Optimal. Leaf size=148 \[ -\frac{2 (80 A-3 B-4 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{A x}{a^4}-\frac{(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
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Rubi [A] time = 0.283938, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {4052, 3922, 3919, 3794} \[ -\frac{2 (80 A-3 B-4 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{A x}{a^4}-\frac{(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4} \]
Antiderivative was successfully verified.
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Rule 4052
Rule 3922
Rule 3919
Rule 3794
Rubi steps
\begin{align*} \int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx &=-\frac{(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{\int \frac{-7 a A+a (3 A-3 B-4 C) \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac{(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}+\frac{\int \frac{35 a^2 A-2 a^2 (10 A-3 B-4 C) \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx}{35 a^4}\\ &=-\frac{(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{\int \frac{-105 a^3 A+a^3 (55 A-6 B-8 C) \sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^6}\\ &=\frac{A x}{a^4}-\frac{(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{(2 (80 A-3 B-4 C)) \int \frac{\sec (c+d x)}{a+a \sec (c+d x)} \, dx}{105 a^3}\\ &=\frac{A x}{a^4}-\frac{(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (1+\sec (c+d x))^2}-\frac{(A-B+C) \tan (c+d x)}{7 d (a+a \sec (c+d x))^4}-\frac{(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a+a \sec (c+d x))^3}-\frac{2 (80 A-3 B-4 C) \tan (c+d x)}{105 d \left (a^4+a^4 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 1.36193, size = 405, normalized size = 2.74 \[ \frac{\sec \left (\frac{c}{2}\right ) \sec ^7\left (\frac{1}{2} (c+d x)\right ) \left (8260 A \sin \left (c+\frac{d x}{2}\right )-7140 A \sin \left (c+\frac{3 d x}{2}\right )+3780 A \sin \left (2 c+\frac{3 d x}{2}\right )-2800 A \sin \left (2 c+\frac{5 d x}{2}\right )+840 A \sin \left (3 c+\frac{5 d x}{2}\right )-520 A \sin \left (3 c+\frac{7 d x}{2}\right )+3675 A d x \cos \left (c+\frac{d x}{2}\right )+2205 A d x \cos \left (c+\frac{3 d x}{2}\right )+2205 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+735 A d x \cos \left (2 c+\frac{5 d x}{2}\right )+735 A d x \cos \left (3 c+\frac{5 d x}{2}\right )+105 A d x \cos \left (3 c+\frac{7 d x}{2}\right )+105 A d x \cos \left (4 c+\frac{7 d x}{2}\right )-9940 A \sin \left (\frac{d x}{2}\right )+3675 A d x \cos \left (\frac{d x}{2}\right )-1260 B \sin \left (c+\frac{d x}{2}\right )+882 B \sin \left (c+\frac{3 d x}{2}\right )-630 B \sin \left (2 c+\frac{3 d x}{2}\right )+294 B \sin \left (2 c+\frac{5 d x}{2}\right )-210 B \sin \left (3 c+\frac{5 d x}{2}\right )+72 B \sin \left (3 c+\frac{7 d x}{2}\right )+1260 B \sin \left (\frac{d x}{2}\right )-350 C \sin \left (c+\frac{d x}{2}\right )+336 C \sin \left (c+\frac{3 d x}{2}\right )-210 C \sin \left (2 c+\frac{3 d x}{2}\right )+182 C \sin \left (2 c+\frac{5 d x}{2}\right )+26 C \sin \left (3 c+\frac{7 d x}{2}\right )+560 C \sin \left (\frac{d x}{2}\right )\right )}{13440 a^4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.085, size = 255, normalized size = 1.7 \begin{align*}{\frac{A}{56\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{7}}-{\frac{B}{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{A}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{3\,B}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{C}{40\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{11\,A}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{B}{8\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{C}{24\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{15\,A}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{B}{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 ) }+2\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.45776, size = 386, normalized size = 2.61 \begin{align*} -\frac{5 \, A{\left (\frac{\frac{315 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{77 \, \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{3 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac{336 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - \frac{C{\left (\frac{105 \, \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{15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} - \frac{3 \, B{\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 = 0.49751, size = 512, normalized size = 3.46 \begin{align*} \frac{105 \, A d x \cos \left (d x + c\right )^{4} + 420 \, A d x \cos \left (d x + c\right )^{3} + 630 \, A d x \cos \left (d x + c\right )^{2} + 420 \, A d x \cos \left (d x + c\right ) + 105 \, A d x -{\left ({\left (260 \, A - 36 \, B - 13 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (620 \, A - 39 \, B - 52 \, C\right )} \cos \left (d x + c\right )^{2} +{\left (535 \, A - 24 \, B - 32 \, C\right )} \cos \left (d x + c\right ) + 160 \, A - 6 \, B - 8 \, C\right )} \sin \left (d x + c\right )}{105 \,{\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\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{A}{\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 + \int \frac{B \sec{\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 + \int \frac{C \sec ^{2}{\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 = 1.17882, size = 297, normalized size = 2.01 \begin{align*} \frac{\frac{840 \,{\left (d x + c\right )} A}{a^{4}} + \frac{15 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, B 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} - 105 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 63 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 21 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 385 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 105 \, B a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 35 \, C a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 1575 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 105 \, B 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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