Optimal. Leaf size=150 \[ -\frac {10 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac {a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {10 a^3 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A]
time = 0.22, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2815, 2759,
2758, 2728, 212} \begin {gather*} -\frac {10 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac {a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {10 a^3 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2758
Rule 2759
Rule 2815
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{3/2}} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}-\frac {1}{2} \left (5 a^3 c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}-\left (5 a^3\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {10 a^3 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}-\frac {\left (10 a^3\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{c}\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {10 a^3 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}+\frac {\left (20 a^3\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{c f}\\ &=-\frac {10 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{c^{3/2} f}+\frac {a^3 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}+\frac {10 a^3 \cos (e+f x)}{c f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.50, size = 173, normalized size = 1.15 \begin {gather*} -\frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (50 \cos \left (\frac {1}{2} (e+f x)\right )+25 \cos \left (\frac {3}{2} (e+f x)\right )+\cos \left (\frac {5}{2} (e+f x)\right )+50 \sin \left (\frac {1}{2} (e+f x)\right )-(120+120 i) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) (-1+\sin (e+f x))-25 \sin \left (\frac {3}{2} (e+f x)\right )+\sin \left (\frac {5}{2} (e+f x)\right )\right )}{6 c f (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.88, size = 189, normalized size = 1.26
method | result | size |
default | \(\frac {2 a^{3} \left (15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c^{2}-\left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \sqrt {c}\, \sin \left (f x +e \right )-12 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {3}{2}} \sin \left (f x +e \right )-15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{2}+\left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \sqrt {c}+18 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {3}{2}}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{3 c^{\frac {7}{2}} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(189\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs.
\(2 (141) = 282\).
time = 0.36, size = 352, normalized size = 2.35 \begin {gather*} \frac {\frac {15 \, \sqrt {2} {\left (a^{3} c \cos \left (f x + e\right )^{2} - a^{3} c \cos \left (f x + e\right ) - 2 \, a^{3} c + {\left (a^{3} c \cos \left (f x + e\right ) + 2 \, a^{3} c\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{\sqrt {c}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt {c}} - 2 \, {\left (a^{3} \cos \left (f x + e\right )^{3} + 13 \, a^{3} \cos \left (f x + e\right )^{2} + 18 \, a^{3} \cos \left (f x + e\right ) + 6 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 12 \, a^{3} \cos \left (f x + e\right ) + 6 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (c^{2} f \cos \left (f x + e\right )^{2} - c^{2} f \cos \left (f x + e\right ) - 2 \, c^{2} f + {\left (c^{2} f \cos \left (f x + e\right ) + 2 \, c^{2} f\right )} \sin \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*} a^{3} \left (\int \frac {3 \sin {\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {3 \sin ^{2}{\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {\sin ^{3}{\left (e + f x \right )}}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx + \int \frac {1}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx\right ) \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. 376 vs.
\(2 (141) = 282\).
time = 0.57, size = 376, normalized size = 2.51 \begin {gather*} -\frac {\frac {30 \, \sqrt {2} a^{3} \log \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}\right )}{c^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {3 \, \sqrt {2} a^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{c^{\frac {3}{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 )} - \frac {3 \, \sqrt {2} {\left (a^{3} + \frac {10 \, a^{3} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1}\right )} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}{c^{\frac {3}{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 )} - \frac {16 \, \sqrt {2} {\left (7 \, a^{3} \sqrt {c} - \frac {12 \, a^{3} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + \frac {9 \, a^{3} \sqrt {c} {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2}}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}}\right )}}{c^{2} {\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} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{6 \, 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 (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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