Optimal. Leaf size=313 \[ -\frac {b \sin (c+d x) \left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right )}{a^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\left (-\left (a^2 (A-2 C)\right )-2 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^2 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {(3 A b-2 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}+\frac {A \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a d \sqrt {a+b \cos (c+d x)}} \]
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Rubi [A] time = 1.07, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ -\frac {b \sin (c+d x) \left (a^2 (-(A-2 C))-2 a b B+3 A b^2\right )}{a^2 d \left (a^2-b^2\right ) \sqrt {a+b \cos (c+d x)}}+\frac {\left (a^2 (-(A-2 C))-2 a b B+3 A b^2\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^2 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {(3 A b-2 a B) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^2 d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}+\frac {A \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a d \sqrt {a+b \cos (c+d x)}} \]
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 ^2(c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx &=\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}+\frac {\int \frac {\left (\frac {1}{2} (-3 A b+2 a B)+a C \cos (c+d x)+\frac {1}{2} A b \cos ^2(c+d x)\right ) \sec (c+d x)}{(a+b \cos (c+d x))^{3/2}} \, dx}{a}\\ &=-\frac {b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}+\frac {2 \int \frac {\left (-\frac {1}{4} \left (a^2-b^2\right ) (3 A b-2 a B)+\frac {1}{2} a \left (A b^2-a (b B-a C)\right ) \cos (c+d x)+\frac {1}{4} b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac {b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}-\frac {2 \int \frac {\left (\frac {1}{4} b \left (a^2-b^2\right ) (3 A b-2 a B)-\frac {1}{4} a A b \left (a^2-b^2\right ) \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{a^2 b \left (a^2-b^2\right )}+\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \int \sqrt {a+b \cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=-\frac {b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}+\frac {A \int \frac {1}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 a}-\frac {(3 A b-2 a B) \int \frac {\sec (c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx}{2 a^2}+\frac {\left (\left (3 A b^2-2 a b B-a^2 (A-2 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}{2 a^2 \left (a^2-b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}\\ &=\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^2 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}-\frac {b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}+\frac {\left (A \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}{2 a \sqrt {a+b \cos (c+d x)}}-\frac {\left ((3 A b-2 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^2 \sqrt {a+b \cos (c+d x)}}\\ &=\frac {\left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sqrt {a+b \cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a^2 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \cos (c+d x)}{a+b}}}+\frac {A \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{a d \sqrt {a+b \cos (c+d x)}}-\frac {(3 A b-2 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^2 d \sqrt {a+b \cos (c+d x)}}-\frac {b \left (3 A b^2-2 a b B-a^2 (A-2 C)\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d \sqrt {a+b \cos (c+d x)}}+\frac {A \tan (c+d x)}{a d \sqrt {a+b \cos (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 7.01, size = 751, normalized size = 2.40 \[ \frac {\cos ^2(c+d x) \sqrt {a+b \cos (c+d x)} \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (\frac {2 A \tan (c+d x)}{a^2}-\frac {4 \left (a^2 b C \sin (c+d x)-a b^2 B \sin (c+d x)+A b^3 \sin (c+d x)\right )}{a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))}\right )}{d (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)}+\frac {\cos ^2(c+d x) \left (A \sec ^2(c+d x)+B \sec (c+d x)+C\right ) \left (-\frac {2 i \sin (c+d x) \cos (2 (c+d x)) \left (-a^2 A b+2 a^2 b C-2 a b^2 B+3 A b^3\right ) \sqrt {\frac {b-b \cos (c+d x)}{a+b}} \sqrt {-\frac {b \cos (c+d x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \cos (c+d x)}\right )|\frac {a+b}{a-b}\right )\right )\right )}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\cos ^2(c+d x)} \sqrt {-\frac {a^2-2 a (a+b \cos (c+d x))+(a+b \cos (c+d x))^2-b^2}{b^2}} \left (2 a^2-4 a (a+b \cos (c+d x))+2 (a+b \cos (c+d x))^2-b^2\right )}+\frac {2 \left (4 a^3 C-4 a^2 b B+4 a A b^2\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}+\frac {2 \left (4 a^3 B-7 a^2 A b+2 a^2 b C-6 a b^2 B+9 A b^3\right ) \sqrt {\frac {a+b \cos (c+d x)}{a+b}} \Pi \left (2;\frac {1}{2} (c+d x)|\frac {2 b}{a+b}\right )}{\sqrt {a+b \cos (c+d x)}}\right )}{2 a^2 d (a-b) (a+b) (2 A+2 B \cos (c+d x)+C \cos (2 c+2 d x)+C)} \]
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 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{2}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 7.61, size = 915, normalized size = 2.92 \[ \text {result too large to display} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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