Optimal. Leaf size=115 \[ -\frac {16 (a \sin (c+d x)+a)^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {8 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}} \]
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Rubi [A] time = 0.22, 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} \[ -\frac {16 (a \sin (c+d x)+a)^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}}+\frac {8 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{3 d e (e \cos (c+d x))^{5/2}} \]
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))^{7/2}} \, dx &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{3 d e (e \cos (c+d x))^{5/2}}+\frac {4 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{7/2}} \, dx}{3 a}\\ &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{3 d e (e \cos (c+d x))^{5/2}}+\frac {8 (a+a \sin (c+d x))^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac {8 \int \frac {(a+a \sin (c+d x))^{5/2}}{(e \cos (c+d x))^{7/2}} \, dx}{3 a^2}\\ &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{3 d e (e \cos (c+d x))^{5/2}}+\frac {8 (a+a \sin (c+d x))^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac {16 (a+a \sin (c+d x))^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 56, normalized size = 0.49 \[ \frac {2 \sqrt {a (\sin (c+d x)+1)} (4 \sin (c+d x)+4 \cos (2 (c+d x))+3)}{15 d e (e \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.08, size = 58, normalized size = 0.50 \[ \frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right ) - 1\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{15 \, d e^{4} \cos \left (d x + c\right )^{3}} \]
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 = 0.21, size = 54, normalized size = 0.47 \[ \frac {2 \left (8 \left (\cos ^{2}\left (d x +c \right )\right )+4 \sin \left (d x +c \right )-1\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right )}{15 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.98, size = 282, normalized size = 2.45 \[ \frac {2 \, {\left (7 \, \sqrt {a} \sqrt {e} + \frac {8 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {25 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {8 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {7 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{15 \, {\left (e^{4} + \frac {3 \, e^{4} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, e^{4} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {e^{4} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.06, size = 97, normalized size = 0.84 \[ \frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (2\,\sin \left (c+d\,x\right )+7\,\cos \left (2\,c+2\,d\,x\right )+2\,\cos \left (4\,c+4\,d\,x\right )+2\,\sin \left (3\,c+3\,d\,x\right )+5\right )}{15\,d\,e^3\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )+3\right )} \]
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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