Optimal. Leaf size=159 \[ -\frac{32 a^2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a \sin (c+d x)+a}}-\frac{1024 a^4 \cos ^5(c+d x)}{3003 d (a \sin (c+d x)+a)^{3/2}}-\frac{4096 a^5 \cos ^5(c+d x)}{15015 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d} \]
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Rubi [A] time = 0.293338, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ -\frac{32 a^2 \cos ^5(c+d x) \sqrt{a \sin (c+d x)+a}}{143 d}-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a \sin (c+d x)+a}}-\frac{1024 a^4 \cos ^5(c+d x)}{3003 d (a \sin (c+d x)+a)^{3/2}}-\frac{4096 a^5 \cos ^5(c+d x)}{15015 d (a \sin (c+d x)+a)^{5/2}}-\frac{2 a \cos ^5(c+d x) (a \sin (c+d x)+a)^{3/2}}{13 d} \]
Antiderivative was successfully verified.
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Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \cos ^4(c+d x) (a+a \sin (c+d x))^{5/2} \, dx &=-\frac{2 a \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{1}{13} (16 a) \int \cos ^4(c+d x) (a+a \sin (c+d x))^{3/2} \, dx\\ &=-\frac{32 a^2 \cos ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}-\frac{2 a \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{1}{143} \left (192 a^2\right ) \int \cos ^4(c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a+a \sin (c+d x)}}-\frac{32 a^2 \cos ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}-\frac{2 a \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{1}{429} \left (512 a^3\right ) \int \frac{\cos ^4(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{1024 a^4 \cos ^5(c+d x)}{3003 d (a+a \sin (c+d x))^{3/2}}-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a+a \sin (c+d x)}}-\frac{32 a^2 \cos ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}-\frac{2 a \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}+\frac{\left (2048 a^4\right ) \int \frac{\cos ^4(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx}{3003}\\ &=-\frac{4096 a^5 \cos ^5(c+d x)}{15015 d (a+a \sin (c+d x))^{5/2}}-\frac{1024 a^4 \cos ^5(c+d x)}{3003 d (a+a \sin (c+d x))^{3/2}}-\frac{128 a^3 \cos ^5(c+d x)}{429 d \sqrt{a+a \sin (c+d x)}}-\frac{32 a^2 \cos ^5(c+d x) \sqrt{a+a \sin (c+d x)}}{143 d}-\frac{2 a \cos ^5(c+d x) (a+a \sin (c+d x))^{3/2}}{13 d}\\ \end{align*}
Mathematica [A] time = 0.309193, size = 79, normalized size = 0.5 \[ -\frac{2 \left (1155 \sin ^4(c+d x)+6300 \sin ^3(c+d x)+14210 \sin ^2(c+d x)+16700 \sin (c+d x)+9683\right ) \cos ^5(c+d x) (a (\sin (c+d x)+1))^{5/2}}{15015 d (\sin (c+d x)+1)^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 87, normalized size = 0.6 \begin{align*}{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ){a}^{3} \left ( \sin \left ( dx+c \right ) -1 \right ) ^{3} \left ( 1155\, \left ( \sin \left ( dx+c \right ) \right ) ^{4}+6300\, \left ( \sin \left ( dx+c \right ) \right ) ^{3}+14210\, \left ( \sin \left ( dx+c \right ) \right ) ^{2}+16700\,\sin \left ( dx+c \right ) +9683 \right ) }{15015\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.68916, size = 595, normalized size = 3.74 \begin{align*} \frac{2 \,{\left (1155 \, a^{2} \cos \left (d x + c\right )^{7} - 2835 \, a^{2} \cos \left (d x + c\right )^{6} - 6230 \, a^{2} \cos \left (d x + c\right )^{5} + 320 \, a^{2} \cos \left (d x + c\right )^{4} - 512 \, a^{2} \cos \left (d x + c\right )^{3} + 1024 \, a^{2} \cos \left (d x + c\right )^{2} - 4096 \, a^{2} \cos \left (d x + c\right ) - 8192 \, a^{2} -{\left (1155 \, a^{2} \cos \left (d x + c\right )^{6} + 3990 \, a^{2} \cos \left (d x + c\right )^{5} - 2240 \, a^{2} \cos \left (d x + c\right )^{4} - 2560 \, a^{2} \cos \left (d x + c\right )^{3} - 3072 \, a^{2} \cos \left (d x + c\right )^{2} - 4096 \, a^{2} \cos \left (d x + c\right ) - 8192 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15015 \,{\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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