Optimal. Leaf size=174 \[ \frac {2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {4 C \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac {4 C x}{a^4}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {2 (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.58, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3042, 2977, 2968, 3023, 12, 2735, 2648} \[ \frac {2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\cos (c+d x)+1)^2}+\frac {4 C \sin (c+d x)}{a^4 d (\cos (c+d x)+1)}-\frac {4 C x}{a^4}-\frac {(A+C) \sin (c+d x) \cos ^4(c+d x)}{7 d (a \cos (c+d x)+a)^4}+\frac {2 (A-6 C) \sin (c+d x) \cos ^3(c+d x)}{35 a d (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2648
Rule 2735
Rule 2968
Rule 2977
Rule 3023
Rule 3042
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^4} \, dx &=-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {\int \frac {\cos ^3(c+d x) (a (3 A-4 C)+a (A+8 C) \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx}{7 a^2}\\ &=-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos ^2(c+d x) \left (6 a^2 (A-6 C)+a^2 (3 A+52 C) \cos (c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx}{35 a^4}\\ &=\frac {(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {\cos (c+d x) \left (2 a^3 (3 A-88 C)+2 a^3 (3 A+122 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac {(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int \frac {2 a^3 (3 A-88 C) \cos (c+d x)+2 a^3 (3 A+122 C) \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^6}\\ &=\frac {2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {\int -\frac {420 a^4 C \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{105 a^7}\\ &=\frac {2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}-\frac {(4 C) \int \frac {\cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=-\frac {4 C x}{a^4}+\frac {2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(4 C) \int \frac {1}{a+a \cos (c+d x)} \, dx}{a^3}\\ &=-\frac {4 C x}{a^4}+\frac {2 (3 A+122 C) \sin (c+d x)}{105 a^4 d}+\frac {(3 A-88 C) \cos ^2(c+d x) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^2}-\frac {(A+C) \cos ^4(c+d x) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {2 (A-6 C) \cos ^3(c+d x) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {4 C \sin (c+d x)}{d \left (a^4+a^4 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [B] time = 0.76, size = 371, normalized size = 2.13 \[ -\frac {\sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {1}{2} (c+d x)\right ) \left (2520 A \sin \left (c+\frac {d x}{2}\right )-1764 A \sin \left (c+\frac {3 d x}{2}\right )+1260 A \sin \left (2 c+\frac {3 d x}{2}\right )-588 A \sin \left (2 c+\frac {5 d x}{2}\right )+420 A \sin \left (3 c+\frac {5 d x}{2}\right )-144 A \sin \left (3 c+\frac {7 d x}{2}\right )-2520 A \sin \left (\frac {d x}{2}\right )+46130 C \sin \left (c+\frac {d x}{2}\right )-46116 C \sin \left (c+\frac {3 d x}{2}\right )+18060 C \sin \left (2 c+\frac {3 d x}{2}\right )-19292 C \sin \left (2 c+\frac {5 d x}{2}\right )+2100 C \sin \left (3 c+\frac {5 d x}{2}\right )-3791 C \sin \left (3 c+\frac {7 d x}{2}\right )-735 C \sin \left (4 c+\frac {7 d x}{2}\right )-105 C \sin \left (4 c+\frac {9 d x}{2}\right )-105 C \sin \left (5 c+\frac {9 d x}{2}\right )+29400 C d x \cos \left (c+\frac {d x}{2}\right )+17640 C d x \cos \left (c+\frac {3 d x}{2}\right )+17640 C d x \cos \left (2 c+\frac {3 d x}{2}\right )+5880 C d x \cos \left (2 c+\frac {5 d x}{2}\right )+5880 C d x \cos \left (3 c+\frac {5 d x}{2}\right )+840 C d x \cos \left (3 c+\frac {7 d x}{2}\right )+840 C d x \cos \left (4 c+\frac {7 d x}{2}\right )-60830 C \sin \left (\frac {d x}{2}\right )+29400 C d x \cos \left (\frac {d x}{2}\right )\right )}{26880 a^4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.73, size = 193, normalized size = 1.11 \[ -\frac {420 \, C d x \cos \left (d x + c\right )^{4} + 1680 \, C d x \cos \left (d x + c\right )^{3} + 2520 \, C d x \cos \left (d x + c\right )^{2} + 1680 \, C d x \cos \left (d x + c\right ) + 420 \, C d x - {\left (105 \, C \cos \left (d x + c\right )^{4} + 4 \, {\left (9 \, A + 296 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (39 \, A + 2636 \, C\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (6 \, A + 559 \, C\right )} \cos \left (d x + c\right ) + 6 \, A + 664 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.48, size = 184, normalized size = 1.06 \[ -\frac {\frac {3360 \, {\left (d x + c\right )} C}{a^{4}} - \frac {1680 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a^{4}} + \frac {15 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 63 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 147 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 805 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 105 \, A a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 5145 \, C a^{24} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{28}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 210, normalized size = 1.21 \[ -\frac {\left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{56 d \,a^{4}}-\frac {C \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 d \,a^{4}}+\frac {3 A \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}+\frac {7 C \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{40 d \,a^{4}}-\frac {\left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{8 d \,a^{4}}-\frac {23 C \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d \,a^{4}}+\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {49 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}+\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \,a^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}-\frac {8 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{d \,a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 246, normalized size = 1.41 \[ \frac {C {\left (\frac {1680 \, \sin \left (d x + c\right )}{{\left (a^{4} + \frac {a^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} + \frac {\frac {5145 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {805 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {147 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{4}} - \frac {6720 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{4}}\right )} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 192, normalized size = 1.10 \[ \frac {2\,C\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^4\,d}-\frac {\left (-\frac {12\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}-\frac {764\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\left (\frac {23\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}+\frac {143\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{105}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\left (-\frac {9\,A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{70}-\frac {8\,C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{35}\right )\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {A\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}+\frac {C\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{56}}{a^4\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}-\frac {4\,C\,x}{a^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 21.16, size = 462, normalized size = 2.66 \[ \begin {cases} - \frac {15 A \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {48 A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {42 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {105 A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 C d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {3360 C d x}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {15 C \tan ^{9}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {132 C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} - \frac {658 C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {4340 C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} + \frac {6825 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{840 a^{4} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 840 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + C \cos ^{2}{\relax (c )}\right ) \cos ^{3}{\relax (c )}}{\left (a \cos {\relax (c )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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