Optimal. Leaf size=294 \[ \frac {2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a b (9 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A] time = 0.63, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4221, 3050, 3033, 3023, 2748, 2635, 2641, 2639} \[ \frac {2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a b (9 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^2}{11 d \sec ^{\frac {5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3050
Rule 4221
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{11} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\frac {1}{2} a (11 A+5 C)+\frac {1}{2} b (11 A+9 C) \cos (c+d x)+2 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{99} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{4} a^2 (11 A+5 C)+\frac {11}{2} a b (9 A+7 C) \cos (c+d x)+\frac {9}{4} \left (4 a^2 C+b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{693} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{8} \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right )+\frac {77}{4} a b (9 A+7 C) \cos (c+d x)\right ) \, dx\\ &=\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {1}{9} \left (2 a b (9 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{77} \left (\left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{15} \left (2 a b (9 A+7 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{231} \left (\left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {4 a b (9 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {8 a b C \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (4 a^2 C+b^2 (11 A+9 C)\right ) \sin (c+d x)}{77 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^2 \sin (c+d x)}{11 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {4 a b (9 A+7 C) \sin (c+d x)}{45 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.42, size = 209, normalized size = 0.71 \[ \frac {\sqrt {\sec (c+d x)} \left (2 \sin (2 (c+d x)) \left (5 \left (36 \left (11 a^2 C+11 A b^2+16 b^2 C\right ) \cos (2 (c+d x))+132 a^2 (14 A+13 C)+308 a b C \cos (3 (c+d x))+3 b^2 (572 A+531 C)+63 b^2 C \cos (4 (c+d x))\right )+308 a b (36 A+43 C) \cos (c+d x)\right )+480 \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+14784 a b (9 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{55440 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{2} \cos \left (d x + c\right )^{4} + 2 \, C a b \cos \left (d x + c\right )^{3} + 2 \, A a b \cos \left (d x + c\right ) + A a^{2} + {\left (C a^{2} + A b^{2}\right )} \cos \left (d x + c\right )^{2}}{\sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.01, size = 649, normalized size = 2.21 \[ -\frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (20160 C \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-24640 C a b -50400 b^{2} C \right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (7920 A \,b^{2}+7920 a^{2} C +49280 C a b +56880 b^{2} C \right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-11088 A a b -11880 A \,b^{2}-11880 a^{2} C -45584 C a b -34920 b^{2} C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (4620 a^{2} A +11088 A a b +9240 A \,b^{2}+9240 a^{2} C +20944 C a b +13860 b^{2} C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-2310 a^{2} A -2772 A a b -2640 A \,b^{2}-2640 a^{2} C -3696 C a b -2790 b^{2} C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-4158 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b +1155 a^{2} A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+825 A \,b^{2} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3234 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a b +825 a^{2} C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+675 b^{2} C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{2}}{\sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^2}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{2}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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