Optimal. Leaf size=197 \[ \frac {2 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 d (a+b)}-\frac {2 \left (-3 a^2 B+3 a A b-b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^3 d}+\frac {2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 B \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}} \]
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Rubi [A] time = 0.56, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2960, 4034, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac {2 \left (-3 a^2 B+3 a A b-b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 b^3 d}+\frac {2 a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right )}{b^3 d (a+b)}+\frac {2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d}+\frac {2 B \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2805
Rule 2960
Rule 3771
Rule 3787
Rule 3849
Rule 4034
Rule 4106
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {B+A \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (b+a \sec (c+d x))} \, dx\\ &=\frac {2 B \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-\frac {3}{2} (A b-a B)-\frac {1}{2} b B \sec (c+d x)-\frac {1}{2} a B \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{3 b}\\ &=\frac {2 B \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}-\frac {2 \int \frac {-\frac {3}{2} b (A b-a B)-\left (\frac {b^2 B}{2}-\frac {3}{2} a (A b-a B)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)}} \, dx}{3 b^3}+\frac {\left (a^2 (A b-a B)\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{b^3}\\ &=\frac {2 B \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}+\frac {(A b-a B) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx}{b^2}-\frac {\left (3 a A b-3 a^2 B-b^2 B\right ) \int \sqrt {\sec (c+d x)} \, dx}{3 b^3}+\frac {\left (a^2 (A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{b^3}\\ &=\frac {2 a^2 (A b-a B) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{b^3 (a+b) d}+\frac {2 B \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}+\frac {\left ((A b-a B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{b^2}-\frac {\left (\left (3 a A b-3 a^2 B-b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b^3}\\ &=\frac {2 (A b-a B) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{b^2 d}-\frac {2 \left (3 a A b-3 a^2 B-b^2 B\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{3 b^3 d}+\frac {2 a^2 (A b-a B) \sqrt {\cos (c+d x)} \Pi \left (\frac {2 b}{a+b};\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{b^3 (a+b) d}+\frac {2 B \sin (c+d x)}{3 b d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 6.66, size = 278, normalized size = 1.41 \[ \frac {2 \csc (c+d x) \left (3 a^2 B \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )+b (-3 a B+3 A b+b B) \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-3 b (A b-a B) \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-3 a A b \sqrt {-\tan ^2(c+d x)} \sqrt {\sec (c+d x)} \Pi \left (-\frac {a}{b};\left .\sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )\right |-1\right )-3 a b B \sec ^2(c+d x)+3 a b B+3 A b^2 \sec ^2(c+d x)-3 A b^2+b^2 B \sin (c+d x) \tan (c+d x)\right )}{3 b^3 d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 160.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.57, size = 786, normalized size = 3.99 \[ \text {result too large to display} \]
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 \[ \int \frac {B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {A+B\,\cos \left (c+d\,x\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,\left (a+b\,\cos \left (c+d\,x\right )\right )} \,d x \]
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 \[ \int \frac {A + B \cos {\left (c + d x \right )}}{\left (a + b \cos {\left (c + d x \right )}\right ) \sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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