Optimal. Leaf size=154 \[ -\frac {32 (a \sin (c+d x)+a)^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}}+\frac {16 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a \sin (c+d x)+a)^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 154, normalized size of antiderivative = 1.00, 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} \[ -\frac {32 (a \sin (c+d x)+a)^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}}+\frac {16 (a \sin (c+d x)+a)^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a \sin (c+d x)+a)^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{5 d e (e \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{9/2}} \, dx &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}+\frac {6 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{9/2}} \, dx}{5 a}\\ &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {24 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{9/2}} \, dx}{5 a^2}\\ &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {16 \int \frac {(a+a \sin (c+d x))^{7/2}}{(e \cos (c+d x))^{9/2}} \, dx}{5 a^3}\\ &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {32 (a+a \sin (c+d x))^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.80, size = 74, normalized size = 0.48 \[ \frac {2 \sec ^4(c+d x) \sqrt {a (\sin (c+d x)+1)} \sqrt {e \cos (c+d x)} (10 \sin (c+d x)+4 \sin (3 (c+d x))-4 \cos (2 (c+d x))-5)}{35 d e^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 70, normalized size = 0.45 \[ -\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} - 2 \, {\left (8 \, \cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) + 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{35 \, d e^{5} \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.23, size = 70, normalized size = 0.45 \[ \frac {2 \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-8 \left (\cos ^{2}\left (d x +c \right )\right )+6 \sin \left (d x +c \right )-1\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right )}{35 d \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 357, normalized size = 2.32 \[ -\frac {2 \, {\left (9 \, \sqrt {a} \sqrt {e} - \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {84 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {44 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {9 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{4}}{35 \, {\left (e^{5} + \frac {4 \, e^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, e^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, e^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {e^{5} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.24, size = 129, normalized size = 0.84 \[ -\frac {16\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (23\,\cos \left (c+d\,x\right )+11\,\cos \left (3\,c+3\,d\,x\right )+2\,\cos \left (5\,c+5\,d\,x\right )-16\,\sin \left (2\,c+2\,d\,x\right )-11\,\sin \left (4\,c+4\,d\,x\right )-2\,\sin \left (6\,c+6\,d\,x\right )\right )}{35\,d\,e^4\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (15\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+\cos \left (6\,c+6\,d\,x\right )+10\right )} \]
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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