Optimal. Leaf size=710 \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{3 b d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{\left (8 A b^4-C \left (-26 a^2 b^2+15 a^4+3 b^4\right )\right ) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )^2}+\frac{2 \left (a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left (21 a^2 b^2 C-5 a^3 b C-15 a^4 C-a b^3 (2 A-3 C)+6 A b^4\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left (8 A b^4-C \left (-26 a^2 b^2+15 a^4+3 b^4\right )\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{5 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{b^4 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 2.4282, antiderivative size = 710, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used = {4221, 3048, 3047, 3061, 3053, 2809, 2998, 2816, 2994} \[ -\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x)}{3 b d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^{3/2}}-\frac{\left (8 A b^4-C \left (-26 a^2 b^2+15 a^4+3 b^4\right )\right ) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}{3 b^3 d \left (a^2-b^2\right )^2}+\frac{2 \left (a^2 b^2 (A+9 C)-5 a^4 C+3 A b^4\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{\sec (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left (21 a^2 b^2 C-5 a^3 b C-15 a^4 C-a b^3 (2 A-3 C)+6 A b^4\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{\left (8 A b^4-C \left (-26 a^2 b^2+15 a^4+3 b^4\right )\right ) \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a b^3 d (a-b) (a+b)^{3/2} \sqrt{\sec (c+d x)}}+\frac{5 a C \sqrt{a+b} \sqrt{\cos (c+d x)} \csc (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{b^4 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3048
Rule 3047
Rule 3061
Rule 3053
Rule 2809
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{5/2}} \, dx\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (\frac{3}{2} \left (A b^2+a^2 C\right )-\frac{3}{2} a b (A+C) \cos (c+d x)-\frac{1}{2} \left (2 A b^2+5 a^2 C-3 b^2 C\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^{3/2}} \, dx}{3 b \left (a^2-b^2\right )}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right )-\frac{1}{2} a b \left (2 A b^2-\left (a^2-3 b^2\right ) C\right ) \cos (c+d x)-\frac{1}{4} \left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} a \left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right )+\frac{1}{2} b \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right ) \cos (c+d x)-\frac{15}{4} a \left (a^2-b^2\right )^2 C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} a \left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right )+\frac{1}{2} b \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{3 b^3 \left (a^2-b^2\right )^2}-\frac{\left (5 a C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx}{2 b^3}\\ &=\frac{5 a \sqrt{a+b} C \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{b^4 d \sqrt{\sec (c+d x)}}-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}-\frac{\left ((a-b) \left (6 A b^4-a b^3 (2 A-3 C)-15 a^4 C-5 a^3 b C+21 a^2 b^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{6 b^3 \left (a^2-b^2\right )^2}+\frac{\left (a \left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{6 b^3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right ) \sqrt{\cos (c+d x)} \csc (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{3 a (a-b) b^3 (a+b)^{3/2} d \sqrt{\sec (c+d x)}}-\frac{\left (6 A b^4-a b^3 (2 A-3 C)-15 a^4 C-5 a^3 b C+21 a^2 b^2 C\right ) \sqrt{\cos (c+d x)} \csc (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{3 a (a-b) b^3 (a+b)^{3/2} d \sqrt{\sec (c+d x)}}+\frac{5 a \sqrt{a+b} C \sqrt{\cos (c+d x)} \csc (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (1+\sec (c+d x))}{a-b}}}{b^4 d \sqrt{\sec (c+d x)}}-\frac{2 \left (A b^2+a^2 C\right ) \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left (3 A b^4-5 a^4 C+a^2 b^2 (A+9 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{\left (8 A b^4-\left (15 a^4-26 a^2 b^2+3 b^4\right ) C\right ) \sqrt{a+b \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{3 b^3 \left (a^2-b^2\right )^2 d}\\ \end{align*}
Mathematica [B] time = 20.706, size = 1609, normalized size = 2.27 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.312, size = 6471, normalized size = 9.1 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{{\left (b^{3} \cos \left (d x + c\right )^{3} + 3 \, a b^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} b \cos \left (d x + c\right ) + a^{3}\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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