Optimal. Leaf size=98 \[ -\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f} \]
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Rubi [A]
time = 0.18, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753,
2752} \begin {gather*} -\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2753
Rule 2815
Rubi steps
\begin {align*} \int \frac {(c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx &=\frac {\int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{a c}\\ &=\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {8 \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{3 a}\\ &=\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {(32 c) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{3 a}\\ &=-\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 102, normalized size = 1.04 \begin {gather*} -\frac {c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (45+\cos (2 (e+f x))+20 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{3 a f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.50, size = 59, normalized size = 0.60
method | result | size |
default | \(-\frac {2 c^{3} \left (\sin \left (f x +e \right )-1\right ) \left (\sin ^{2}\left (f x +e \right )-10 \sin \left (f x +e \right )-23\right )}{3 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(59\) |
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. 208 vs.
\(2 (92) = 184\).
time = 0.60, size = 208, normalized size = 2.12 \begin {gather*} \frac {2 \, {\left (23 \, c^{\frac {5}{2}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {40 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {23 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )}}{3 \, {\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 62, normalized size = 0.63 \begin {gather*} -\frac {2 \, {\left (c^{2} \cos \left (f x + e\right )^{2} + 10 \, c^{2} \sin \left (f x + e\right ) + 22 \, c^{2}\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a f \cos \left (f x + e\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 {c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}}{\sin {\left (e + f x \right )} + 1}\, dx + \int \left (- \frac {2 c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\right )\, dx + \int \frac {c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}}{\sin {\left (e + f x \right )} + 1}\, dx}{a} \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. 237 vs.
\(2 (92) = 184\).
time = 0.57, size = 237, normalized size = 2.42 \begin {gather*} \frac {8 \, \sqrt {2} \sqrt {c} {\left (\frac {3 \, c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a {\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 )}} - \frac {5 \, c^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \frac {12 \, c^{2} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {3 \, c^{2} {\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}}}{a {\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}}\right )}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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