Optimal. Leaf size=65 \[ \frac {\log (\cos (c+d x))}{a^2 d}+\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}+\frac {\sec ^4(c+d x)}{4 a^2 d} \]
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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, 76}
\begin {gather*} \frac {\sec ^4(c+d x)}{4 a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}+\frac {2 \sec (c+d x)}{a^2 d}+\frac {\log (\cos (c+d x))}{a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 3964
Rubi steps
\begin {align*} \int \frac {\tan ^7(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-a x)^3 (a+a x)}{x^5} \, dx,x,\cos (c+d x)\right )}{a^6 d}\\ &=-\frac {\text {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^6 d}\\ &=\frac {\log (\cos (c+d x))}{a^2 d}+\frac {2 \sec (c+d x)}{a^2 d}-\frac {2 \sec ^3(c+d x)}{3 a^2 d}+\frac {\sec ^4(c+d x)}{4 a^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 83, normalized size = 1.28 \begin {gather*} \frac {(20 \cos (c+d x)+3 (2+4 \cos (3 (c+d x))+3 \log (\cos (c+d x))+4 \cos (2 (c+d x)) \log (\cos (c+d x))+\cos (4 (c+d x)) \log (\cos (c+d x)))) \sec ^4(c+d x)}{24 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 45, normalized size = 0.69
method | result | size |
derivativedivides | \(-\frac {-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{4}+\frac {2 \left (\sec ^{3}\left (d x +c \right )\right )}{3}-2 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{2}}\) | \(45\) |
default | \(-\frac {-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{4}+\frac {2 \left (\sec ^{3}\left (d x +c \right )\right )}{3}-2 \sec \left (d x +c \right )+\ln \left (\sec \left (d x +c \right )\right )}{d \,a^{2}}\) | \(45\) |
risch | \(-\frac {i x}{a^{2}}-\frac {2 i c}{a^{2} d}+\frac {4 \,{\mathrm e}^{7 i \left (d x +c \right )}+\frac {20 \,{\mathrm e}^{5 i \left (d x +c \right )}}{3}+4 \,{\mathrm e}^{4 i \left (d x +c \right )}+\frac {20 \,{\mathrm e}^{3 i \left (d x +c \right )}}{3}+4 \,{\mathrm e}^{i \left (d x +c \right )}}{d \,a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{a^{2} d}\) | \(115\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 50, normalized size = 0.77 \begin {gather*} \frac {\frac {12 \, \log \left (\cos \left (d x + c\right )\right )}{a^{2}} + \frac {24 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right ) + 3}{a^{2} \cos \left (d x + c\right )^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.51, size = 55, normalized size = 0.85 \begin {gather*} \frac {12 \, \cos \left (d x + c\right )^{4} \log \left (-\cos \left (d x + c\right )\right ) + 24 \, \cos \left (d x + c\right )^{3} - 8 \, \cos \left (d x + c\right ) + 3}{12 \, a^{2} d \cos \left (d x + c\right )^{4}} \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 ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \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. 180 vs.
\(2 (61) = 122\).
time = 3.69, size = 180, normalized size = 2.77 \begin {gather*} -\frac {\frac {12 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}} - \frac {12 \, \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{2}} - \frac {\frac {4 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {54 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {124 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {25 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 7}{a^{2} {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{4}}}{12 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.87, size = 135, normalized size = 2.08 \begin {gather*} \frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {8}{3}}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^2\right )}-\frac {2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{a^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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