Optimal. Leaf size=34 \[ \frac {\tan (c+d x)}{a^2 d}-\frac {2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac {x}{a^2} \]
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Rubi [A] time = 0.07, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3888, 3773, 3770, 3767, 8} \[ \frac {\tan (c+d x)}{a^2 d}-\frac {2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac {x}{a^2} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3773
Rule 3888
Rubi steps
\begin {align*} \int \frac {\tan ^4(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=\frac {\int (-a+a \sec (c+d x))^2 \, dx}{a^4}\\ &=\frac {x}{a^2}+\frac {\int \sec ^2(c+d x) \, dx}{a^2}-\frac {2 \int \sec (c+d x) \, dx}{a^2}\\ &=\frac {x}{a^2}-\frac {2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac {\operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{a^2 d}\\ &=\frac {x}{a^2}-\frac {2 \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac {\tan (c+d x)}{a^2 d}\\ \end {align*}
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Mathematica [B] time = 0.52, size = 177, normalized size = 5.21 \[ \frac {4 \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \left (\frac {\sin (d x)}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-2 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+d x\right )}{a^2 d (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 66, normalized size = 1.94 \[ \frac {d x \cos \left (d x + c\right ) - \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) + \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + \sin \left (d x + c\right )}{a^{2} d \cos \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.66, size = 79, normalized size = 2.32 \[ \frac {\frac {d x + c}{a^{2}} - \frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a^{2}} + \frac {2 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a^{2}} - \frac {2 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )} a^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.40, size = 102, normalized size = 3.00 \[ -\frac {1}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2} d}-\frac {1}{a^{2} d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {2 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2} d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 123, normalized size = 3.62 \[ \frac {2 \, {\left (\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{2}} + \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{2}} + \frac {\sin \left (d x + c\right )}{{\left (a^{2} - \frac {a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.19, size = 61, normalized size = 1.79 \[ \frac {x}{a^2}-\frac {4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-a^2\right )} \]
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 \[ \frac {\int \frac {\tan ^{4}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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