Optimal. Leaf size=73 \[ \frac {8 (a+a \sin (c+d x))^{11/2}}{11 a^3 d}-\frac {8 (a+a \sin (c+d x))^{13/2}}{13 a^4 d}+\frac {2 (a+a \sin (c+d x))^{15/2}}{15 a^5 d} \]
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Rubi [A]
time = 0.05, 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 = {2746, 45}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=\frac {\text {Subst}\left (\int (a-x)^2 (a+x)^{9/2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\text {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 = 0.12, size = 51, normalized size = 0.70 \begin {gather*} \frac {2 (1+\sin (c+d x))^3 (a (1+\sin (c+d x)))^{5/2} \left (263-374 \sin (c+d x)+143 \sin ^2(c+d x)\right )}{2145 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 41, normalized size = 0.56
method | result | size |
default | \(-\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {11}{2}} \left (143 \left (\cos ^{2}\left (d x +c \right )\right )+374 \sin \left (d x +c \right )-406\right )}{2145 a^{3} d}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 55, normalized size = 0.75 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 114, normalized size = 1.56 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 7.29, size = 108, normalized size = 1.48 \begin {gather*} \frac {256 \, \sqrt {2} {\left (143 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{15} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 330 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 195 \, a^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )\right )} \sqrt {a}}{2145 \, d} \end {gather*}
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 \begin {gather*} \int {\cos \left (c+d\,x\right )}^5\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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