Optimal. Leaf size=229 \[ \frac{8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}+\frac{(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{x (8 A-21 B)}{2 a^4}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.672043, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {2977, 2734} \[ \frac{8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}+\frac{(52 A-129 B) \sin (c+d x) \cos ^3(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac{4 (83 A-216 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)}-\frac{(8 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{x (8 A-21 B)}{2 a^4}+\frac{(A-B) \sin (c+d x) \cos ^5(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac{(A-2 B) \sin (c+d x) \cos ^4(c+d x)}{5 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2977
Rule 2734
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^4} \, dx &=\frac{(A-B) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{\int \frac{\cos ^4(c+d x) (5 a (A-B)-a (2 A-9 B) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=\frac{(A-B) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(A-2 B) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\cos ^3(c+d x) \left (28 a^2 (A-2 B)-a^2 (24 A-73 B) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac{(52 A-129 B) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac{(A-B) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(A-2 B) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac{\int \frac{\cos ^2(c+d x) \left (3 a^3 (52 A-129 B)-a^3 (176 A-477 B) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac{(52 A-129 B) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac{(A-B) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(A-2 B) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac{4 (83 A-216 B) \cos ^2(c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}+\frac{\int \cos (c+d x) \left (8 a^4 (83 A-216 B)-105 a^4 (8 A-21 B) \cos (c+d x)\right ) \, dx}{105 a^8}\\ &=-\frac{(8 A-21 B) x}{2 a^4}+\frac{8 (83 A-216 B) \sin (c+d x)}{105 a^4 d}-\frac{(8 A-21 B) \cos (c+d x) \sin (c+d x)}{2 a^4 d}+\frac{(52 A-129 B) \cos ^3(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}+\frac{(A-B) \cos ^5(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac{(A-2 B) \cos ^4(c+d x) \sin (c+d x)}{5 a d (a+a \cos (c+d x))^3}+\frac{4 (83 A-216 B) \cos ^2(c+d x) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [B] time = 1.25512, size = 555, normalized size = 2.42 \[ \frac{\sec \left (\frac{c}{2}\right ) \cos \left (\frac{1}{2} (c+d x)\right ) \left (-14700 d x (8 A-21 B) \cos \left (c+\frac{d x}{2}\right )-14700 d x (8 A-21 B) \cos \left (\frac{d x}{2}\right )-184520 A \sin \left (c+\frac{d x}{2}\right )+184464 A \sin \left (c+\frac{3 d x}{2}\right )-72240 A \sin \left (2 c+\frac{3 d x}{2}\right )+77168 A \sin \left (2 c+\frac{5 d x}{2}\right )-8400 A \sin \left (3 c+\frac{5 d x}{2}\right )+15164 A \sin \left (3 c+\frac{7 d x}{2}\right )+2940 A \sin \left (4 c+\frac{7 d x}{2}\right )+420 A \sin \left (4 c+\frac{9 d x}{2}\right )+420 A \sin \left (5 c+\frac{9 d x}{2}\right )-70560 A d x \cos \left (c+\frac{3 d x}{2}\right )-70560 A d x \cos \left (2 c+\frac{3 d x}{2}\right )-23520 A d x \cos \left (2 c+\frac{5 d x}{2}\right )-23520 A d x \cos \left (3 c+\frac{5 d x}{2}\right )-3360 A d x \cos \left (3 c+\frac{7 d x}{2}\right )-3360 A d x \cos \left (4 c+\frac{7 d x}{2}\right )+243320 A \sin \left (\frac{d x}{2}\right )+386190 B \sin \left (c+\frac{d x}{2}\right )-422478 B \sin \left (c+\frac{3 d x}{2}\right )+132930 B \sin \left (2 c+\frac{3 d x}{2}\right )-181461 B \sin \left (2 c+\frac{5 d x}{2}\right )+3675 B \sin \left (3 c+\frac{5 d x}{2}\right )-36003 B \sin \left (3 c+\frac{7 d x}{2}\right )-9555 B \sin \left (4 c+\frac{7 d x}{2}\right )-945 B \sin \left (4 c+\frac{9 d x}{2}\right )-945 B \sin \left (5 c+\frac{9 d x}{2}\right )+105 B \sin \left (5 c+\frac{11 d x}{2}\right )+105 B \sin \left (6 c+\frac{11 d x}{2}\right )+185220 B d x \cos \left (c+\frac{3 d x}{2}\right )+185220 B d x \cos \left (2 c+\frac{3 d x}{2}\right )+61740 B d x \cos \left (2 c+\frac{5 d x}{2}\right )+61740 B d x \cos \left (3 c+\frac{5 d x}{2}\right )+8820 B d x \cos \left (3 c+\frac{7 d x}{2}\right )+8820 B d x \cos \left (4 c+\frac{7 d x}{2}\right )-539490 B \sin \left (\frac{d x}{2}\right )\right )}{6720 a^4 d (\cos (c+d x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 332, 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{B}{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{9\,B}{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{13\,B}{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{111\,B}{8\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}A}{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-9\,{\frac{B \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}}{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}+2\,{\frac{A\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-7\,{\frac{B\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) ^{2}}}-8\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) A}{d{a}^{4}}}+21\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{d{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48977, size = 491, normalized size = 2.14 \begin{align*} -\frac{3 \, B{\left (\frac{280 \,{\left (\frac{7 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{9 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{4} + \frac{2 \, a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{3885 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{455 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{63 \, \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}}}{a^{4}} - \frac{5880 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} - 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{6720 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )}}{840 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42473, size = 647, normalized size = 2.83 \begin{align*} -\frac{105 \,{\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{4} + 420 \,{\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{3} + 630 \,{\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right )^{2} + 420 \,{\left (8 \, A - 21 \, B\right )} d x \cos \left (d x + c\right ) + 105 \,{\left (8 \, A - 21 \, B\right )} d x -{\left (105 \, B \cos \left (d x + c\right )^{5} + 210 \,{\left (A - 2 \, B\right )} \cos \left (d x + c\right )^{4} + 4 \,{\left (592 \, A - 1509 \, B\right )} \cos \left (d x + c\right )^{3} + 4 \,{\left (1318 \, A - 3411 \, B\right )} \cos \left (d x + c\right )^{2} +{\left (4472 \, A - 11619 \, B\right )} \cos \left (d x + c\right ) + 1328 \, A - 3456 \, B\right )} \sin \left (d x + c\right )}{210 \,{\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 [A] time = 91.6891, size = 1085, normalized size = 4.74 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28697, size = 315, normalized size = 1.38 \begin{align*} -\frac{\frac{420 \,{\left (d x + c\right )}{\left (8 \, A - 21 \, B\right )}}{a^{4}} - \frac{840 \,{\left (2 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 7 \, 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^{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} - 147 \, A a^{24} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 189 \, B 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} - 1365 \, B 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 ) + 11655 \, B 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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