Optimal. Leaf size=115 \[ -\frac {2}{5 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {8 \sqrt {a+a \sin (c+d x)}}{5 a d e (e \cos (c+d x))^{3/2}}+\frac {16 (a+a \sin (c+d x))^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.14, 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 = {2751, 2750}
\begin {gather*} \frac {16 (a \sin (c+d x)+a)^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}-\frac {8 \sqrt {a \sin (c+d x)+a}}{5 a d e (e \cos (c+d x))^{3/2}}-\frac {2}{5 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2751
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}} \, dx &=-\frac {2}{5 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}+\frac {4 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{5 a}\\ &=-\frac {2}{5 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {8 \sqrt {a+a \sin (c+d x)}}{5 a d e (e \cos (c+d x))^{3/2}}+\frac {8 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{5 a^2}\\ &=-\frac {2}{5 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {8 \sqrt {a+a \sin (c+d x)}}{5 a d e (e \cos (c+d x))^{3/2}}+\frac {16 (a+a \sin (c+d x))^{3/2}}{15 a^2 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 56, normalized size = 0.49 \begin {gather*} \frac {2 \left (-7+4 \sin (c+d x)+8 \sin ^2(c+d x)\right )}{15 d e (e \cos (c+d x))^{3/2} \sqrt {a (1+\sin (c+d x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 54, normalized size = 0.47
method | result | size |
default | \(-\frac {2 \left (8 \left (\cos ^{2}\left (d x +c \right )\right )-4 \sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right )}{15 d \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}\) | \(54\) |
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. 258 vs.
\(2 (88) = 176\).
time = 0.57, size = 258, normalized size = 2.24 \begin {gather*} -\frac {2 \, {\left (7 \, \sqrt {a} - \frac {8 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {25 \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {25 \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {8 \, \sqrt {a} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {7 \, \sqrt {a} \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} e^{\left (-\frac {5}{2}\right )}}{15 \, {\left (a + \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a \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 {7}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 77, normalized size = 0.67 \begin {gather*} -\frac {2 \, {\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} \sqrt {\cos \left (d x + c\right )}}{15 \, {\left (a d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \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 = 6.68, size = 120, normalized size = 1.04 \begin {gather*} -\frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (8\,\cos \left (c+d\,x\right )+6\,\cos \left (3\,c+3\,d\,x\right )-\sin \left (2\,c+2\,d\,x\right )+2\,\sin \left (4\,c+4\,d\,x\right )\right )}{15\,a\,d\,e^2\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\sin \left (c+d\,x\right )+4\,\cos \left (2\,c+2\,d\,x\right )-\cos \left (4\,c+4\,d\,x\right )+4\,\sin \left (3\,c+3\,d\,x\right )+5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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