Optimal. Leaf size=66 \[ -\frac {5 \tan (c+d x)}{2 a^3 d}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac {\tan (c+d x) (1-\sec (c+d x))}{2 a^3 d}-\frac {x}{a^3} \]
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Rubi [A] time = 0.09, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3888, 3775, 3914, 3767, 8, 3770} \[ -\frac {5 \tan (c+d x)}{2 a^3 d}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac {\tan (c+d x) (1-\sec (c+d x))}{2 a^3 d}-\frac {x}{a^3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3775
Rule 3888
Rule 3914
Rubi steps
\begin {align*} \int \frac {\tan ^6(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=\frac {\int (-a+a \sec (c+d x))^3 \, dx}{a^6}\\ &=-\frac {(1-\sec (c+d x)) \tan (c+d x)}{2 a^3 d}-\frac {\int (-a+a \sec (c+d x)) (-2 a+5 a \sec (c+d x)) \, dx}{2 a^5}\\ &=-\frac {x}{a^3}-\frac {(1-\sec (c+d x)) \tan (c+d x)}{2 a^3 d}-\frac {5 \int \sec ^2(c+d x) \, dx}{2 a^3}+\frac {7 \int \sec (c+d x) \, dx}{2 a^3}\\ &=-\frac {x}{a^3}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac {(1-\sec (c+d x)) \tan (c+d x)}{2 a^3 d}+\frac {5 \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 a^3 d}\\ &=-\frac {x}{a^3}+\frac {7 \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac {5 \tan (c+d x)}{2 a^3 d}-\frac {(1-\sec (c+d x)) \tan (c+d x)}{2 a^3 d}\\ \end {align*}
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Mathematica [B] time = 0.96, size = 241, normalized size = 3.65 \[ \frac {2 \cos ^6\left (\frac {1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (-\frac {12 \sin (d x)}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {1}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {1}{d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {14 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {14 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}-4 x\right )}{a^3 (\sec (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 87, normalized size = 1.32 \[ -\frac {4 \, d x \cos \left (d x + c\right )^{2} - 7 \, \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) + 7 \, \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{4 \, a^{3} d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 7.54, size = 97, normalized size = 1.47 \[ -\frac {\frac {2 \, {\left (d x + c\right )}}{a^{3}} - \frac {7 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} + \frac {7 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} - \frac {2 \, {\left (7 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.60, size = 144, normalized size = 2.18 \[ \frac {1}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {7}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a^{3} d}-\frac {1}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {7}{2 a^{3} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {7 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a^{3} d}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 171, normalized size = 2.59 \[ -\frac {\frac {2 \, {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {4 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}} - \frac {7 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac {7 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.31, size = 92, normalized size = 1.39 \[ \frac {7\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^3\,d}-\frac {x}{a^3}-\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\left (a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a^3\right )} \]
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 \[ \frac {\int \frac {\tan ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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