Optimal. Leaf size=65 \[ \frac {\log (\cos (c+d x))}{a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {3 \sec ^2(c+d x)}{2 a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d} \]
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Rubi [A]
time = 0.04, 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 = {3964, 45}
\begin {gather*} \frac {\sec ^3(c+d x)}{3 a^3 d}-\frac {3 \sec ^2(c+d x)}{2 a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}+\frac {\log (\cos (c+d x))}{a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3964
Rubi steps
\begin {align*} \int \frac {\tan ^7(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-a x)^3}{x^4} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\text {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 )}{a^6 d}\\ &=\frac {\log (\cos (c+d x))}{a^3 d}+\frac {3 \sec (c+d x)}{a^3 d}-\frac {3 \sec ^2(c+d x)}{2 a^3 d}+\frac {\sec ^3(c+d x)}{3 a^3 d}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 64, normalized size = 0.98 \begin {gather*} \frac {(22+18 \cos (2 (c+d x))+9 \cos (c+d x) (-2+\log (\cos (c+d x)))+3 \cos (3 (c+d x)) \log (\cos (c+d x))) \sec ^3(c+d x)}{12 a^3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 45, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sec ^{2}\left (d x +c \right )\right )}{2}-3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}\) | \(45\) |
default | \(-\frac {-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{3}+\frac {3 \left (\sec ^{2}\left (d x +c \right )\right )}{2}-3 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{3}}\) | \(45\) |
risch | \(-\frac {i x}{a^{3}}-\frac {2 i c}{a^{3} d}+\frac {6 \,{\mathrm e}^{5 i \left (d x +c \right )}-6 \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {44 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}-6 \,{\mathrm e}^{2 i \left (d x +c \right )}+6 \,{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{3} d}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 50, normalized size = 0.77 \begin {gather*} \frac {\frac {6 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} + \frac {18 \, \cos \left (d x + c\right )^{2} - 9 \, \cos \left (d x + c\right ) + 2}{a^{3} \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 = 3.67, size = 55, normalized size = 0.85 \begin {gather*} \frac {6 \, \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) + 18 \, \cos \left (d x + c\right )^{2} - 9 \, \cos \left (d x + c\right ) + 2}{6 \, a^{3} 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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\tan ^{7}{\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. 158 vs.
\(2 (61) = 122\).
time = 3.72, size = 158, normalized size = 2.43 \begin {gather*} -\frac {\frac {6 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{3}} - \frac {6 \, \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 {75 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {51 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {11 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 29}{a^{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 = 2.03, size = 109, normalized size = 1.68 \begin {gather*} -\frac {14\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {20}{3}}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^3\right )}-\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^3\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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