Optimal. Leaf size=57 \[ \frac {a \log (\cos (c+d x))}{d}-\frac {a \sec (c+d x)}{d}+\frac {a \sec ^2(c+d x)}{2 d}+\frac {a \sec ^3(c+d x)}{3 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 57, 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 = {3964, 76}
\begin {gather*} \frac {a \sec ^3(c+d x)}{3 d}+\frac {a \sec ^2(c+d x)}{2 d}-\frac {a \sec (c+d x)}{d}+\frac {a \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 3964
Rubi steps
\begin {align*} \int (a+a \sec (c+d x)) \tan ^3(c+d x) \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-a x) (a+a x)^2}{x^4} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {a^3}{x^4}+\frac {a^3}{x^3}-\frac {a^3}{x^2}-\frac {a^3}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac {a \log (\cos (c+d x))}{d}-\frac {a \sec (c+d x)}{d}+\frac {a \sec ^2(c+d x)}{2 d}+\frac {a \sec ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 55, normalized size = 0.96 \begin {gather*} -\frac {a \sec (c+d x)}{d}+\frac {a \sec ^3(c+d x)}{3 d}+\frac {a \left (2 \log (\cos (c+d x))+\tan ^2(c+d x)\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 83, normalized size = 1.46
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+a \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(83\) |
default | \(\frac {a \left (\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{3 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}\right )+a \left (\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}+\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(83\) |
risch | \(-i a x -\frac {2 i a c}{d}-\frac {2 a \left (3 \,{\mathrm e}^{5 i \left (d x +c \right )}-3 \,{\mathrm e}^{4 i \left (d x +c \right )}+2 \,{\mathrm e}^{3 i \left (d x +c \right )}-3 \,{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(107\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 50, normalized size = 0.88 \begin {gather*} \frac {6 \, a \log \left (\cos \left (d x + c\right )\right ) - \frac {6 \, a \cos \left (d x + c\right )^{2} - 3 \, a \cos \left (d x + c\right ) - 2 \, a}{\cos \left (d x + c\right )^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.61, size = 57, normalized size = 1.00 \begin {gather*} \frac {6 \, a \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) - 6 \, a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + 2 \, a}{6 \, d \cos \left (d x + c\right )^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 76, normalized size = 1.33 \begin {gather*} \begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{3 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} - \frac {2 a \sec {\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\left (c \right )} + a\right ) \tan ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 155 vs.
\(2 (53) = 106\).
time = 0.81, size = 155, normalized size = 2.72 \begin {gather*} -\frac {6 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 6 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {19 \, a + \frac {69 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {45 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {11 \, a {\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.99, size = 96, normalized size = 1.68 \begin {gather*} \frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {4\,a}{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 )}-\frac {2\,a\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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