Optimal. Leaf size=173 \[ \frac {904 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {128 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^4 \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}+\frac {8 a^4 \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {150 a^4 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{77 d}+\frac {128 a^4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {904 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d} \]
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Rubi [A] time = 0.20, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2757, 2635, 2641, 2639} \[ \frac {904 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {128 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a^4 \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{11 d}+\frac {8 a^4 \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {150 a^4 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{77 d}+\frac {128 a^4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {904 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{231 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2757
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4 \, dx &=\int \left (a^4 \cos ^{\frac {3}{2}}(c+d x)+4 a^4 \cos ^{\frac {5}{2}}(c+d x)+6 a^4 \cos ^{\frac {7}{2}}(c+d x)+4 a^4 \cos ^{\frac {9}{2}}(c+d x)+a^4 \cos ^{\frac {11}{2}}(c+d x)\right ) \, dx\\ &=a^4 \int \cos ^{\frac {3}{2}}(c+d x) \, dx+a^4 \int \cos ^{\frac {11}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \cos ^{\frac {9}{2}}(c+d x) \, dx+\left (6 a^4\right ) \int \cos ^{\frac {7}{2}}(c+d x) \, dx\\ &=\frac {2 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {8 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {12 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {8 a^4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 a^4 \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{3} a^4 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{11} \left (9 a^4\right ) \int \cos ^{\frac {7}{2}}(c+d x) \, dx+\frac {1}{5} \left (12 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{9} \left (28 a^4\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{7} \left (30 a^4\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {24 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {74 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {128 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {150 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {8 a^4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 a^4 \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{77} \left (45 a^4\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{7} \left (10 a^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\frac {1}{15} \left (28 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {128 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {74 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {904 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {128 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {150 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {8 a^4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 a^4 \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}+\frac {1}{77} \left (15 a^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {128 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {904 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {904 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {128 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {150 a^4 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{77 d}+\frac {8 a^4 \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{9 d}+\frac {2 a^4 \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{11 d}\\ \end {align*}
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Mathematica [C] time = 3.62, size = 271, normalized size = 1.57 \[ \frac {a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (\frac {59136 \sec (c) \left (\csc (c) \sqrt {\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )} \left (3 \cos \left (c-\tan ^{-1}(\tan (c))-d x\right )+\cos \left (c+\tan ^{-1}(\tan (c))+d x\right )\right )-2 \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right )\right )}{\sqrt {\sec ^2(c)} \sqrt {\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )}}-108480 \sin (c) \sqrt {\csc ^2(c)} \cos (c+d x) \sqrt {\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )+\cos (c+d x) (122610 \sin (c+d x)+45584 \sin (2 (c+d x))+14445 \sin (3 (c+d x))+3080 \sin (4 (c+d x))+315 \sin (5 (c+d x))-236544 \cot (c))\right )}{443520 d \sqrt {\cos (c+d x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} \cos \left (d x + c\right )^{5} + 4 \, a^{4} \cos \left (d x + c\right )^{4} + 6 \, a^{4} \cos \left (d x + c\right )^{3} + 4 \, a^{4} \cos \left (d x + c\right )^{2} + a^{4} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.49, size = 273, normalized size = 1.58 \[ -\frac {8 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{4} \left (5040 \left (\cos ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5320 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1740 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+326 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+678 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4465 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1695 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3696 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2001 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3465 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cos \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.88, size = 221, normalized size = 1.28 \[ \frac {2\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3\,d}+\frac {2\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3\,d}-\frac {8\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a^4\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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