Optimal. Leaf size=218 \[ -\frac{4 (34 A-19 B) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(23 A-13 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{x (23 A-13 B)}{2 a^3}-\frac{(13 A-8 B) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
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Rubi [A] time = 0.49492, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {4020, 3787, 2633, 2635, 8} \[ -\frac{4 (34 A-19 B) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(23 A-13 B) \sin (c+d x) \cos ^2(c+d x)}{3 d \left (a^3 \sec (c+d x)+a^3\right )}-\frac{x (23 A-13 B)}{2 a^3}-\frac{(13 A-8 B) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 4020
Rule 3787
Rule 2633
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx &=-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}+\frac{\int \frac{\cos ^3(c+d x) (a (8 A-3 B)-5 a (A-B) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}+\frac{\int \frac{\cos ^3(c+d x) \left (3 a^2 (21 A-11 B)-4 a^2 (13 A-8 B) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{\int \cos ^3(c+d x) \left (12 a^3 (34 A-19 B)-15 a^3 (23 A-13 B) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{(4 (34 A-19 B)) \int \cos ^3(c+d x) \, dx}{5 a^3}-\frac{(23 A-13 B) \int \cos ^2(c+d x) \, dx}{a^3}\\ &=-\frac{(23 A-13 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{(23 A-13 B) \int 1 \, dx}{2 a^3}-\frac{(4 (34 A-19 B)) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a^3 d}\\ &=-\frac{(23 A-13 B) x}{2 a^3}+\frac{4 (34 A-19 B) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B) \cos (c+d x) \sin (c+d x)}{2 a^3 d}-\frac{(A-B) \cos ^2(c+d x) \sin (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(13 A-8 B) \cos ^2(c+d x) \sin (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(23 A-13 B) \cos ^2(c+d x) \sin (c+d x)}{3 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{4 (34 A-19 B) \sin ^3(c+d x)}{15 a^3 d}\\ \end{align*}
Mathematica [B] time = 1.04269, size = 491, normalized size = 2.25 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-600 d x (23 A-13 B) \cos \left (c+\frac{d x}{2}\right )-600 d x (23 A-13 B) \cos \left (\frac{d x}{2}\right )-11110 A \sin \left (c+\frac{d x}{2}\right )+15380 A \sin \left (c+\frac{3 d x}{2}\right )-380 A \sin \left (2 c+\frac{3 d x}{2}\right )+4777 A \sin \left (2 c+\frac{5 d x}{2}\right )+1625 A \sin \left (3 c+\frac{5 d x}{2}\right )+230 A \sin \left (3 c+\frac{7 d x}{2}\right )+230 A \sin \left (4 c+\frac{7 d x}{2}\right )-20 A \sin \left (4 c+\frac{9 d x}{2}\right )-20 A \sin \left (5 c+\frac{9 d x}{2}\right )+5 A \sin \left (5 c+\frac{11 d x}{2}\right )+5 A \sin \left (6 c+\frac{11 d x}{2}\right )-6900 A d x \cos \left (c+\frac{3 d x}{2}\right )-6900 A d x \cos \left (2 c+\frac{3 d x}{2}\right )-1380 A d x \cos \left (2 c+\frac{5 d x}{2}\right )-1380 A d x \cos \left (3 c+\frac{5 d x}{2}\right )+20410 A \sin \left (\frac{d x}{2}\right )+7560 B \sin \left (c+\frac{d x}{2}\right )-9230 B \sin \left (c+\frac{3 d x}{2}\right )+930 B \sin \left (2 c+\frac{3 d x}{2}\right )-2782 B \sin \left (2 c+\frac{5 d x}{2}\right )-750 B \sin \left (3 c+\frac{5 d x}{2}\right )-105 B \sin \left (3 c+\frac{7 d x}{2}\right )-105 B \sin \left (4 c+\frac{7 d x}{2}\right )+15 B \sin \left (4 c+\frac{9 d x}{2}\right )+15 B \sin \left (5 c+\frac{9 d x}{2}\right )+3900 B d x \cos \left (c+\frac{3 d x}{2}\right )+3900 B d x \cos \left (2 c+\frac{3 d x}{2}\right )+780 B d x \cos \left (2 c+\frac{5 d x}{2}\right )+780 B d x \cos \left (3 c+\frac{5 d x}{2}\right )-12760 B \sin \left (\frac{d x}{2}\right )\right )}{480 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 362, normalized size = 1.7 \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{5\,A}{6\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{2\,B}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}+{\frac{49\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{31\,B}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+17\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}A}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}-7\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{5}B}{d{a}^{3} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{3}}}+{\frac{76\,A}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3} \left ( 1+ \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2} \right ) ^{-3}}-12\,{\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 ) ^{3}}}+11\,{\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 ) ^{3}}}-5\,{\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 ) ^{3}}}-23\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}+13\,{\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 [B] time = 1.50998, size = 556, normalized size = 2.55 \begin{align*} \frac{A{\left (\frac{20 \,{\left (\frac{33 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{76 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{51 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a^{3} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} + \frac{\frac{735 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{50 \, \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{1380 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} - B{\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 )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.498422, size = 540, normalized size = 2.48 \begin{align*} -\frac{15 \,{\left (23 \, A - 13 \, B\right )} d x \cos \left (d x + c\right )^{3} + 45 \,{\left (23 \, A - 13 \, B\right )} d x \cos \left (d x + c\right )^{2} + 45 \,{\left (23 \, A - 13 \, B\right )} d x \cos \left (d x + c\right ) + 15 \,{\left (23 \, A - 13 \, B\right )} d x -{\left (10 \, A \cos \left (d x + c\right )^{5} - 15 \,{\left (A - B\right )} \cos \left (d x + c\right )^{4} + 5 \,{\left (19 \, A - 9 \, B\right )} \cos \left (d x + c\right )^{3} +{\left (869 \, A - 479 \, B\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (429 \, A - 239 \, B\right )} \cos \left (d x + c\right ) + 544 \, A - 304 \, 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(-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 = 1.34235, size = 308, normalized size = 1.41 \begin{align*} -\frac{\frac{30 \,{\left (d x + c\right )}{\left (23 \, A - 13 \, B\right )}}{a^{3}} - \frac{20 \,{\left (51 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 76 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 33 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, 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 )}^{3} 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} - 50 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, B a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 735 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 465 \, 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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