Optimal. Leaf size=66 \[ \frac {a^3 \sec ^3(c+d x)}{3 d}+\frac {3 a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 66, 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 = {3879, 43} \[ \frac {a^3 \sec ^3(c+d x)}{3 d}+\frac {3 a^3 \sec ^2(c+d x)}{2 d}+\frac {3 a^3 \sec (c+d x)}{d}-\frac {a^3 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 3879
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^3 \tan (c+d x) \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {(a+a x)^3}{x^4} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}+\frac {3 a^3}{x^3}+\frac {3 a^3}{x^2}+\frac {a^3}{x}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac {a^3 \log (\cos (c+d x))}{d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {3 a^3 \sec ^2(c+d x)}{2 d}+\frac {a^3 \sec ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 64, normalized size = 0.97 \[ -\frac {a^3 \sec ^3(c+d x) (-18 \cos (2 (c+d x))+9 \cos (c+d x) (\log (\cos (c+d x))-2)+3 \cos (3 (c+d x)) \log (\cos (c+d x))-22)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.66, size = 65, normalized size = 0.98 \[ -\frac {6 \, a^{3} \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) - 18 \, a^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{3} \cos \left (d x + c\right ) - 2 \, a^{3}}{6 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.51, size = 167, normalized size = 2.53 \[ \frac {6 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 6 \, a^{3} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {51 \, a^{3} + \frac {69 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 62, normalized size = 0.94 \[ \frac {a^{3} \left (\sec ^{3}\left (d x +c \right )\right )}{3 d}+\frac {3 a^{3} \left (\sec ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a^{3} \sec \left (d x +c \right )}{d}+\frac {a^{3} \ln \left (\sec \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.68, size = 58, normalized size = 0.88 \[ -\frac {6 \, a^{3} \log \left (\cos \left (d x + c\right )\right ) - \frac {18 \, a^{3} \cos \left (d x + c\right )^{2} + 9 \, a^{3} \cos \left (d x + c\right ) + 2 \, a^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.94, size = 105, normalized size = 1.59 \[ \frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {20\,a^3}{3}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.92, size = 76, normalized size = 1.15 \[ \begin {cases} \frac {a^{3} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{3} \sec ^{3}{\left (c + d x \right )}}{3 d} + \frac {3 a^{3} \sec ^{2}{\left (c + d x \right )}}{2 d} + \frac {3 a^{3} \sec {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a \sec {\relax (c )} + a\right )^{3} \tan {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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