Optimal. Leaf size=416 \[ -\frac {5 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^3 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}-\frac {b \left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sin (c+d x)}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}+\frac {\left (-\left (a^4 (3 A-8 C)\right )+26 a^2 A b^2-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 1.40, antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3056, 3055, 3059, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {b \left (26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right ) \sin (c+d x)}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {a+b \cos (c+d x)}}-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}-\frac {\left (5 A b^2-a^2 (3 A-2 C)\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\left (26 a^2 A b^2+a^4 (-(3 A-8 C))-15 A b^4\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 d \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {5 A b \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^3 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}} \]
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 3056
Rule 3059
Rubi steps
\begin {align*} \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx &=\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}+\frac {\int \frac {\left (-\frac {5 A b}{2}+a C \cos (c+d x)+\frac {3}{2} A b \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}+\frac {2 \int \frac {\left (-\frac {15}{4} A b \left (a^2-b^2\right )+\frac {3}{2} a \left (A b^2+a^2 C\right ) \cos (c+d x)-\frac {1}{4} b \left (5 A b^2-a^2 (3 A-2 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}+\frac {4 \int \frac {\left (-\frac {15}{8} A b \left (a^2-b^2\right )^2-\frac {1}{4} a \left (5 A b^4-3 a^4 C-a^2 b^2 (9 A+C)\right ) \cos (c+d x)+\frac {1}{8} b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {4 \int \frac {\left (\frac {15}{8} A b^2 \left (a^2-b^2\right )^2-\frac {1}{8} a b \left (a^2-b^2\right ) \left (3 a^2 A-5 A b^2-2 a^2 C\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{3 a^3 b \left (a^2-b^2\right )^2}+\frac {\left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {(5 A b) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 a^3}-\frac {\left (5 A b^2-a^2 (3 A-2 C)\right ) \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{6 a^2 \left (a^2-b^2\right )}+\frac {\left (\left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 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}{6 a^3 \left (a^2-b^2\right )^2 \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ &=\frac {\left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}-\frac {\left (5 A b \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}{2 a^3 \sqrt {a+b \cos (c+d x)}}-\frac {\left (\left (5 A b^2-a^2 (3 A-2 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}{6 a^2 \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}\\ &=\frac {\left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {\left (5 A b^2-a^2 (3 A-2 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 )}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}-\frac {5 A b \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 )}{a^3 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (5 A b^2-a^2 (3 A-2 C)\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac {b \left (26 a^2 A b^2-15 A b^4-a^4 (3 A-8 C)\right ) \sin (c+d x)}{3 a^3 \left (a^2-b^2\right )^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d (a+b \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 7.14, size = 786, normalized size = 1.89 \[ \frac {\cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A \sec ^2(c+d x)+C\right ) \left (\frac {2 A \tan (c+d x)}{a^3}-\frac {4 \left (a^2 b C \sin (c+d x)+A b^3 \sin (c+d x)\right )}{3 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {8 \left (2 a^4 b C \sin (c+d x)+5 a^2 A b^3 \sin (c+d x)-3 A b^5 \sin (c+d x)\right )}{3 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}\right )}{d (2 A+C \cos (2 c+2 d x)+C)}+\frac {\cos ^2(c+d x) \left (A \sec ^2(c+d x)+C\right ) \left (\frac {2 \left (12 a^5 C+36 a^3 A b^2+4 a^3 b^2 C-20 a A b^4\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (-33 a^4 A b+8 a^4 b C+86 a^2 A b^3-45 A b^5\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}-\frac {2 i \left (-3 a^4 A b+8 a^4 b C+26 a^2 A b^3-15 A b^5\right ) \sin (c+d x) \cos (2 (c+d x)) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b \cos (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right )}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left (2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right )}\right )}{6 a^3 d (b-a)^2 (a+b)^2 (2 A+C \cos (2 c+2 d x)+C)} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 11.50, size = 1331, normalized size = 3.20 \[ \text {result too large to display} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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