Optimal. Leaf size=150 \[ \frac{4 \left (3 a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{8 a \left (a^2-b^2\right )}{b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right )^2}{3 b^5 d (a+b \sin (c+d x))^{3/2}}+\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{3/2}}{3 b^5 d} \]
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Rubi [A] time = 0.12241, antiderivative size = 150, 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} \[ \frac{4 \left (3 a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}{b^5 d}+\frac{8 a \left (a^2-b^2\right )}{b^5 d \sqrt{a+b \sin (c+d x)}}-\frac{2 \left (a^2-b^2\right )^2}{3 b^5 d (a+b \sin (c+d x))^{3/2}}+\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}-\frac{8 a (a+b \sin (c+d x))^{3/2}}{3 b^5 d} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 697
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{(a+x)^{5/2}} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\left (a^2-b^2\right )^2}{(a+x)^{5/2}}-\frac{4 \left (a^3-a b^2\right )}{(a+x)^{3/2}}+\frac{2 \left (3 a^2-b^2\right )}{\sqrt{a+x}}-4 a \sqrt{a+x}+(a+x)^{3/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=-\frac{2 \left (a^2-b^2\right )^2}{3 b^5 d (a+b \sin (c+d x))^{3/2}}+\frac{8 a \left (a^2-b^2\right )}{b^5 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \left (3 a^2-b^2\right ) \sqrt{a+b \sin (c+d x)}}{b^5 d}-\frac{8 a (a+b \sin (c+d x))^{3/2}}{3 b^5 d}+\frac{2 (a+b \sin (c+d x))^{5/2}}{5 b^5 d}\\ \end{align*}
Mathematica [A] time = 0.29451, size = 117, normalized size = 0.78 \[ \frac{16 \left (\left (6 a^2 b^2-3 b^4\right ) \sin ^2(c+d x)+3 a b \left (8 a^2-5 b^2\right ) \sin (c+d x)-10 a^2 b^2+16 a^4-a b^3 \sin ^3(c+d x)-b^4\right )+6 b^4 \cos ^4(c+d x)}{15 b^5 d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.332, size = 116, normalized size = 0.8 \begin{align*}{\frac{16\,a{b}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +2\, \left ( 192\,{a}^{3}b-128\,a{b}^{3} \right ) \sin \left ( dx+c \right ) +6\,{b}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{4}+2\, \left ( -48\,{a}^{2}{b}^{2}+24\,{b}^{4} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+256\,{a}^{4}-64\,{a}^{2}{b}^{2}-64\,{b}^{4}}{15\,{b}^{5}d} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953407, size = 165, normalized size = 1.1 \begin{align*} \frac{2 \,{\left (\frac{3 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 20 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 30 \,{\left (3 \, a^{2} - b^{2}\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{b^{4}} - \frac{5 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - 12 \,{\left (a^{3} - a b^{2}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} b^{4}}\right )}}{15 \, b d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.95455, size = 344, normalized size = 2.29 \begin{align*} -\frac{2 \,{\left (3 \, b^{4} \cos \left (d x + c\right )^{4} + 128 \, a^{4} - 32 \, a^{2} b^{2} - 32 \, b^{4} - 24 \,{\left (2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (a b^{3} \cos \left (d x + c\right )^{2} + 24 \, a^{3} b - 16 \, a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt{b \sin \left (d x + c\right ) + a}}{15 \,{\left (b^{7} d \cos \left (d x + c\right )^{2} - 2 \, a b^{6} d \sin \left (d x + c\right ) -{\left (a^{2} b^{5} + b^{7}\right )} d\right )}} \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 [A] time = 1.11167, size = 184, normalized size = 1.23 \begin{align*} \frac{2 \,{\left (3 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} - 20 \,{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}} a + 90 \, \sqrt{b \sin \left (d x + c\right ) + a} a^{2} - 30 \, \sqrt{b \sin \left (d x + c\right ) + a} b^{2} + \frac{5 \,{\left (12 \,{\left (b \sin \left (d x + c\right ) + a\right )} a^{3} - a^{4} - 12 \,{\left (b \sin \left (d x + c\right ) + a\right )} a b^{2} + 2 \, a^{2} b^{2} - b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\right )}}{15 \, b^{5} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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