Optimal. Leaf size=194 \[ \frac {2 a \left (7 a^2+15 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac {40 a b^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d} \]
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Rubi [A] time = 0.22, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2793, 3023, 2748, 2635, 2641, 2639} \[ \frac {2 a \left (7 a^2+15 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{45 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 d}+\frac {2 b^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))}{9 d}+\frac {40 a b^2 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{63 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2793
Rule 3023
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \, dx &=\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {2}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {1}{2} a \left (9 a^2+5 b^2\right )+\frac {1}{2} b \left (27 a^2+7 b^2\right ) \cos (c+d x)+10 a b^2 \cos ^2(c+d x)\right ) \, dx\\ &=\frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {4}{63} \int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {9}{4} a \left (7 a^2+15 b^2\right )+\frac {7}{4} b \left (27 a^2+7 b^2\right ) \cos (c+d x)\right ) \, dx\\ &=\frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {1}{9} \left (b \left (27 a^2+7 b^2\right )\right ) \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\frac {1}{7} \left (a \left (7 a^2+15 b^2\right )\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx\\ &=\frac {2 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}+\frac {1}{15} \left (b \left (27 a^2+7 b^2\right )\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (a \left (7 a^2+15 b^2\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 b \left (27 a^2+7 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {2 a \left (7 a^2+15 b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 a \left (7 a^2+15 b^2\right ) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 b \left (27 a^2+7 b^2\right ) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {40 a b^2 \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{63 d}+\frac {2 b^2 \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x)) \sin (c+d x)}{9 d}\\ \end {align*}
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Mathematica [A] time = 1.03, size = 137, normalized size = 0.71 \[ \frac {60 \left (7 a^3+15 a b^2\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+84 \left (27 a^2 b+7 b^3\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt {\cos (c+d x)} \left (5 \left (84 a^3+54 a b^2 \cos (2 (c+d x))+234 a b^2+7 b^3 \cos (3 (c+d x))\right )+7 b \left (108 a^2+43 b^2\right ) \cos (c+d x)\right )}{630 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \cos \left (d x + c\right )^{4} + 3 \, a b^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} b \cos \left (d x + c\right )^{2} + a^{3} \cos \left (d x + c\right )\right )} \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.75, size = 470, normalized size = 2.42 \[ -\frac {2 \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 )}\, \left (-1120 b^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2160 b^{2} a +2240 b^{3}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-1512 a^{2} b -3240 b^{2} a -2072 b^{3}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (420 a^{3}+1512 a^{2} b +2520 b^{2} a +952 b^{3}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-210 a^{3}-378 a^{2} b -720 b^{2} a -168 b^{3}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+105 a^{3} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )+225 b^{2} a \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 )-567 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 ) a^{2} b -147 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{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 ) b^{3}\right )}{315 \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 \[ \int {\left (b \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.05, size = 178, normalized size = 0.92 \[ \frac {2\,a^3\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3\,d}+\frac {2\,a^3\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3\,d}-\frac {2\,b^3\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {6\,a^2\,b\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a\,b^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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