Optimal. Leaf size=204 \[ \frac{4 a^3 (13 A+11 B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (21 A+17 B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a^3 (24 A+23 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (9 A+13 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{63 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
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Rubi [A] time = 0.447287, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {2976, 2968, 3023, 2748, 2639, 2635, 2641} \[ \frac{4 a^3 (13 A+11 B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (21 A+17 B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a^3 (24 A+23 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (9 A+13 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{63 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d} \]
Antiderivative was successfully verified.
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Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 (A+B \cos (c+d x)) \, dx &=\frac{2 a B \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2}{9} \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2 \left (\frac{3}{2} a (3 A+B)+\frac{1}{2} a (9 A+13 B) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a B \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \cos ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{4}{63} \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x)) \left (\frac{15}{2} a^2 (3 A+2 B)+\frac{3}{2} a^2 (24 A+23 B) \cos (c+d x)\right ) \, dx\\ &=\frac{2 a B \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \cos ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{4}{63} \int \sqrt{\cos (c+d x)} \left (\frac{15}{2} a^3 (3 A+2 B)+\left (\frac{15}{2} a^3 (3 A+2 B)+\frac{3}{2} a^3 (24 A+23 B)\right ) \cos (c+d x)+\frac{3}{2} a^3 (24 A+23 B) \cos ^2(c+d x)\right ) \, dx\\ &=\frac{4 a^3 (24 A+23 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \cos ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{8}{315} \int \sqrt{\cos (c+d x)} \left (\frac{21}{4} a^3 (21 A+17 B)+\frac{45}{4} a^3 (13 A+11 B) \cos (c+d x)\right ) \, dx\\ &=\frac{4 a^3 (24 A+23 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \cos ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{1}{7} \left (2 a^3 (13 A+11 B)\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\frac{1}{15} \left (2 a^3 (21 A+17 B)\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^3 (21 A+17 B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{4 a^3 (24 A+23 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \cos ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}+\frac{1}{21} \left (2 a^3 (13 A+11 B)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^3 (21 A+17 B) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{15 d}+\frac{4 a^3 (13 A+11 B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{4 a^3 (13 A+11 B) \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{4 a^3 (24 A+23 B) \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac{2 a B \cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{9 d}+\frac{2 (9 A+13 B) \cos ^{\frac{3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{63 d}\\ \end{align*}
Mathematica [C] time = 6.25951, size = 944, normalized size = 4.63 \[ \sqrt{\cos (c+d x)} (\cos (c+d x) a+a)^3 \left (-\frac{(21 A+17 B) \cot (c)}{30 d}+\frac{(107 A+97 B) \cos (d x) \sin (c)}{336 d}+\frac{(54 A+73 B) \cos (2 d x) \sin (2 c)}{720 d}+\frac{(A+3 B) \cos (3 d x) \sin (3 c)}{112 d}+\frac{B \cos (4 d x) \sin (4 c)}{288 d}+\frac{(107 A+97 B) \cos (c) \sin (d x)}{336 d}+\frac{(54 A+73 B) \cos (2 c) \sin (2 d x)}{720 d}+\frac{(A+3 B) \cos (3 c) \sin (3 d x)}{112 d}+\frac{B \cos (4 c) \sin (4 d x)}{288 d}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )-\frac{7 A (\cos (c+d x) a+a)^3 \csc (c) \left (\frac{\, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right ) \sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{1-\cos \left (d x+\tan ^{-1}(\tan (c))\right )} \sqrt{\cos \left (d x+\tan ^{-1}(\tan (c))\right )+1} \sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}}}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{20 d}-\frac{17 B (\cos (c+d x) a+a)^3 \csc (c) \left (\frac{\, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right ) \sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{1-\cos \left (d x+\tan ^{-1}(\tan (c))\right )} \sqrt{\cos \left (d x+\tan ^{-1}(\tan (c))\right )+1} \sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left (d x+\tan ^{-1}(\tan (c))\right ) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left (d x+\tan ^{-1}(\tan (c))\right ) \sqrt{\tan ^2(c)+1}}}\right ) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{60 d}-\frac{13 A (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{42 d \sqrt{\cot ^2(c)+1}}-\frac{11 B (\cos (c+d x) a+a)^3 \csc (c) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right )}{42 d \sqrt{\cot ^2(c)+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 3.167, size = 413, normalized size = 2. \begin{align*} \text{result 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 (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \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 (B a^{3} \cos \left (d x + c\right )^{4} +{\left (A + 3 \, B\right )} a^{3} \cos \left (d x + c\right )^{3} + 3 \,{\left (A + B\right )} a^{3} \cos \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right ) + A a^{3}\right )} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \cos \left (d x + c\right ) + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \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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