Optimal. Leaf size=97 \[ -\frac {2 (a \sin (c+d x)+a)^{17/2}}{17 a^7 d}+\frac {4 (a \sin (c+d x)+a)^{15/2}}{5 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d} \]
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Rubi [A] time = 0.09, 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 = {2667, 43} \[ -\frac {2 (a \sin (c+d x)+a)^{17/2}}{17 a^7 d}+\frac {4 (a \sin (c+d x)+a)^{15/2}}{5 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{13/2}}{13 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{11/2}}{11 a^4 d} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2667
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+a \sin (c+d x))^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^{9/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{9/2}-12 a^2 (a+x)^{11/2}+6 a (a+x)^{13/2}-(a+x)^{15/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {16 (a+a \sin (c+d x))^{11/2}}{11 a^4 d}-\frac {24 (a+a \sin (c+d x))^{13/2}}{13 a^5 d}+\frac {4 (a+a \sin (c+d x))^{15/2}}{5 a^6 d}-\frac {2 (a+a \sin (c+d x))^{17/2}}{17 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.45, size = 61, normalized size = 0.63 \[ -\frac {2 (\sin (c+d x)+1)^4 \left (715 \sin ^3(c+d x)-2717 \sin ^2(c+d x)+3641 \sin (c+d x)-1767\right ) (a (\sin (c+d x)+1))^{3/2}}{12155 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 110, normalized size = 1.13 \[ -\frac {2 \, {\left (715 \, a \cos \left (d x + c\right )^{8} - 66 \, a \cos \left (d x + c\right )^{6} - 112 \, a \cos \left (d x + c\right )^{4} - 256 \, a \cos \left (d x + c\right )^{2} - 2 \, {\left (429 \, a \cos \left (d x + c\right )^{6} + 504 \, a \cos \left (d x + c\right )^{4} + 640 \, a \cos \left (d x + c\right )^{2} + 1024 \, a\right )} \sin \left (d x + c\right ) - 2048 \, a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{12155 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.11, size = 505, normalized size = 5.21 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 57, normalized size = 0.59 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {11}{2}} \left (715 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-2717 \left (\cos ^{2}\left (d x +c \right )\right )-4356 \sin \left (d x +c \right )+4484\right )}{12155 a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 72, normalized size = 0.74 \[ -\frac {2 \, {\left (715 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {17}{2}} - 4862 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a + 11220 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a^{2} - 8840 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a^{3}\right )}}{12155 \, a^{7} d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^7\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2} \,d x \]
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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