Optimal. Leaf size=187 \[ -\frac{8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{4 (19 A-9 B) \sin (c+d x) \cos (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{x (13 A-6 B)}{2 a^3}-\frac{(11 A-6 B) \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.469506, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {4020, 3787, 2635, 8, 2637} \[ -\frac{8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{4 (19 A-9 B) \sin (c+d x) \cos (c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{x (13 A-6 B)}{2 a^3}-\frac{(11 A-6 B) \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4020
Rule 3787
Rule 2635
Rule 8
Rule 2637
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx &=-\frac{(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos ^2(c+d x) (a (7 A-2 B)-4 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{\int \frac{\cos ^2(c+d x) \left (a^2 (43 A-18 B)-3 a^2 (11 A-6 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\int \cos ^2(c+d x) \left (15 a^3 (13 A-6 B)-8 a^3 (19 A-9 B) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(8 (19 A-9 B)) \int \cos (c+d x) \, dx}{15 a^3}+\frac{(13 A-6 B) \int \cos ^2(c+d x) \, dx}{a^3}\\ &=-\frac{8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{(13 A-6 B) \int 1 \, dx}{2 a^3}\\ &=\frac{(13 A-6 B) x}{2 a^3}-\frac{8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A-B) \cos (c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(11 A-6 B) \cos (c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{4 (19 A-9 B) \cos (c+d x) \sin (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 0.786978, size = 435, normalized size = 2.33 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (600 d x (13 A-6 B) \cos \left (c+\frac{d x}{2}\right )+600 d x (13 A-6 B) \cos \left (\frac{d x}{2}\right )+7560 A \sin \left (c+\frac{d x}{2}\right )-9230 A \sin \left (c+\frac{3 d x}{2}\right )+930 A \sin \left (2 c+\frac{3 d x}{2}\right )-2782 A \sin \left (2 c+\frac{5 d x}{2}\right )-750 A \sin \left (3 c+\frac{5 d x}{2}\right )-105 A \sin \left (3 c+\frac{7 d x}{2}\right )-105 A \sin \left (4 c+\frac{7 d x}{2}\right )+15 A \sin \left (4 c+\frac{9 d x}{2}\right )+15 A \sin \left (5 c+\frac{9 d x}{2}\right )+3900 A d x \cos \left (c+\frac{3 d x}{2}\right )+3900 A d x \cos \left (2 c+\frac{3 d x}{2}\right )+780 A d x \cos \left (2 c+\frac{5 d x}{2}\right )+780 A d x \cos \left (3 c+\frac{5 d x}{2}\right )-12760 A \sin \left (\frac{d x}{2}\right )-4500 B \sin \left (c+\frac{d x}{2}\right )+4860 B \sin \left (c+\frac{3 d x}{2}\right )-900 B \sin \left (2 c+\frac{3 d x}{2}\right )+1452 B \sin \left (2 c+\frac{5 d x}{2}\right )+300 B \sin \left (3 c+\frac{5 d x}{2}\right )+60 B \sin \left (3 c+\frac{7 d x}{2}\right )+60 B \sin \left (4 c+\frac{7 d x}{2}\right )-1800 B d x \cos \left (c+\frac{3 d x}{2}\right )-1800 B d x \cos \left (2 c+\frac{3 d x}{2}\right )-360 B d x \cos \left (2 c+\frac{5 d x}{2}\right )-360 B d x \cos \left (3 c+\frac{5 d x}{2}\right )+7020 B \sin \left (\frac{d x}{2}\right )\right )}{480 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.101, size = 292, normalized size = 1.6 \begin{align*} -{\frac{A}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{B}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}+{\frac{2\,A}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{B}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{31\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{17\,B}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-7\,{\frac{A \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{B \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-5\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+13\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-6\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{d{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50664, size = 435, normalized size = 2.33 \begin{align*} -\frac{A{\left (\frac{60 \,{\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{\frac{465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{780 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - 3 \, B{\left (\frac{40 \, \sin \left (d x + c\right )}{{\left (a^{3} + \frac{a^{3} \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{85 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{10 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{\sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{120 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.484462, size = 495, normalized size = 2.65 \begin{align*} \frac{15 \,{\left (13 \, A - 6 \, B\right )} d x \cos \left (d x + c\right )^{3} + 45 \,{\left (13 \, A - 6 \, B\right )} d x \cos \left (d x + c\right )^{2} + 45 \,{\left (13 \, A - 6 \, B\right )} d x \cos \left (d x + c\right ) + 15 \,{\left (13 \, A - 6 \, B\right )} d x +{\left (15 \, A \cos \left (d x + c\right )^{4} - 15 \,{\left (3 \, A - 2 \, B\right )} \cos \left (d x + c\right )^{3} -{\left (479 \, A - 234 \, B\right )} \cos \left (d x + c\right )^{2} - 3 \,{\left (239 \, A - 114 \, B\right )} \cos \left (d x + c\right ) - 304 \, A + 144 \, B\right )} \sin \left (d x + c\right )}{30 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} 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 \cos ^{2}{\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 + \int \frac{B \cos ^{2}{\left (c + d x \right )} \sec{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27541, size = 270, normalized size = 1.44 \begin{align*} \frac{\frac{30 \,{\left (d x + c\right )}{\left (13 \, A - 6 \, B\right )}}{a^{3}} - \frac{60 \,{\left (7 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 2 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 5 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \, B \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}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 3 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 40 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 30 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 465 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 255 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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