Optimal. Leaf size=115 \[ -\frac {16 \sqrt {e \cos (c+d x)}}{45 a^2 d e \sqrt {a \sin (c+d x)+a}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.20, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ -\frac {16 \sqrt {e \cos (c+d x)}}{45 a^2 d e \sqrt {a \sin (c+d x)+a}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a \sin (c+d x)+a)^{3/2}}-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}+\frac {4 \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx}{9 a}\\ &=-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}+\frac {8 \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}} \, dx}{45 a^2}\\ &=-\frac {2 \sqrt {e \cos (c+d x)}}{9 d e (a+a \sin (c+d x))^{5/2}}-\frac {8 \sqrt {e \cos (c+d x)}}{45 a d e (a+a \sin (c+d x))^{3/2}}-\frac {16 \sqrt {e \cos (c+d x)}}{45 a^2 d e \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 69, normalized size = 0.60 \[ -\frac {2 \left (8 \sin ^2(c+d x)+20 \sin (c+d x)+17\right ) \sqrt {a (\sin (c+d x)+1)} \sqrt {e \cos (c+d x)}}{45 a^3 d e (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.88, size = 98, normalized size = 0.85 \[ -\frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} - 20 \, \sin \left (d x + c\right ) - 25\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{45 \, {\left (3 \, a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e + {\left (a^{3} d e \cos \left (d x + c\right )^{2} - 4 \, a^{3} d e\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 54, normalized size = 0.47 \[ -\frac {2 \left (-8 \left (\cos ^{2}\left (d x +c \right )\right )+20 \sin \left (d x +c \right )+25\right ) \cos \left (d x +c \right )}{45 d \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}} \sqrt {e \cos \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 287, normalized size = 2.50 \[ -\frac {2 \, {\left (17 \, \sqrt {a} \sqrt {e} + \frac {40 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {49 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {49 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {40 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {17 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{45 \, {\left (a^{3} e + \frac {3 \, a^{3} e \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{3} e \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} e \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {11}{2}} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.66, size = 137, normalized size = 1.19 \[ -\frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (137\,\cos \left (c+d\,x\right )-71\,\cos \left (3\,c+3\,d\,x\right )+2\,\cos \left (5\,c+5\,d\,x\right )+144\,\sin \left (2\,c+2\,d\,x\right )-18\,\sin \left (4\,c+4\,d\,x\right )\right )}{45\,a^3\,d\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (210\,\sin \left (c+d\,x\right )-120\,\cos \left (2\,c+2\,d\,x\right )+10\,\cos \left (4\,c+4\,d\,x\right )-45\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+126\right )} \]
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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