Optimal. Leaf size=97 \[ \frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^4 d}-\frac {24 (a+a \sin (c+d x))^{7/2}}{7 a^5 d}+\frac {4 (a+a \sin (c+d x))^{9/2}}{3 a^6 d}-\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^7 d} \]
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Rubi [A]
time = 0.06, antiderivative size = 97, 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)^{11/2}}{11 a^7 d}+\frac {4 (a \sin (c+d x)+a)^{9/2}}{3 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{7/2}}{7 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{5/2}}{5 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 2746
Rubi steps
\begin {align*} \int \frac {\cos ^7(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac {\text {Subst}\left (\int (a-x)^3 (a+x)^{3/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \left (8 a^3 (a+x)^{3/2}-12 a^2 (a+x)^{5/2}+6 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^4 d}-\frac {24 (a+a \sin (c+d x))^{7/2}}{7 a^5 d}+\frac {4 (a+a \sin (c+d x))^{9/2}}{3 a^6 d}-\frac {2 (a+a \sin (c+d x))^{11/2}}{11 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 54, normalized size = 0.56 \begin {gather*} -\frac {2 (a (1+\sin (c+d x)))^{5/2} \left (-533+755 \sin (c+d x)-455 \sin ^2(c+d x)+105 \sin ^3(c+d x)\right )}{1155 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 57, normalized size = 0.59
method | result | size |
default | \(\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}} \left (105 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-455 \left (\cos ^{2}\left (d x +c \right )\right )-860 \sin \left (d x +c \right )+988\right )}{1155 a^{4} d}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 72, normalized size = 0.74 \begin {gather*} -\frac {2 \, {\left (105 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 770 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 1980 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a^{2} - 1848 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{3}\right )}}{1155 \, a^{7} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 72, normalized size = 0.74 \begin {gather*} \frac {2 \, {\left (245 \, \cos \left (d x + c\right )^{4} + 32 \, \cos \left (d x + c\right )^{2} - {\left (105 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{2} - 256\right )} \sin \left (d x + c\right ) + 256\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{1155 \, a^{2} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.31, size = 112, normalized size = 1.15 \begin {gather*} -\frac {64 \, {\left (105 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 385 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 495 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 231 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5}\right )}}{1155 \, a^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \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 \frac {{\cos \left (c+d\,x\right )}^7}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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