Optimal. Leaf size=73 \[ \frac {2 (a \sin (c+d x)+a)^{15/2}}{15 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{13/2}}{13 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2667, 43} \[ \frac {2 (a \sin (c+d x)+a)^{15/2}}{15 a^5 d}-\frac {8 (a \sin (c+d x)+a)^{13/2}}{13 a^4 d}+\frac {8 (a \sin (c+d x)+a)^{11/2}}{11 a^3 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^2 (a+x)^{9/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (4 a^2 (a+x)^{9/2}-4 a (a+x)^{11/2}+(a+x)^{13/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {8 (a+a \sin (c+d x))^{11/2}}{11 a^3 d}-\frac {8 (a+a \sin (c+d x))^{13/2}}{13 a^4 d}+\frac {2 (a+a \sin (c+d x))^{15/2}}{15 a^5 d}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 51, normalized size = 0.70 \[ \frac {2 (\sin (c+d x)+1)^3 \left (143 \sin ^2(c+d x)-374 \sin (c+d x)+263\right ) (a (\sin (c+d x)+1))^{5/2}}{2145 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 114, normalized size = 1.56 \[ -\frac {2 \, {\left (341 \, a^{2} \cos \left (d x + c\right )^{6} - 28 \, a^{2} \cos \left (d x + c\right )^{4} - 64 \, a^{2} \cos \left (d x + c\right )^{2} - 512 \, a^{2} + {\left (143 \, a^{2} \cos \left (d x + c\right )^{6} - 252 \, a^{2} \cos \left (d x + c\right )^{4} - 320 \, a^{2} \cos \left (d x + c\right )^{2} - 512 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{2145 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.77, size = 471, normalized size = 6.45 \[ -\frac {1}{2882880} \, \sqrt {2} {\left (\frac {16380 \, a^{2} \cos \left (\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {102960 \, a^{2} \cos \left (\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {300300 \, a^{2} \cos \left (\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {13860 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {13}{2} \, d x + \frac {13}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {80080 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {180180 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} + \frac {3465 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {13}{2} \, d x + \frac {13}{2} \, c\right )}{d} - \frac {5005 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {9}{2} \, d x + \frac {9}{2} \, c\right )}{d} - \frac {171171 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {5}{2} \, d x + \frac {5}{2} \, c\right )}{d} - \frac {2027025 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{d} + \frac {3003 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {15}{2} \, d x + \frac {15}{2} \, c\right )}{d} - \frac {4095 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {11}{2} \, d x + \frac {11}{2} \, c\right )}{d} - \frac {122265 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {7}{2} \, d x + \frac {7}{2} \, c\right )}{d} - \frac {675675 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {3}{2} \, d x + \frac {3}{2} \, c\right )}{d}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 41, normalized size = 0.56 \[ -\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {11}{2}} \left (143 \left (\cos ^{2}\left (d x +c \right )\right )+374 \sin \left (d x +c \right )-406\right )}{2145 a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 55, normalized size = 0.75 \[ \frac {2 \, {\left (143 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} - 660 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a + 780 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a^{2}\right )}}{2145 \, a^{5} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
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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