Optimal. Leaf size=572 \[ -\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {\left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}+\frac {b \sin (c+d x) \left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right )}{12 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\left (12 a^3 B-a^2 (27 A b-8 b C)-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^3 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {b \sin (c+d x) \left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {\left (-12 a^5 B+a^4 b (33 A-56 C)+104 a^3 b^2 B-2 a^2 b^3 (85 A-12 C)-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 2.31, antiderivative size = 572, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.209, Rules used = {3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {b \sin (c+d x) \left (-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {b \sin (c+d x) \left (-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right )}{12 a^3 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac {\left (-a^2 (27 A b-8 b C)+12 a^3 B-20 a b^2 B+35 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^3 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\left (-2 a^2 b^3 (85 A-12 C)+a^4 b (33 A-56 C)+104 a^3 b^2 B-12 a^5 B-60 a b^4 B+105 A b^5\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (4 a^2 (A+2 C)-20 a b B+35 A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \tan (c+d x) \sec (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2805
Rule 2807
Rule 3002
Rule 3055
Rule 3059
Rubi steps
\begin {align*} \int \frac {\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx &=\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {1}{2} (-7 A b+4 a B)+a (A+2 C) \cos (c+d x)+\frac {5}{2} A b \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{2 a}\\ &=-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {1}{4} \left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right )+\frac {5}{2} a A b \cos (c+d x)-\frac {3}{4} b (7 A b-4 a B) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{2 a^2}\\ &=\frac {b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (\frac {3}{8} \left (a^2-b^2\right ) \left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right )-\frac {3}{4} a b \left (7 A b^2-4 a b B-a^2 (3 A-4 C)\right ) \cos (c+d x)+\frac {1}{8} b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 a^3 \left (a^2-b^2\right )}\\ &=\frac {b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {2 \int \frac {\left (\frac {3}{16} \left (a^2-b^2\right )^2 \left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right )+\frac {1}{8} a b \left (35 A b^4+36 a^3 b B-20 a b^3 B+3 a^4 (A-8 C)-2 a^2 b^2 (27 A-4 C)\right ) \cos (c+d x)+\frac {1}{16} b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B-2 a^2 b^3 (85 A-12 C)+a^4 (33 A b-56 b C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}-\frac {2 \int \frac {\left (-\frac {3}{16} b \left (a^2-b^2\right )^2 \left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right )-\frac {1}{16} a b \left (a^2-b^2\right ) \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^4 b \left (a^2-b^2\right )^2}+\frac {\left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{24 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{8 a^4}+\frac {\left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )}+\frac {\left (\left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sqrt {a+b \cos (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}} \, dx}{24 a^4 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ &=\frac {\left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}+\frac {\left (\left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{8 a^4 \sqrt {a+b \cos (c+d x)}}+\frac {\left (\left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cos (c+d x)}{a+b}}} \, dx}{24 a^3 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\\ &=\frac {\left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {\left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{12 a^3 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {\left (35 A b^2-20 a b B+4 a^2 (A+2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{4 a^4 d \sqrt {a+b \cos (c+d x)}}+\frac {b \left (35 A b^3+12 a^3 B-20 a b^2 B-a^2 (27 A b-8 b C)\right ) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (105 A b^5-12 a^5 B+104 a^3 b^2 B-60 a b^4 B+a^4 b (33 A-56 C)-2 a^2 b^3 (85 A-12 C)\right ) \sin (c+d x)}{12 a^4 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}-\frac {(7 A b-4 a B) \tan (c+d x)}{4 a^2 d (a+b \cos (c+d x))^{3/2}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a d (a+b \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 7.74, size = 922, normalized size = 1.61 \[ \frac {\frac {2 \left (12 A b a^5-96 b C a^5+144 b^2 B a^4-216 A b^3 a^3+32 b^3 C a^3-80 b^4 B a^2+140 A b^5 a\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (24 A a^6+48 C a^6-132 b B a^5+195 A b^2 a^4-152 b^2 C a^4+344 b^3 B a^3-566 A b^4 a^2+72 b^4 C a^2-180 b^5 B a+315 A b^6\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (105 A b^6-60 a B b^5-170 a^2 A b^4+24 a^2 C b^4+104 a^3 B b^3+33 a^4 A b^2-56 a^4 C b^2-12 a^5 B b\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {\cos (c+d x) b+b}{a-b}} \cos (2 (c+d x)) \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right ) \sin (c+d x)}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-2 (a+b \cos (c+d x)) a-b^2+(a+b \cos (c+d x))^2}{b^2}} \left (2 a^2-4 (a+b \cos (c+d x)) a-b^2+2 (a+b \cos (c+d x))^2\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac {\sqrt {a+b \cos (c+d x)} \left (\frac {\sec (c+d x) (4 a B \sin (c+d x)-11 A b \sin (c+d x))}{4 a^4}+\frac {2 \left (A \sin (c+d x) b^4-a B \sin (c+d x) b^3+a^2 C \sin (c+d x) b^2\right )}{3 a^3 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac {2 \left (-9 A \sin (c+d x) b^6+6 a B \sin (c+d x) b^5+13 a^2 A \sin (c+d x) b^4-3 a^2 C \sin (c+d x) b^4-10 a^3 B \sin (c+d x) b^3+7 a^4 C \sin (c+d x) b^2\right )}{3 a^4 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac {A \sec (c+d x) \tan (c+d x)}{2 a^3}\right )}{d} \]
Warning: Unable to verify antiderivative.
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fricas [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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{3}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 21.47, size = 2019, normalized size = 3.53 \[ \text {Expression too large to display} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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