Optimal. Leaf size=115 \[ -\frac {2 \sqrt {a+a \sin (c+d x)}}{3 d e (e \cos (c+d x))^{5/2}}+\frac {8 (a+a \sin (c+d x))^{3/2}}{3 a d e (e \cos (c+d x))^{5/2}}-\frac {16 (a+a \sin (c+d x))^{5/2}}{15 a^2 d e (e \cos (c+d x))^{5/2}} \]
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Rubi [A]
time = 0.15, 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)^{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}} \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))^{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 = 0.36, size = 56, normalized size = 0.49 \begin {gather*} \frac {2 \sqrt {a (1+\sin (c+d x))} (3+4 \cos (2 (c+d x))+4 \sin (c+d x))}{15 d e (e \cos (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 54, normalized size = 0.47
method | result | size |
default | \(\frac {2 \left (8 \left (\cos ^{2}\left (d x +c \right )\right )+4 \sin \left (d x +c \right )-1\right ) \cos \left (d x +c \right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}{15 d \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}\) | \(54\) |
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. 255 vs.
\(2 (88) = 176\).
time = 0.55, size = 255, normalized size = 2.22 \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 {7}{2}\right )}}{15 \, 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}} {\left (\frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 47, normalized size = 0.41 \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} e^{\left (-\frac {7}{2}\right )}}{15 \, d \cos \left (d x + c\right )^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = 6.06, size = 97, normalized size = 0.84 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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