Optimal. Leaf size=174 \[ \frac {(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}-\frac {(3 A-4 B+4 C) \sin (c+d x)}{a d}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac {(4 A-4 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {3 (4 A-4 B+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac {3 x (4 A-4 B+5 C)}{8 a} \]
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Rubi [A] time = 0.23, antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3041, 2748, 2633, 2635, 8} \[ \frac {(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}-\frac {(3 A-4 B+4 C) \sin (c+d x)}{a d}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{d (a \cos (c+d x)+a)}+\frac {(4 A-4 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac {3 (4 A-4 B+5 C) \sin (c+d x) \cos (c+d x)}{8 a d}+\frac {3 x (4 A-4 B+5 C)}{8 a} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2633
Rule 2635
Rule 2748
Rule 3041
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+a \cos (c+d x)} \, dx &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int \cos ^3(c+d x) (-a (3 A-4 B+4 C)+a (4 A-4 B+5 C) \cos (c+d x)) \, dx}{a^2}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {(3 A-4 B+4 C) \int \cos ^3(c+d x) \, dx}{a}+\frac {(4 A-4 B+5 C) \int \cos ^4(c+d x) \, dx}{a}\\ &=\frac {(4 A-4 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 (4 A-4 B+5 C)) \int \cos ^2(c+d x) \, dx}{4 a}+\frac {(3 A-4 B+4 C) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{a d}\\ &=-\frac {(3 A-4 B+4 C) \sin (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(4 A-4 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}+\frac {(3 (4 A-4 B+5 C)) \int 1 \, dx}{8 a}\\ &=\frac {3 (4 A-4 B+5 C) x}{8 a}-\frac {(3 A-4 B+4 C) \sin (c+d x)}{a d}+\frac {3 (4 A-4 B+5 C) \cos (c+d x) \sin (c+d x)}{8 a d}+\frac {(4 A-4 B+5 C) \cos ^3(c+d x) \sin (c+d x)}{4 a d}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac {(3 A-4 B+4 C) \sin ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [B] time = 0.78, size = 393, normalized size = 2.26 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (72 d x (4 A-4 B+5 C) \cos \left (c+\frac {d x}{2}\right )+72 d x (4 A-4 B+5 C) \cos \left (\frac {d x}{2}\right )-96 A \sin \left (c+\frac {d x}{2}\right )-72 A \sin \left (c+\frac {3 d x}{2}\right )-72 A \sin \left (2 c+\frac {3 d x}{2}\right )+24 A \sin \left (2 c+\frac {5 d x}{2}\right )+24 A \sin \left (3 c+\frac {5 d x}{2}\right )-480 A \sin \left (\frac {d x}{2}\right )+168 B \sin \left (c+\frac {d x}{2}\right )+144 B \sin \left (c+\frac {3 d x}{2}\right )+144 B \sin \left (2 c+\frac {3 d x}{2}\right )-16 B \sin \left (2 c+\frac {5 d x}{2}\right )-16 B \sin \left (3 c+\frac {5 d x}{2}\right )+8 B \sin \left (3 c+\frac {7 d x}{2}\right )+8 B \sin \left (4 c+\frac {7 d x}{2}\right )+552 B \sin \left (\frac {d x}{2}\right )-168 C \sin \left (c+\frac {d x}{2}\right )-120 C \sin \left (c+\frac {3 d x}{2}\right )-120 C \sin \left (2 c+\frac {3 d x}{2}\right )+40 C \sin \left (2 c+\frac {5 d x}{2}\right )+40 C \sin \left (3 c+\frac {5 d x}{2}\right )-5 C \sin \left (3 c+\frac {7 d x}{2}\right )-5 C \sin \left (4 c+\frac {7 d x}{2}\right )+3 C \sin \left (4 c+\frac {9 d x}{2}\right )+3 C \sin \left (5 c+\frac {9 d x}{2}\right )-552 C \sin \left (\frac {d x}{2}\right )\right )}{192 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 134, normalized size = 0.77 \[ \frac {9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} d x \cos \left (d x + c\right ) + 9 \, {\left (4 \, A - 4 \, B + 5 \, C\right )} d x + {\left (6 \, C \cos \left (d x + c\right )^{4} + 2 \, {\left (4 \, B - C\right )} \cos \left (d x + c\right )^{3} + {\left (12 \, A - 4 \, B + 13 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (12 \, A - 28 \, B + 19 \, C\right )} \cos \left (d x + c\right ) - 48 \, A + 64 \, B - 64 \, C\right )} \sin \left (d x + c\right )}{24 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 249, normalized size = 1.43 \[ \frac {\frac {9 \, {\left (d x + c\right )} {\left (4 \, A - 4 \, B + 5 \, C\right )}}{a} - \frac {24 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} - \frac {2 \, {\left (36 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 60 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 75 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 124 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 115 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 60 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 100 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 109 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 21 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4} a}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 526, normalized size = 3.02 \[ -\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {3 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {25 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {5 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {7 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {115 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{12 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {31 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {5 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {109 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{12 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {25 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{3 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {7 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) A}{a d}-\frac {3 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d}+\frac {15 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{4 a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 525, normalized size = 3.02 \[ -\frac {C {\left (\frac {\frac {21 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {109 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {115 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {75 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a + \frac {4 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {a \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}} - \frac {45 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {12 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 4 \, B {\left (\frac {\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {3 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} + 12 \, A {\left (\frac {\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a + \frac {2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac {3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} + \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.03, size = 189, normalized size = 1.09 \[ \frac {3\,x\,\left (4\,A-4\,B+5\,C\right )}{8\,a}-\frac {\left (3\,A-5\,B+\frac {25\,C}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (7\,A-\frac {31\,B}{3}+\frac {115\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (5\,A-\frac {25\,B}{3}+\frac {109\,C}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (A-3\,B+\frac {7\,C}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.68, size = 2688, normalized size = 15.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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