Optimal. Leaf size=159 \[ \frac{244 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{88 \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{4 \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{12 \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.368714, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3816, 4019, 4008, 3787, 3770, 3767, 8} \[ \frac{244 \tan (c+d x)}{105 a^4 d}-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{88 \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{4 \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{\tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{12 \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 3816
Rule 4019
Rule 4008
Rule 3787
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\sec ^6(c+d x)}{(a+a \sec (c+d x))^4} \, dx &=-\frac{\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{\int \frac{\sec ^4(c+d x) (4 a-8 a \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec ^3(c+d x) \left (36 a^2-52 a^2 \sec (c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec ^2(c+d x) \left (176 a^3-244 a^3 \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=-\frac{88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac{\int \sec (c+d x) \left (-420 a^4+244 a^4 \sec (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac{88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}+\frac{244 \int \sec ^2(c+d x) \, dx}{105 a^4}-\frac{4 \int \sec (c+d x) \, dx}{a^4}\\ &=-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}-\frac{244 \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=-\frac{4 \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{244 \tan (c+d x)}{105 a^4 d}-\frac{88 \sec ^2(c+d x) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{\sec ^4(c+d x) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{12 \sec ^3(c+d x) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{4 \tan (c+d x)}{d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 1.22005, size = 349, normalized size = 2.19 \[ \frac{\cos \left (\frac{1}{2} (c+d x)\right ) \sec ^4(c+d x) \left (\sec \left (\frac{c}{2}\right ) \sec (c) \left (-20524 \sin \left (c-\frac{d x}{2}\right )+14644 \sin \left (c+\frac{d x}{2}\right )-16660 \sin \left (2 c+\frac{d x}{2}\right )-4690 \sin \left (c+\frac{3 d x}{2}\right )+14378 \sin \left (2 c+\frac{3 d x}{2}\right )-9100 \sin \left (3 c+\frac{3 d x}{2}\right )+11668 \sin \left (c+\frac{5 d x}{2}\right )-630 \sin \left (2 c+\frac{5 d x}{2}\right )+9358 \sin \left (3 c+\frac{5 d x}{2}\right )-2940 \sin \left (4 c+\frac{5 d x}{2}\right )+4228 \sin \left (2 c+\frac{7 d x}{2}\right )+315 \sin \left (3 c+\frac{7 d x}{2}\right )+3493 \sin \left (4 c+\frac{7 d x}{2}\right )-420 \sin \left (5 c+\frac{7 d x}{2}\right )+664 \sin \left (3 c+\frac{9 d x}{2}\right )+105 \sin \left (4 c+\frac{9 d x}{2}\right )+559 \sin \left (5 c+\frac{9 d x}{2}\right )-10780 \sin \left (\frac{d x}{2}\right )+18788 \sin \left (\frac{3 d x}{2}\right )\right ) \sec (c+d x)+107520 \cos ^7\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{1680 a^4 d (\sec (c+d x)+1)^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.04, size = 158, normalized size = 1. \begin{align*}{\frac{1}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{7}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{23}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{49}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{1}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-4\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }{d{a}^{4}}}-{\frac{1}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+4\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21242, size = 251, normalized size = 1.58 \begin{align*} \frac{\frac{1680 \, \sin \left (d x + c\right )}{{\left (a^{4} - \frac{a^{4} \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 )}} + \frac{\frac{5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{3360 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{4}} + \frac{3360 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{4}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04913, size = 632, normalized size = 3.97 \begin{align*} -\frac{210 \,{\left (\cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 210 \,{\left (\cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} + 6 \, \cos \left (d x + c\right )^{3} + 4 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (664 \, \cos \left (d x + c\right )^{4} + 2236 \, \cos \left (d x + c\right )^{3} + 2636 \, \cos \left (d x + c\right )^{2} + 1184 \, \cos \left (d x + c\right ) + 105\right )} \sin \left (d x + c\right )}{105 \,{\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\sec ^{6}{\left (c + d x \right )}}{\sec ^{4}{\left (c + d x \right )} + 4 \sec ^{3}{\left (c + d x \right )} + 6 \sec ^{2}{\left (c + d x \right )} + 4 \sec{\left (c + d x \right )} + 1}\, dx}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37133, size = 188, normalized size = 1.18 \begin{align*} -\frac{\frac{3360 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{3360 \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac{1680 \, \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^{4}} - \frac{15 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 147 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 805 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5145 \, a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{28}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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