Optimal. Leaf size=152 \[ \frac{32 (a \sin (c+d x)+a)^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}-\frac{16 (a \sin (c+d x)+a)^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a \sin (c+d x)+a)^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}} \]
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Rubi [A] time = 0.308801, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac{32 (a \sin (c+d x)+a)^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}-\frac{16 (a \sin (c+d x)+a)^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a \sin (c+d x)+a)^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{2 (a \sin (c+d x)+a)^{3/2}}{3 d e (e \cos (c+d x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2672
Rule 2671
Rubi steps
\begin{align*} \int \frac{(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{11/2}} \, dx &=-\frac{2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac{2 \int \frac{(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{11/2}} \, dx}{a}\\ &=-\frac{2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a+a \sin (c+d x))^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{8 \int \frac{(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{11/2}} \, dx}{a^2}\\ &=-\frac{2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a+a \sin (c+d x))^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{16 (a+a \sin (c+d x))^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac{16 \int \frac{(a+a \sin (c+d x))^{9/2}}{(e \cos (c+d x))^{11/2}} \, dx}{5 a^3}\\ &=-\frac{2 (a+a \sin (c+d x))^{3/2}}{3 d e (e \cos (c+d x))^{9/2}}+\frac{4 (a+a \sin (c+d x))^{5/2}}{a d e (e \cos (c+d x))^{9/2}}-\frac{16 (a+a \sin (c+d x))^{7/2}}{5 a^2 d e (e \cos (c+d x))^{9/2}}+\frac{32 (a+a \sin (c+d x))^{9/2}}{45 a^3 d e (e \cos (c+d x))^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.248208, size = 74, normalized size = 0.49 \[ \frac{2 \sec ^5(c+d x) (a (\sin (c+d x)+1))^{3/2} \sqrt{e \cos (c+d x)} (6 \sin (c+d x)-4 \sin (3 (c+d x))+12 \cos (2 (c+d x))+7)}{45 d e^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.112, size = 70, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 32\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) -48\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-20\,\sin \left ( dx+c \right ) +10 \right ) \cos \left ( dx+c \right ) }{45\,d} \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( e\cos \left ( dx+c \right ) \right ) ^{-{\frac{11}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.66688, size = 482, normalized size = 3.17 \begin{align*} \frac{2 \,{\left (19 \, a^{\frac{3}{2}} \sqrt{e} - \frac{12 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{58 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{116 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac{116 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac{58 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{12 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac{19 \, a^{\frac{3}{2}} \sqrt{e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )}{\left (\frac{\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{45 \,{\left (e^{6} + \frac{4 \, e^{6} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{6 \, e^{6} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac{4 \, e^{6} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac{e^{6} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{5}{2}}{\left (-\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38017, size = 248, normalized size = 1.63 \begin{align*} -\frac{2 \,{\left (24 \, a \cos \left (d x + c\right )^{2} - 2 \,{\left (8 \, a \cos \left (d x + c\right )^{2} - 5 \, a\right )} \sin \left (d x + c\right ) - 5 \, a\right )} \sqrt{e \cos \left (d x + c\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{45 \,{\left (d e^{6} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) - d e^{6} \cos \left (d x + c\right )^{3}\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(-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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