Optimal. Leaf size=36 \[ -\frac {2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f} \]
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Rubi [A]
time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2815, 2752}
\begin {gather*} -\frac {2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2815
Rubi steps
\begin {align*} \int \frac {\sqrt {c-c \sin (e+f x)}}{(a+a \sin (e+f x))^2} \, dx &=\frac {\int \sec ^4(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{a^2 c^2}\\ &=-\frac {2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(36)=72\).
time = 0.09, size = 73, normalized size = 2.03 \begin {gather*} -\frac {2 \sqrt {c-c \sin (e+f x)}}{3 a^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.46, size = 49, normalized size = 1.36
method | result | size |
default | \(\frac {2 c \left (\sin \left (f x +e \right )-1\right )}{3 a^{2} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(49\) |
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. 161 vs.
\(2 (34) = 68\).
time = 0.52, size = 161, normalized size = 4.47 \begin {gather*} \frac {2 \, {\left (\sqrt {c} + \frac {2 \, \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {\sqrt {c} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} f \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 50, normalized size = 1.39 \begin {gather*} -\frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\right )\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*} \frac {\int \frac {\sqrt {- c \sin {\left (e + f x \right )} + c}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 116 vs.
\(2 (34) = 68\).
time = 0.50, size = 116, normalized size = 3.22 \begin {gather*} \frac {\sqrt {2} \sqrt {c} {\left (\frac {3 \, {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )}}{3 \, a^{2} f {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.35, size = 227, normalized size = 6.31 \begin {gather*} -\frac {4\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (\sin \left (2\,e+2\,f\,x\right )-4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-{\sin \left (e+f\,x\right )}^2\,2{}\mathrm {i}+2+2{}\mathrm {i}\right )}{3\,a^2\,f\,\left (-4\,{\sin \left (e+f\,x\right )}^2+\sin \left (e+f\,x\right )+\sin \left (3\,e+3\,f\,x\right )+4\right )}+\frac {4\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (-{\sin \left (e+f\,x\right )}^2\,4{}\mathrm {i}+\sin \left (e+f\,x\right )\,1{}\mathrm {i}-2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2+2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\,1{}\mathrm {i}+4{}\mathrm {i}\right )}{3\,a^2\,f\,\left (-8\,{\sin \left (e+f\,x\right )}^2+4\,\sin \left (e+f\,x\right )+2\,{\sin \left (2\,e+2\,f\,x\right )}^2+4\,\sin \left (3\,e+3\,f\,x\right )+8\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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