Optimal. Leaf size=834 \[ \frac{C \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d}-\frac{\sqrt{a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d}+\frac{\sqrt{a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cot (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d} \]
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Rubi [A] time = 3.79186, antiderivative size = 834, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.156, Rules used = {3049, 3061, 3053, 2809, 2998, 2816, 2994} \[ \frac{C \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{7/2}}{5 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{5/2}}{40 b d}+\frac{\left (-15 C a^2+50 b B a+80 A b^2+64 b^2 C\right ) \sqrt{\cos (c+d x)} \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{240 b d}+\frac{\left (-15 C a^3+50 b B a^2+4 b^2 (60 A+43 C) a+120 b^3 B\right ) \sqrt{\cos (c+d x)} \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{320 b d}+\frac{\left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \sin (c+d x) \sqrt{a+b \cos (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}-\frac{(a-b) \sqrt{a+b} \left (-45 C a^4+150 b B a^3+12 b^2 (220 A+141 C) a^2+2840 b^3 B a+256 b^4 (5 A+4 C)\right ) \cot (c+d x) E\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 a b^2 d}-\frac{\sqrt{a+b} \left (45 C a^4-30 b (5 B+C) a^3-4 b^2 (660 A+295 B+423 C) a^2-8 b^3 (260 A+355 B+193 C) a-16 b^4 (80 A+45 B+64 C)\right ) \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{1920 b^2 d}+\frac{\sqrt{a+b} \left (-3 C a^5+10 b B a^4-40 b^2 (2 A+C) a^3-240 b^3 B a^2-80 b^4 (4 A+3 C) a-96 b^5 B\right ) \cot (c+d x) \Pi \left (\frac{a+b}{b};\sin ^{-1}\left (\frac{\sqrt{a+b \cos (c+d x)}}{\sqrt{a+b} \sqrt{\cos (c+d x)}}\right )|-\frac{a+b}{a-b}\right ) \sqrt{\frac{a (1-\sec (c+d x))}{a+b}} \sqrt{\frac{a (\sec (c+d x)+1)}{a-b}}}{128 b^3 d} \]
Antiderivative was successfully verified.
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Rule 3049
Rule 3061
Rule 3053
Rule 2809
Rule 2998
Rule 2816
Rule 2994
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{(a+b \cos (c+d x))^{5/2} \left (\frac{a C}{2}+b (5 A+4 C) \cos (c+d x)+\frac{1}{2} (10 b B-3 a C) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{5 b}\\ &=\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{(a+b \cos (c+d x))^{3/2} \left (\frac{5}{4} a (2 b B+a C)+\frac{1}{2} b (40 a A+30 b B+27 a C) \cos (c+d x)+\frac{1}{4} \left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{20 b}\\ &=\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{\sqrt{a+b \cos (c+d x)} \left (\frac{1}{8} a \left (80 A b^2+110 a b B+15 a^2 C+64 b^2 C\right )+\frac{1}{4} b \left (310 a b B+32 b^2 (5 A+4 C)+3 a^2 (80 A+49 C)\right ) \cos (c+d x)+\frac{3}{8} \left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \cos ^2(c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{60 b}\\ &=\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{\frac{1}{16} a \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right )+\frac{1}{8} b \left (1610 a^2 b B+360 b^3 B+4 a b^2 (380 A+289 C)+a^3 (960 A+573 C)\right ) \cos (c+d x)+\frac{1}{16} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{120 b}\\ &=\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{-\frac{1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac{1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \cos (c+d x)-\frac{15}{16} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{240 b^2}\\ &=\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}+\frac{\int \frac{-\frac{1}{16} a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right )+\frac{1}{8} a b \left (590 a^2 b B+360 b^3 B+15 a^3 C+4 a b^2 (260 A+193 C)\right ) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \cos (c+d x)}} \, dx}{240 b^2}-\frac{\left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 C)\right ) \int \frac{\sqrt{\cos (c+d x)}}{\sqrt{a+b \cos (c+d x)}} \, dx}{256 b^2}\\ &=\frac{\sqrt{a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 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}}}{128 b^3 d}+\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}-\frac{\left (a \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 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}{3840 b^2}-\frac{\left (a \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 C)\right )\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)}} \, dx}{3840 b^2}\\ &=-\frac{(a-b) \sqrt{a+b} \left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 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}}}{1920 a b^2 d}-\frac{\sqrt{a+b} \left (45 a^4 C-30 a^3 b (5 B+C)-16 b^4 (80 A+45 B+64 C)-8 a b^3 (260 A+355 B+193 C)-4 a^2 b^2 (660 A+295 B+423 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}}}{1920 b^2 d}+\frac{\sqrt{a+b} \left (10 a^4 b B-240 a^2 b^3 B-96 b^5 B-3 a^5 C-40 a^3 b^2 (2 A+C)-80 a b^4 (4 A+3 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}}}{128 b^3 d}+\frac{\left (150 a^3 b B+2840 a b^3 B-45 a^4 C+256 b^4 (5 A+4 C)+12 a^2 b^2 (220 A+141 C)\right ) \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left (50 a^2 b B+120 b^3 B-15 a^3 C+4 a b^2 (60 A+43 C)\right ) \sqrt{\cos (c+d x)} \sqrt{a+b \cos (c+d x)} \sin (c+d x)}{320 b d}+\frac{\left (80 A b^2+50 a b B-15 a^2 C+64 b^2 C\right ) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{240 b d}+\frac{(10 b B-3 a C) \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{5/2} \sin (c+d x)}{40 b d}+\frac{C \sqrt{\cos (c+d x)} (a+b \cos (c+d x))^{7/2} \sin (c+d x)}{5 b d}\\ \end{align*}
Mathematica [C] time = 6.71755, size = 1410, normalized size = 1.69 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.873, size = 7062, normalized size = 8.5 \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{\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}} \sqrt{\cos \left (d x + c\right )}\,{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 ({\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} \sqrt{\cos \left (d x + c\right )}, 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(-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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