Optimal. Leaf size=97 \[ -\frac {2 (a \sin (c+d x)+a)^{21/2}}{21 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{19/2}}{19 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d} \]
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Rubi [A] time = 0.08, 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} \[ -\frac {2 (a \sin (c+d x)+a)^{21/2}}{21 a^7 d}+\frac {12 (a \sin (c+d x)+a)^{19/2}}{19 a^6 d}-\frac {24 (a \sin (c+d x)+a)^{17/2}}{17 a^5 d}+\frac {16 (a \sin (c+d x)+a)^{15/2}}{15 a^4 d} \]
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))^{7/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a-x)^3 (a+x)^{13/2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^3 (a+x)^{13/2}-12 a^2 (a+x)^{15/2}+6 a (a+x)^{17/2}-(a+x)^{19/2}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {16 (a+a \sin (c+d x))^{15/2}}{15 a^4 d}-\frac {24 (a+a \sin (c+d x))^{17/2}}{17 a^5 d}+\frac {12 (a+a \sin (c+d x))^{19/2}}{19 a^6 d}-\frac {2 (a+a \sin (c+d x))^{21/2}}{21 a^7 d}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 64, normalized size = 0.66 \[ -\frac {2 a^3 (\sin (c+d x)+1)^7 \left (1615 \sin ^3(c+d x)-5865 \sin ^2(c+d x)+7365 \sin (c+d x)-3243\right ) \sqrt {a (\sin (c+d x)+1)}}{33915 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 154, normalized size = 1.59 \[ \frac {2 \, {\left (1615 \, a^{3} \cos \left (d x + c\right )^{10} - 8300 \, a^{3} \cos \left (d x + c\right )^{8} + 264 \, a^{3} \cos \left (d x + c\right )^{6} + 448 \, a^{3} \cos \left (d x + c\right )^{4} + 1024 \, a^{3} \cos \left (d x + c\right )^{2} + 8192 \, a^{3} - 8 \, {\left (680 \, a^{3} \cos \left (d x + c\right )^{8} - 429 \, a^{3} \cos \left (d x + c\right )^{6} - 504 \, a^{3} \cos \left (d x + c\right )^{4} - 640 \, a^{3} \cos \left (d x + c\right )^{2} - 1024 \, a^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{33915 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 6.14, size = 669, normalized size = 6.90 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 57, normalized size = 0.59 \[ \frac {2 \left (a +a \sin \left (d x +c \right )\right )^{\frac {15}{2}} \left (1615 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-5865 \left (\cos ^{2}\left (d x +c \right )\right )-8980 \sin \left (d x +c \right )+9108\right )}{33915 a^{4} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 72, normalized size = 0.74 \[ -\frac {2 \, {\left (1615 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {21}{2}} - 10710 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {19}{2}} a + 23940 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {17}{2}} a^{2} - 18088 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {15}{2}} a^{3}\right )}}{33915 \, a^{7} d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^7\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{7/2} \,d x \]
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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