Optimal. Leaf size=97 \[ \frac {16 (a+a \sin (c+d x))^{3/2}}{3 a^4 d}-\frac {24 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}+\frac {12 (a+a \sin (c+d x))^{7/2}}{7 a^6 d}-\frac {2 (a+a \sin (c+d x))^{9/2}}{9 a^7 d} \]
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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)^{9/2}}{9 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{7/2}}{7 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{5/2}}{5 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{3/2}}{3 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))^{5/2}} \, dx &=\frac {\text {Subst}\left (\int (a-x)^3 \sqrt {a+x} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\text {Subst}\left (\int \left (8 a^3 \sqrt {a+x}-12 a^2 (a+x)^{3/2}+6 a (a+x)^{5/2}-(a+x)^{7/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {16 (a+a \sin (c+d x))^{3/2}}{3 a^4 d}-\frac {24 (a+a \sin (c+d x))^{5/2}}{5 a^5 d}+\frac {12 (a+a \sin (c+d x))^{7/2}}{7 a^6 d}-\frac {2 (a+a \sin (c+d x))^{9/2}}{9 a^7 d}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 54, normalized size = 0.56 \begin {gather*} -\frac {2 (a (1+\sin (c+d x)))^{3/2} \left (-319+321 \sin (c+d x)-165 \sin ^2(c+d x)+35 \sin ^3(c+d x)\right )}{315 a^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.34, size = 57, normalized size = 0.59
method | result | size |
default | \(\frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}} \left (35 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-165 \left (\cos ^{2}\left (d x +c \right )\right )-356 \sin \left (d x +c \right )+484\right )}{315 a^{4} d}\) | \(57\) |
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 (35 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} - 270 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} a + 756 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{2} - 840 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{3}\right )}}{315 \, a^{7} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 62, normalized size = 0.64 \begin {gather*} -\frac {2 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 226 \, \cos \left (d x + c\right )^{2} + 2 \, {\left (65 \, \cos \left (d x + c\right )^{2} - 64\right )} \sin \left (d x + c\right ) - 128\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{315 \, a^{3} 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 = 7.42, size = 112, normalized size = 1.15 \begin {gather*} -\frac {32 \, {\left (35 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 135 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 189 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 105 \, \sqrt {2} \sqrt {a} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}\right )}}{315 \, a^{3} 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 )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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