{4} \log \left (-\cos \left (d x + c\right )\right ) - 24 \, a^{2} \cos \left (d x + c\right )^{3} + 8 \, a^{2} \cos \left (d x + c\right ) + 3 \, a^{2}}{12 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.43, size = 192, normalized size = 2.95 \[ -\frac {12 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 12 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {57 \, a^{2} + \frac {252 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {246 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {124 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {25 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.72, size = 140, normalized size = 2.15 \[ \frac {a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {2 a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}-\frac {2 a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )}-\frac {2 a^{2} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3 d}-\frac {4 a^{2} \cos \left (d x +c \right )}{3 d}+\frac {a^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 58, normalized size = 0.89 \[ \frac {12 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {24 \, a^{2} \cos \left (d x + c\right )^{3} - 8 \, a^{2} \cos \left (d x + c\right ) - 3 \, a^{2}}{\cos \left (d x + c\right )^{4}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.87, size = 133, normalized size = 2.05 \[ \frac {2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-8\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {38\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-\frac {8\,a^2}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {2\,a^2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.82, size = 126, normalized size = 1.94 \[ \begin {cases} - \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{2} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{4 d} + \frac {2 a^{2} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{3 d} + \frac {a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {a^{2} \sec ^{2}{\left (c + d x \right )}}{4 d} - \frac {4 a^{2} \sec {\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{2} \tan ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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