Optimal. Leaf size=236 \[ \frac {2 (21 A+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}-\frac {2 (21 A+29 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 a d}+\frac {4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}-\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}-\frac {2 C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.82, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3046, 2983, 2968, 3023, 2751, 2649, 206} \[ \frac {2 (21 A+19 C) \sin (c+d x) \cos ^2(c+d x)}{105 d \sqrt {a \cos (c+d x)+a}}-\frac {2 (21 A+29 C) \sin (c+d x) \sqrt {a \cos (c+d x)+a}}{315 a d}+\frac {4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt {a \cos (c+d x)+a}}-\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}+\frac {2 C \sin (c+d x) \cos ^4(c+d x)}{9 d \sqrt {a \cos (c+d x)+a}}-\frac {2 C \sin (c+d x) \cos ^3(c+d x)}{63 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2751
Rule 2968
Rule 2983
Rule 3023
Rule 3046
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \cos ^2(c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx &=\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {2 \int \frac {\cos ^3(c+d x) \left (\frac {1}{2} a (9 A+8 C)-\frac {1}{2} a C \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{9 a}\\ &=-\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {4 \int \frac {\cos ^2(c+d x) \left (-\frac {3 a^2 C}{2}+\frac {3}{4} a^2 (21 A+19 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{63 a^2}\\ &=\frac {2 (21 A+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {8 \int \frac {\cos (c+d x) \left (\frac {3}{2} a^3 (21 A+19 C)-\frac {3}{8} a^3 (21 A+29 C) \cos (c+d x)\right )}{\sqrt {a+a \cos (c+d x)}} \, dx}{315 a^3}\\ &=\frac {2 (21 A+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}+\frac {8 \int \frac {\frac {3}{2} a^3 (21 A+19 C) \cos (c+d x)-\frac {3}{8} a^3 (21 A+29 C) \cos ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{315 a^3}\\ &=\frac {2 (21 A+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (21 A+29 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 a d}+\frac {16 \int \frac {-\frac {3}{16} a^4 (21 A+29 C)+\frac {3}{8} a^4 (147 A+143 C) \cos (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{945 a^4}\\ &=\frac {4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (21 A+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (21 A+29 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 a d}+(-A-C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx\\ &=\frac {4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (21 A+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (21 A+29 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 a d}+\frac {(2 (A+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 )}{d}\\ &=-\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}+\frac {4 (147 A+143 C) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {2 (21 A+19 C) \cos ^2(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}-\frac {2 C \cos ^3(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 C \cos ^4(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}}-\frac {2 (21 A+29 C) \sqrt {a+a \cos (c+d x)} \sin (c+d x)}{315 a d}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 121, normalized size = 0.51 \[ \frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (2 \sin \left (\frac {1}{2} (c+d x)\right ) (-2 (84 A+131 C) \cos (c+d x)+4 (63 A+92 C) \cos (2 (c+d x))+2436 A-10 C \cos (3 (c+d x))+35 C \cos (4 (c+d x))+2389 C)-2520 (A+C) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{1260 d \sqrt {a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 191, normalized size = 0.81 \[ \frac {4 \, {\left (35 \, C \cos \left (d x + c\right )^{4} - 5 \, C \cos \left (d x + c\right )^{3} + 3 \, {\left (21 \, A + 19 \, C\right )} \cos \left (d x + c\right )^{2} - {\left (21 \, A + 29 \, C\right )} \cos \left (d x + c\right ) + 273 \, A + 257 \, C\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac {315 \, \sqrt {2} {\left ({\left (A + C\right )} a \cos \left (d x + c\right ) + {\left (A + C\right )} a\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + \frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt {a}}}{630 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.88, size = 227, normalized size = 0.96 \[ \frac {\frac {315 \, {\left (\sqrt {2} A + \sqrt {2} C\right )} \log \left ({\left | -\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt {a}} + \frac {2 \, {\left (315 \, \sqrt {2} A a^{4} + 315 \, \sqrt {2} C a^{4} + {\left (1050 \, \sqrt {2} A a^{4} + 840 \, \sqrt {2} C a^{4} + {\left (1512 \, \sqrt {2} A a^{4} + 1638 \, \sqrt {2} C a^{4} + {\left (1134 \, \sqrt {2} A a^{4} + 936 \, \sqrt {2} C a^{4} + {\left (357 \, \sqrt {2} A a^{4} + 383 \, \sqrt {2} C a^{4}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a\right )}^{\frac {9}{2}}}}{315 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.17, size = 340, normalized size = 1.44 \[ \frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (1120 C \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2160 C \sqrt {a}\, \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 \sqrt {2}\, \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (A +4 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-420 \sqrt {2}\, \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (A +2 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-315 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a A -315 \sqrt {2}\, \ln \left (\frac {4 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+4 a}{\cos \left (\frac {d x}{2}+\frac {c}{2}\right )}\right ) a C +630 A \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}+630 C \sqrt {2}\, \sqrt {a \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \sqrt {a}\right )}{315 a^{\frac {3}{2}} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {a \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, d} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
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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