Optimal. Leaf size=563 \[ \frac{2 \left (a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (-a^2 b C-3 a^3 C+3 a A b^2+6 a b^2 C-A b^3\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 \left (a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a^2 b^2 d \sqrt{a+b} \left (a^2-b^2\right )}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{b^3 d} \]
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Rubi [A] time = 1.61337, antiderivative size = 563, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.189, Rules used = {3048, 3051, 2809, 2993, 2998, 2816, 2994} \[ \frac{2 \left (a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right ) \sin (c+d x)}{3 b^2 d \left (a^2-b^2\right )^2 \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (-a^2 b C-3 a^3 C+3 a A b^2+6 a b^2 C-A b^3\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a b^2 d (a-b) (a+b)^{3/2}}-\frac{2 \left (a^2 b^2 (3 A+7 C)-3 a^4 C+A b^4\right ) \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{3 a^2 b^2 d \sqrt{a+b} \left (a^2-b^2\right )}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{b^3 d} \]
Antiderivative was successfully verified.
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Rule 3048
Rule 3051
Rule 2809
Rule 2993
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \frac{\sqrt{\cos (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 ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (A b^2+a^2 C\right )-\frac{3}{2} a b (A+C) \cos (c+d x)-\frac{3}{2} \left (a^2-b^2\right ) C \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (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 ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}-\frac{2 \int \frac{\frac{1}{2} b \left (A b^2+a^2 C\right )+\left (\frac{3}{2} a \left (a^2-b^2\right ) C-\frac{3}{2} a b^2 (A+C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}} \, dx}{3 b^2 \left (a^2-b^2\right )}+\frac{C \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx}{b^2}\\ &=-\frac{2 \sqrt{a+b} C \cot (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^3 d}-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (A b^4-3 a^4 C+a^2 b^2 (3 A+7 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{2 \int \frac{\frac{1}{2} b^2 \left (A b^2+a^2 C\right )-a \left (\frac{3}{2} a \left (a^2-b^2\right ) C-\frac{3}{2} a b^2 (A+C)\right )+\left (\frac{1}{2} a b \left (A b^2+a^2 C\right )-b \left (\frac{3}{2} a \left (a^2-b^2\right ) C-\frac{3}{2} a b^2 (A+C)\right )\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )^2}\\ &=-\frac{2 \sqrt{a+b} C \cot (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^3 d}-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (A b^4-3 a^4 C+a^2 b^2 (3 A+7 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}-\frac{\left (A b^3+3 a^3 C+a^2 b C-3 a b^2 (A+2 C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{3 (a-b) b^2 (a+b)^2}-\frac{\left (A b^4-3 a^4 C+a^2 b^2 (3 A+7 C)\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(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^4-3 a^4 C+a^2 b^2 (3 A+7 C)\right ) \cot (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^2 (a-b) b^2 (a+b)^{3/2} d}-\frac{2 \left (A b^3+3 a^3 C+a^2 b C-3 a b^2 (A+2 C)\right ) \cot (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^2 (a+b)^{3/2} d}-\frac{2 \sqrt{a+b} C \cot (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^3 d}-\frac{2 \left (A b^2+a^2 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^{3/2}}+\frac{2 \left (A b^4-3 a^4 C+a^2 b^2 (3 A+7 C)\right ) \sin (c+d x)}{3 b^2 \left (a^2-b^2\right )^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.53945, size = 1388, normalized size = 2.47 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.326, size = 6425, normalized size = 11.4 \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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{\cos \left (d x + c\right )}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{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} \sqrt{\cos \left (d x + c\right )}}{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}}, 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{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{\cos \left (d x + c\right )}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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