Optimal. Leaf size=154 \[ -\frac {2 \sqrt {a+a \sin (c+d x)}}{5 d e (e \cos (c+d x))^{7/2}}-\frac {12 (a+a \sin (c+d x))^{3/2}}{5 a d e (e \cos (c+d x))^{7/2}}+\frac {16 (a+a \sin (c+d x))^{5/2}}{5 a^2 d e (e \cos (c+d x))^{7/2}}-\frac {32 (a+a \sin (c+d x))^{7/2}}{35 a^3 d e (e \cos (c+d x))^{7/2}} \]
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Rubi [A]
time = 0.20, 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 = {2751, 2750}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2751
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 = 0.85, size = 74, normalized size = 0.48 \begin {gather*} \frac {2 \sqrt {e \cos (c+d x)} \sec ^4(c+d x) \sqrt {a (1+\sin (c+d x))} (-5-4 \cos (2 (c+d x))+10 \sin (c+d x)+4 \sin (3 (c+d x)))}{35 d e^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 70, normalized size = 0.45
method | result | size |
default | \(\frac {2 \left (16 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-8 \left (\cos ^{2}\left (d x +c \right )\right )+6 \sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}{35 d \left (e \cos \left (d x +c \right )\right )^{\frac {9}{2}}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 321 vs.
\(2 (118) = 236\).
time = 0.55, size = 321, normalized size = 2.08 \begin {gather*} -\frac {2 \, {\left (9 \, \sqrt {a} - \frac {44 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14 \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {84 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {44 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {9 \, \sqrt {a} \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} e^{\left (-\frac {9}{2}\right )}}{35 \, 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}} {\left (\frac {4 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {6 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {4 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {\sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 59, normalized size = 0.38 \begin {gather*} -\frac {2 \, {\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} e^{\left (-\frac {9}{2}\right )}}{35 \, d \cos \left (d x + c\right )^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} -\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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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