Optimal. Leaf size=83 \[ -\frac{2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d} \]
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Rubi [A] time = 0.0910357, antiderivative size = 83, normalized size of antiderivative = 1., 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} \[ -\frac{2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \cos ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac{\operatorname{Subst}\left (\int (a+x)^{5/2} \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 ) (a+x)^{5/2}+2 a (a+x)^{7/2}-(a+x)^{9/2}\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))^{7/2}}{7 b^3 d}+\frac{4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d}-\frac{2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}\\ \end{align*}
Mathematica [A] time = 0.0858061, size = 58, normalized size = 0.7 \[ -\frac{2 (a+b \sin (c+d x))^{7/2} \left (8 a^2-28 a b \sin (c+d x)+63 b^2 \sin ^2(c+d x)-99 b^2\right )}{693 b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.224, size = 55, normalized size = 0.7 \begin{align*} -{\frac{-126\,{b}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}-56\,ab\sin \left ( dx+c \right ) +16\,{a}^{2}-72\,{b}^{2}}{693\,{b}^{3}d} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951021, size = 82, normalized size = 0.99 \begin{align*} -\frac{2 \,{\left (63 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{11}{2}} - 154 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{9}{2}} a + 99 \,{\left (a^{2} - b^{2}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{7}{2}}\right )}}{693 \, b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25444, size = 342, normalized size = 4.12 \begin{align*} -\frac{2 \,{\left (161 \, a b^{4} \cos \left (d x + c\right )^{4} + 8 \, a^{5} - 96 \, a^{3} b^{2} - 136 \, a b^{4} -{\left (3 \, a^{3} b^{2} + 25 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} +{\left (63 \, b^{5} \cos \left (d x + c\right )^{4} - 4 \, a^{4} b - 184 \, a^{2} b^{3} - 36 \, b^{5} -{\left (113 \, a^{2} b^{3} + 27 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{693 \, b^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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