Optimal. Leaf size=277 \[ \frac {(11 A-15 B+19 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {(245 A-273 B+397 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{210 a^2 d}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(7 A-7 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt {a \cos (c+d x)+a}}-\frac {(35 A-63 B+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt {a \cos (c+d x)+a}}-\frac {(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.87, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3041, 2983, 2968, 3023, 2751, 2649, 206} \[ \frac {(245 A-273 B+397 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{210 a^2 d}+\frac {(11 A-15 B+19 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B+C) \sin (c+d x) \cos ^4(c+d x)}{2 d (a \cos (c+d x)+a)^{3/2}}+\frac {(7 A-7 B+11 C) \sin (c+d x) \cos ^3(c+d x)}{14 a d \sqrt {a \cos (c+d x)+a}}-\frac {(35 A-63 B+67 C) \sin (c+d x) \cos ^2(c+d x)}{70 a d \sqrt {a \cos (c+d x)+a}}-\frac {(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2751
Rule 2968
Rule 2983
Rule 3023
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))^{3/2}} \, dx &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {\int \frac {\cos ^3(c+d x) \left (-2 a (A-2 B+2 C)+\frac {1}{2} a (7 A-7 B+11 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}+\frac {(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\cos ^2(c+d x) \left (\frac {3}{2} a^2 (7 A-7 B+11 C)-\frac {1}{4} a^2 (35 A-63 B+67 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{7 a^3}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}+\frac {(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {2 \int \frac {\cos (c+d x) \left (-\frac {1}{2} a^3 (35 A-63 B+67 C)+\frac {1}{8} a^3 (245 A-273 B+397 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}+\frac {(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {2 \int \frac {-\frac {1}{2} a^3 (35 A-63 B+67 C) \cos (c+d x)+\frac {1}{8} a^3 (245 A-273 B+397 C) \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{35 a^4}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}+\frac {(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {(245 A-273 B+397 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac {4 \int \frac {\frac {1}{16} a^4 (245 A-273 B+397 C)-\frac {1}{8} a^4 (455 A-651 B+799 C) \cos (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{105 a^5}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt {a+a \cos (c+d x)}}-\frac {(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}+\frac {(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {(245 A-273 B+397 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}+\frac {(11 A-15 B+19 C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx}{4 a}\\ &=-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt {a+a \cos (c+d x)}}-\frac {(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}+\frac {(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {(245 A-273 B+397 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}-\frac {(11 A-15 B+19 C) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{2 a d}\\ &=\frac {(11 A-15 B+19 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A-B+C) \cos ^4(c+d x) \sin (c+d x)}{2 d (a+a \cos (c+d x))^{3/2}}-\frac {(455 A-651 B+799 C) \sin (c+d x)}{105 a d \sqrt {a+a \cos (c+d x)}}-\frac {(35 A-63 B+67 C) \cos ^2(c+d x) \sin (c+d x)}{70 a d \sqrt {a+a \cos (c+d x)}}+\frac {(7 A-7 B+11 C) \cos ^3(c+d x) \sin (c+d x)}{14 a d \sqrt {a+a \cos (c+d x)}}+\frac {(245 A-273 B+397 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{210 a^2 d}\\ \end {align*}
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Mathematica [A] time = 1.54, size = 180, normalized size = 0.65 \[ \frac {\frac {1}{2} \sin \left (\frac {1}{2} (c+d x)\right ) \cos ^3\left (\frac {1}{2} (c+d x)\right ) (6 (140 A-273 B+277 C) \cos (c+d x)-4 (35 A-21 B+64 C) \cos (2 (c+d x))+1190 A-42 B \cos (3 (c+d x))-1974 B+18 C \cos (3 (c+d x))-15 C \cos (4 (c+d x))+2161 C)-105 (11 A-15 B+19 C) \cos ^5\left (\frac {1}{2} (c+d x)\right ) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{105 d \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )-1\right ) (a (\cos (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 259, normalized size = 0.94 \[ \frac {105 \, \sqrt {2} {\left ({\left (11 \, A - 15 \, B + 19 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (11 \, A - 15 \, B + 19 \, C\right )} \cos \left (d x + c\right ) + 11 \, A - 15 \, B + 19 \, C\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \, {\left (60 \, C \cos \left (d x + c\right )^{4} + 12 \, {\left (7 \, B - 3 \, C\right )} \cos \left (d x + c\right )^{3} + 28 \, {\left (5 \, A - 3 \, B + 7 \, C\right )} \cos \left (d x + c\right )^{2} - 12 \, {\left (35 \, A - 63 \, B + 67 \, C\right )} \cos \left (d x + c\right ) - 665 \, A + 1029 \, B - 1201 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{840 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.50, size = 305, normalized size = 1.10 \[ -\frac {\frac {105 \, {\left (11 \, \sqrt {2} A - 15 \, \sqrt {2} B + 19 \, \sqrt {2} C\right )} \log \left ({\left | -\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{a^{\frac {3}{2}}} + \frac {{\left ({\left ({\left ({\left (\frac {105 \, {\left (\sqrt {2} A a^{5} - \sqrt {2} B a^{5} + \sqrt {2} C a^{5}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{3}} + \frac {4 \, {\left (455 \, \sqrt {2} A a^{5} - 693 \, \sqrt {2} B a^{5} + 877 \, \sqrt {2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {14 \, {\left (305 \, \sqrt {2} A a^{5} - 453 \, \sqrt {2} B a^{5} + 517 \, \sqrt {2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {140 \, {\left (25 \, \sqrt {2} A a^{5} - 39 \, \sqrt {2} B a^{5} + 47 \, \sqrt {2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \frac {105 \, {\left (9 \, \sqrt {2} A a^{5} - 17 \, \sqrt {2} B a^{5} + 17 \, \sqrt {2} C a^{5}\right )}}{a^{3}}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {7}{2}}}}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.77, size = 577, normalized size = 2.08 \[ \frac {\sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (960 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-96 \sqrt {2}\, \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (7 B +17 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+112 \sqrt {2}\, \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (5 A +6 B +16 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \sqrt {2}\, \left (8 A \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-33 A \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a -48 B \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+45 B \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a +16 C \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-57 \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1155 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a A -1575 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a B +1995 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a C -945 A \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+1785 B \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}-1785 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\right )}{420 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{\frac {5}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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