Optimal. Leaf size=76 \[ -\frac {2 \sqrt {e \cos (c+d x)}}{5 d e (a+a \sin (c+d x))^{3/2}}-\frac {4 \sqrt {e \cos (c+d x)}}{5 a d e \sqrt {a+a \sin (c+d x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 2, 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 {4 \sqrt {e \cos (c+d x)}}{5 a d e \sqrt {a \sin (c+d x)+a}}-\frac {2 \sqrt {e \cos (c+d x)}}{5 d e (a \sin (c+d x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2750
Rule 2751
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {2 \sqrt {e \cos (c+d x)}}{5 d e (a+a \sin (c+d x))^{3/2}}+\frac {2 \int \frac {1}{\sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}} \, dx}{5 a}\\ &=-\frac {2 \sqrt {e \cos (c+d x)}}{5 d e (a+a \sin (c+d x))^{3/2}}-\frac {4 \sqrt {e \cos (c+d x)}}{5 a d e \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 59, normalized size = 0.78 \begin {gather*} -\frac {2 \sqrt {e \cos (c+d x)} \sqrt {a (1+\sin (c+d x))} (3+2 \sin (c+d x))}{5 a^2 d e (1+\sin (c+d x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 44, normalized size = 0.58
method | result | size |
default | \(-\frac {2 \left (2 \sin \left (d x +c \right )+3\right ) \cos \left (d x +c \right )}{5 d \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}} \sqrt {e \cos \left (d x +c \right )}}\) | \(44\) |
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. 197 vs.
\(2 (58) = 116\).
time = 0.56, size = 197, normalized size = 2.59 \begin {gather*} -\frac {2 \, {\left (3 \, \sqrt {a} + \frac {4 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3 \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2} e^{\left (-\frac {1}{2}\right )}}{5 \, {\left (a^{2} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}} \sqrt {-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 72, normalized size = 0.95 \begin {gather*} \frac {2 \, \sqrt {a \sin \left (d x + c\right ) + a} {\left (2 \, \sin \left (d x + c\right ) + 3\right )} \sqrt {\cos \left (d x + c\right )}}{5 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} e^{\frac {1}{2}} - 2 \, a^{2} d e^{\frac {1}{2}} \sin \left (d x + c\right ) - 2 \, a^{2} d e^{\frac {1}{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}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {e \cos {\left (c + d x \right )}}}\, 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 = 6.38, size = 95, normalized size = 1.25 \begin {gather*} -\frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (7\,\cos \left (c+d\,x\right )-\cos \left (3\,c+3\,d\,x\right )+5\,\sin \left (2\,c+2\,d\,x\right )\right )}{5\,a^2\,d\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (15\,\sin \left (c+d\,x\right )-6\,\cos \left (2\,c+2\,d\,x\right )-\sin \left (3\,c+3\,d\,x\right )+10\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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