Optimal. Leaf size=161 \[ \frac{2 (332 A+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 A \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
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Rubi [A] time = 0.616374, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3042, 2978, 2748, 3767, 8, 3770} \[ \frac{2 (332 A+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}-\frac{4 A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{4 A \tan (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2978
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx &=-\frac{(A+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{(a (8 A+C)-a (4 A-3 C) \cos (c+d x)) \sec ^2(c+d x)}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (a^2 (52 A+3 C)-6 a^2 (6 A-C) \cos (c+d x)\right ) \sec ^2(c+d x)}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\left (2 a^3 (122 A+3 C)-2 a^3 (88 A-3 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{4 A \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \left (2 a^4 (332 A+3 C)-420 a^4 A \cos (c+d x)\right ) \sec ^2(c+d x) \, dx}{105 a^8}\\ &=-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{4 A \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{(4 A) \int \sec (c+d x) \, dx}{a^4}+\frac{(2 (332 A+3 C)) \int \sec ^2(c+d x) \, dx}{105 a^4}\\ &=-\frac{4 A \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{4 A \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}-\frac{(2 (332 A+3 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 a^4 d}\\ &=-\frac{4 A \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{2 (332 A+3 C) \tan (c+d x)}{105 a^4 d}-\frac{(88 A-3 C) \tan (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac{(A+C) \tan (c+d x)}{7 d (a+a \cos (c+d x))^4}-\frac{2 (6 A-C) \tan (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac{4 A \tan (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 6.41192, size = 680, normalized size = 4.22 \[ \frac{\frac{\sec \left (\frac{c}{2}\right ) \sec (c) \cos (c+d x) \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (-20524 A \sin \left (c-\frac{d x}{2}\right )+14644 A \sin \left (c+\frac{d x}{2}\right )-16660 A \sin \left (2 c+\frac{d x}{2}\right )-4690 A \sin \left (c+\frac{3 d x}{2}\right )+14378 A \sin \left (2 c+\frac{3 d x}{2}\right )-9100 A \sin \left (3 c+\frac{3 d x}{2}\right )+11668 A \sin \left (c+\frac{5 d x}{2}\right )-630 A \sin \left (2 c+\frac{5 d x}{2}\right )+9358 A \sin \left (3 c+\frac{5 d x}{2}\right )-2940 A \sin \left (4 c+\frac{5 d x}{2}\right )+4228 A \sin \left (2 c+\frac{7 d x}{2}\right )+315 A \sin \left (3 c+\frac{7 d x}{2}\right )+3493 A \sin \left (4 c+\frac{7 d x}{2}\right )-420 A \sin \left (5 c+\frac{7 d x}{2}\right )+664 A \sin \left (3 c+\frac{9 d x}{2}\right )+105 A \sin \left (4 c+\frac{9 d x}{2}\right )+559 A \sin \left (5 c+\frac{9 d x}{2}\right )-10780 A \sin \left (\frac{d x}{2}\right )+18788 A \sin \left (\frac{3 d x}{2}\right )-126 C \sin \left (c-\frac{d x}{2}\right )+126 C \sin \left (c+\frac{d x}{2}\right )-210 C \sin \left (2 c+\frac{d x}{2}\right )+252 C \sin \left (2 c+\frac{3 d x}{2}\right )+132 C \sin \left (c+\frac{5 d x}{2}\right )+132 C \sin \left (3 c+\frac{5 d x}{2}\right )+42 C \sin \left (2 c+\frac{7 d x}{2}\right )+42 C \sin \left (4 c+\frac{7 d x}{2}\right )+6 C \sin \left (3 c+\frac{9 d x}{2}\right )+6 C \sin \left (5 c+\frac{9 d x}{2}\right )-210 C \sin \left (\frac{d x}{2}\right )+252 C \sin \left (\frac{3 d x}{2}\right )\right ) \left (A \sec ^2(c+d x)+C\right )}{840 d (\cos (c+d x)+1)^4 (2 A+C \cos (2 c+2 d x)+C)}+\frac{128 A \cos ^2(c+d x) \cos ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sec ^2(c+d x)+C\right ) \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )-\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (\cos (c+d x)+1)^4 (2 A+C \cos (2 c+2 d x)+C)}-\frac{128 A \cos ^2(c+d x) \cos ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (A \sec ^2(c+d x)+C\right ) \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )+\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d (\cos (c+d x)+1)^4 (2 A+C \cos (2 c+2 d x)+C)}}{a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 244, normalized size = 1.5 \begin{align*}{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{C}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}+{\frac{7\,A}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3\,C}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{23\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{C}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{49\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{A}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+4\,{\frac{A\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) -1 \right ) }{d{a}^{4}}}-{\frac{A}{d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-4\,{\frac{A\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 = 1.1494, size = 370, normalized size = 2.3 \begin{align*} \frac{A{\left (\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}}\right )} + \frac{3 \, C{\left (\frac{35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44844, size = 722, normalized size = 4.48 \begin{align*} -\frac{210 \,{\left (A \cos \left (d x + c\right )^{5} + 4 \, A \cos \left (d x + c\right )^{4} + 6 \, A \cos \left (d x + c\right )^{3} + 4 \, A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 210 \,{\left (A \cos \left (d x + c\right )^{5} + 4 \, A \cos \left (d x + c\right )^{4} + 6 \, A \cos \left (d x + c\right )^{3} + 4 \, A \cos \left (d x + c\right )^{2} + A \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) -{\left (2 \,{\left (332 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (559 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (2636 \, A + 39 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \,{\left (296 \, A + 9 \, C\right )} \cos \left (d x + c\right ) + 105 \, A\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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26265, size = 286, normalized size = 1.78 \begin{align*} -\frac{\frac{3360 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{4}} - \frac{3360 \, A \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{4}} + \frac{1680 \, A \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 a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 147 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 63 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 805 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5145 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 105 \, C 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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