Optimal. Leaf size=647 \[ \frac{\sin (c+d x) \left (a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left (a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{24 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{24 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left (30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{8 b d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 2.36875, antiderivative size = 647, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.178, Rules used = {3047, 3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{\sin (c+d x) \left (a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \cos (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left (a^2 (48 A-48 B-33 C)-2 a b (72 A+27 B+13 C)-4 b^2 (6 A+3 B+4 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{24 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left (a^2 (-(48 A-33 C))+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{24 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left (30 a^2 b B+5 a^3 C+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}} \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right )}{8 b d}-\frac{b \sin (c+d x) \sqrt{\cos (c+d x)} (8 a A-3 a C-2 b B) \sqrt{a+b \cos (c+d x)}}{4 d}-\frac{b (6 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3047
Rule 3049
Rule 3061
Rule 3053
Rule 2809
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \frac{(a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+2 \int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{1}{2} (5 A b+a B)+\frac{1}{2} (b B-a (A-C)) \cos (c+d x)-\frac{1}{2} b (6 A-C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{b (6 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2}{3} \int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{4} a (24 A b+6 a B+b C)+\frac{1}{2} \left (6 a b B-3 a^2 (A-C)+b^2 (3 A+2 C)\right ) \cos (c+d x)-\frac{3}{4} b (8 a A-2 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{b (8 a A-2 b B-3 a C) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac{b (6 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{1}{3} \int \frac{\frac{1}{8} a \left (24 a^2 B+6 b^2 B+a b (72 A+13 C)\right )+\frac{1}{4} \left (36 a^2 b B+6 b^3 B-12 a^3 (A-C)+a b^2 (36 A+19 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx\\ &=\frac{\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}-\frac{b (8 a A-2 b B-3 a C) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac{b (6 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{\int \frac{-\frac{1}{8} a b \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right )+\frac{1}{4} a b \left (24 a^2 B+6 b^2 B+a b (72 A+13 C)\right ) \cos (c+d x)+\frac{3}{8} b \left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{6 b}\\ &=\frac{\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}-\frac{b (8 a A-2 b B-3 a C) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac{b (6 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{\int \frac{-\frac{1}{8} a b \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right )+\frac{1}{4} a b \left (24 a^2 B+6 b^2 B+a b (72 A+13 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{6 b}+\frac{1}{16} \left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx\\ &=-\frac{\sqrt{a+b} \left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \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}}}{8 b d}+\frac{\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}-\frac{b (8 a A-2 b B-3 a C) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac{b (6 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{1}{48} \left (a \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right )\right ) \int \frac{1+\cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx-\frac{1}{48} \left (a \left (a^2 (48 A-48 B-33 C)-4 b^2 (6 A+3 B+4 C)-2 a b (72 A+27 B+13 C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx\\ &=-\frac{(a-b) \sqrt{a+b} \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 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}}}{24 a d}-\frac{\sqrt{a+b} \left (a^2 (48 A-48 B-33 C)-4 b^2 (6 A+3 B+4 C)-2 a b (72 A+27 B+13 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}}}{24 d}-\frac{\sqrt{a+b} \left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \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}}}{8 b d}+\frac{\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{24 d \sqrt{\cos (c+d x)}}-\frac{b (8 a A-2 b B-3 a C) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{4 d}-\frac{b (6 A-C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 d}+\frac{2 A (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.81999, size = 1302, normalized size = 2.01 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.513, size = 5130, normalized size = 7.9 \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} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \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 b^{2} \cos \left (d x + c\right )^{4} +{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{3} + A a^{2} +{\left (C a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} +{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt{b \cos \left (d x + c\right ) + a}}{\cos \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{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )}{\left (b \cos \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}{\cos \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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