Optimal. Leaf size=126 \[ -\frac {(3 c+5 d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+5 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2829, 2729,
2728, 212} \begin {gather*} -\frac {(3 c+5 d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(3 c+5 d) \cos (e+f x)}{16 a f (a \sin (e+f x)+a)^{3/2}}-\frac {(c-d) \cos (e+f x)}{4 f (a \sin (e+f x)+a)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2728
Rule 2729
Rule 2829
Rubi steps
\begin {align*} \int \frac {c+d \sin (e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {(c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}+\frac {(3 c+5 d) \int \frac {1}{(a+a \sin (e+f x))^{3/2}} \, dx}{8 a}\\ &=-\frac {(c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+5 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}+\frac {(3 c+5 d) \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{32 a^2}\\ &=-\frac {(c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+5 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}-\frac {(3 c+5 d) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{16 a^2 f}\\ &=-\frac {(3 c+5 d) \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{16 \sqrt {2} a^{5/2} f}-\frac {(c-d) \cos (e+f x)}{4 f (a+a \sin (e+f x))^{5/2}}-\frac {(3 c+5 d) \cos (e+f x)}{16 a f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.25, size = 227, normalized size = 1.80 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (8 (c-d) \sin \left (\frac {1}{2} (e+f x)\right )+4 (-c+d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 (3 c+5 d) \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-(3 c+5 d) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+(1+i) (-1)^{3/4} (3 c+5 d) \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4\right )}{16 f (a (1+\sin (e+f x)))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs.
\(2(107)=214\).
time = 3.87, size = 278, normalized size = 2.21
method | result | size |
default | \(\frac {\left (-2 \sin \left (f x +e \right ) \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (3 c +5 d \right )+\sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} \left (3 c +5 d \right ) \left (\cos ^{2}\left (f x +e \right )\right )-20 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {5}{2}} c -12 \sqrt {a -a \sin \left (f x +e \right )}\, a^{\frac {5}{2}} d +6 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} c +10 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} a^{\frac {3}{2}} d -6 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} c -10 \sqrt {2}\, \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{3} d \right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{32 a^{\frac {11}{2}} \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(278\) |
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. 420 vs.
\(2 (113) = 226\).
time = 0.37, size = 420, normalized size = 3.33 \begin {gather*} \frac {\sqrt {2} {\left ({\left (3 \, c + 5 \, d\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (3 \, c + 5 \, d\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (3 \, c + 5 \, d\right )} \cos \left (f x + e\right ) + {\left ({\left (3 \, c + 5 \, d\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (3 \, c + 5 \, d\right )} \cos \left (f x + e\right ) - 12 \, c - 20 \, d\right )} \sin \left (f x + e\right ) - 12 \, c - 20 \, d\right )} \sqrt {a} \log \left (-\frac {a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )} + 3 \, a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) + 2 \, a}{\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 ) + 4 \, {\left ({\left (3 \, c + 5 \, d\right )} \cos \left (f x + e\right )^{2} + {\left (7 \, c + d\right )} \cos \left (f x + e\right ) + {\left ({\left (3 \, c + 5 \, d\right )} \cos \left (f x + e\right ) - 4 \, c + 4 \, d\right )} \sin \left (f x + e\right ) + 4 \, c - 4 \, d\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{64 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} 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*} \int \frac {c + d \sin {\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \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. 246 vs.
\(2 (113) = 226\).
time = 0.61, size = 246, normalized size = 1.95 \begin {gather*} \frac {\frac {\sqrt {2} {\left (3 \, \sqrt {a} c + 5 \, \sqrt {a} d\right )} \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {\sqrt {2} {\left (3 \, \sqrt {a} c + 5 \, \sqrt {a} d\right )} \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {2 \, {\left (3 \, \sqrt {2} \sqrt {a} c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 5 \, \sqrt {2} \sqrt {a} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, \sqrt {2} \sqrt {a} c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, \sqrt {2} \sqrt {a} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{2} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{64 \, 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 {c+d\,\sin \left (e+f\,x\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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