Optimal. Leaf size=46 \[ \frac {3 \log (\cos (c+d x))}{a^3 d}-\frac {4 \log (1+\cos (c+d x))}{a^3 d}+\frac {\sec (c+d x)}{a^3 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 46, 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 = {3964, 90}
\begin {gather*} \frac {\sec (c+d x)}{a^3 d}+\frac {3 \log (\cos (c+d x))}{a^3 d}-\frac {4 \log (\cos (c+d x)+1)}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 3964
Rubi steps
\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-a x)^2}{x^2 (a+a x)} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {a}{x^2}-\frac {3 a}{x}+\frac {4 a}{1+x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=\frac {3 \log (\cos (c+d x))}{a^3 d}-\frac {4 \log (1+\cos (c+d x))}{a^3 d}+\frac {\sec (c+d x)}{a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 36, normalized size = 0.78 \begin {gather*} \frac {-8 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+3 \log (\cos (c+d x))+\sec (c+d x)}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 33, normalized size = 0.72
method | result | size |
derivativedivides | \(\frac {\sec \left (d x +c \right )-4 \ln \left (1+\sec \left (d x +c \right )\right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}\) | \(33\) |
default | \(\frac {\sec \left (d x +c \right )-4 \ln \left (1+\sec \left (d x +c \right )\right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}\) | \(33\) |
risch | \(\frac {i x}{a^{3}}+\frac {2 i c}{a^{3} d}+\frac {2 \,{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {8 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a^{3} d}+\frac {3 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{3} d}\) | \(89\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 45, normalized size = 0.98 \begin {gather*} -\frac {\frac {4 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} - \frac {3 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} - \frac {1}{a^{3} \cos \left (d x + c\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.73, size = 53, normalized size = 1.15 \begin {gather*} \frac {3 \, \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 4 \, \cos \left (d x + c\right ) \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 1}{a^{3} d \cos \left (d x + c\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\tan ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \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. 112 vs.
\(2 (46) = 92\).
time = 1.81, size = 112, normalized size = 2.43 \begin {gather*} \frac {\frac {\log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} + \frac {3 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3}} - \frac {\frac {3 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 1}{a^{3} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.27, size = 72, normalized size = 1.57 \begin {gather*} \frac {3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}{a^3\,d}-\frac {2}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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