Optimal. Leaf size=336 \[ \frac {2 b \sin (c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}+\frac {2 (2 a C+3 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt {\sec (c+d x)}} \]
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Rubi [A] time = 1.01, antiderivative size = 336, 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 = {4221, 3049, 3033, 3023, 2748, 2641, 2639} \[ \frac {2 b \sin (c+d x) \left (24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right )}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right )}{63 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )}{21 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right )}{15 d}+\frac {2 (2 a C+3 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^3}{9 d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3049
Rule 4221
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{2} a (9 A+C)+\frac {1}{2} (9 A b+9 a B+7 b C) \cos (c+d x)+\frac {3}{2} (3 b B+2 a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 (3 b B+2 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {1}{63} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{4} a (63 a A+9 b B+13 a C)+\frac {1}{4} \left (126 a A b+63 a^2 B+45 b^2 B+86 a b C\right ) \cos (c+d x)+\frac {1}{4} \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (3 b B+2 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {1}{315} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {5}{8} a^2 (63 a A+9 b B+13 a C)+\frac {21}{8} \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \cos (c+d x)+\frac {15}{8} \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (3 b B+2 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {1}{945} \left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {45}{16} \left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right )+\frac {63}{16} \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (3 b B+2 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}+\frac {1}{21} \left (\left (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (\left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (15 a^3 B+27 a b^2 B+9 a^2 b (5 A+3 C)+b^3 (9 A+7 C)\right ) \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 (21 a^2 b B+5 b^3 B+7 a^3 (3 A+C)+3 a b^2 (7 A+5 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}+\frac {2 b \left (63 A b^2+99 a b B+24 a^2 C+49 b^2 C\right ) \sin (c+d x)}{315 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (54 a^2 b B+15 b^3 B+8 a^3 C+9 a b^2 (7 A+5 C)\right ) \sin (c+d x)}{63 d \sqrt {\sec (c+d x)}}+\frac {2 (3 b B+2 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{21 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{9 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 1.75, size = 253, normalized size = 0.75 \[ \frac {\sqrt {\sec (c+d x)} \left (2 \sin (2 (c+d x)) \left (7 b \cos (c+d x) \left (108 a^2 C+108 a b B+36 A b^2+43 b^2 C\right )+5 \left (84 a^3 C+252 a^2 b B+18 a b^2 (14 A+13 C)+18 b^2 (3 a C+b B) \cos (2 (c+d x))+78 b^3 B+7 b^3 C \cos (3 (c+d x))\right )\right )+240 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right )+336 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right )\right )}{2520 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{3} \cos \left (d x + c\right )^{5} + {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{4} + A a^{3} + {\left (3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sqrt {\sec \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.10, size = 975, normalized size = 2.90 \[ \text {result too large to display} \]
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 \[ \int {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
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 \[ \int \sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
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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