Optimal. Leaf size=83 \[ -\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^3 d}-\frac {2 (a+b \sin (c+d x))^{7/2}}{7 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{5/2}}{5 b^3 d} \]
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Rubi [A] time = 0.08, antiderivative size = 83, 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 = {2668, 697} \[ -\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^3 d}-\frac {2 (a+b \sin (c+d x))^{7/2}}{7 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{5/2}}{5 b^3 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \sqrt {a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \sqrt {a+x} \left (b^2-x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (-a^2+b^2\right ) \sqrt {a+x}+2 a (a+x)^{3/2}-(a+x)^{5/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}{3 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{5/2}}{5 b^3 d}-\frac {2 (a+b \sin (c+d x))^{7/2}}{7 b^3 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 58, normalized size = 0.70 \[ \frac {(a+b \sin (c+d x))^{3/2} \left (-16 a^2+24 a b \sin (c+d x)+15 b^2 \cos (2 (c+d x))+55 b^2\right )}{105 b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 78, normalized size = 0.94 \[ \frac {2 \, {\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - 8 \, a^{3} + 32 \, a b^{2} + {\left (15 \, b^{3} \cos \left (d x + c\right )^{2} + 4 \, a^{2} b + 20 \, b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{105 \, b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.17, size = 78, normalized size = 0.94 \[ -\frac {2 \, {\left (\frac {15 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}}{b^{3}} - \frac {42 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a}{b^{3}} + \frac {35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2}}{b^{3}} - \frac {35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}{b}\right )}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 55, normalized size = 0.66 \[ -\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {3}{2}} \left (-15 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )-12 a b \sin \left (d x +c \right )+8 a^{2}-20 b^{2}\right )}{105 b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 61, normalized size = 0.73 \[ -\frac {2 \, {\left (15 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 42 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 35 \, {\left (a^{2} - b^{2}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\right )}}{105 \, b^{3} d} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,\sqrt {a+b\,\sin \left (c+d\,x\right )} \,d x \]
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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