Optimal. Leaf size=166 \[ \frac {a^{5/2} (c-d) (3 c+5 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{5/2} (c+d)^{3/2} f}-\frac {a^3 (3 c+d) \cos (e+f x)}{d^2 (c+d) f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.27, antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2841, 3060,
2852, 214} \begin {gather*} \frac {a^{5/2} (c-d) (3 c+5 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{5/2} f (c+d)^{3/2}}-\frac {a^3 (3 c+d) \cos (e+f x)}{d^2 f (c+d) \sqrt {a \sin (e+f x)+a}}+\frac {a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{d f (c+d) (c+d \sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2841
Rule 2852
Rule 3060
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^2} \, dx &=\frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}-\frac {a \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (c-5 d)-\frac {1}{2} a (3 c+d) \sin (e+f x)\right )}{c+d \sin (e+f x)} \, dx}{d (c+d)}\\ &=-\frac {a^3 (3 c+d) \cos (e+f x)}{d^2 (c+d) f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}-\frac {\left (a^2 (c-d) (3 c+5 d)\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 d^2 (c+d)}\\ &=-\frac {a^3 (3 c+d) \cos (e+f x)}{d^2 (c+d) f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}+\frac {\left (a^3 (c-d) (3 c+5 d)\right ) \text {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d^2 (c+d) f}\\ &=\frac {a^{5/2} (c-d) (3 c+5 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{5/2} (c+d)^{3/2} f}-\frac {a^3 (3 c+d) \cos (e+f x)}{d^2 (c+d) f \sqrt {a+a \sin (e+f x)}}+\frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{d (c+d) f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(350\) vs. \(2(166)=332\).
time = 2.83, size = 350, normalized size = 2.11 \begin {gather*} \frac {(a (1+\sin (e+f x)))^{5/2} \left (-8 \sqrt {d} \cos \left (\frac {1}{2} (e+f x)\right )+\frac {\left (3 c^2+2 c d-5 d^2\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+2 \log \left (-\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (c+d+\sqrt {d} \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {d} \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )}{(c+d)^{3/2}}+\frac {\left (-3 c^2-2 c d+5 d^2\right ) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )+2 \log \left ((c+d) \sec ^2\left (\frac {1}{4} (e+f x)\right )+\sqrt {d} \sqrt {c+d} \left (-1+2 \tan \left (\frac {1}{4} (e+f x)\right )+\tan ^2\left (\frac {1}{4} (e+f x)\right )\right )\right )\right )}{(c+d)^{3/2}}+8 \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right )-\frac {4 (c-d)^2 \sqrt {d} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d) (c+d \sin (e+f x))}\right )}{4 d^{5/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \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. \(392\) vs.
\(2(148)=296\).
time = 5.43, size = 393, normalized size = 2.37
method | result | size |
default | \(-\frac {a^{2} \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-\sin \left (f x +e \right ) d \left (3 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +d^{2} a}}\right ) a \,c^{2}+2 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +d^{2} a}}\right ) a c d -5 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +d^{2} a}}\right ) a \,d^{2}-2 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, c -2 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, d \right )-3 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +d^{2} a}}\right ) a \,c^{3}-2 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +d^{2} a}}\right ) a \,c^{2} d +5 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +d^{2} a}}\right ) a c \,d^{2}+3 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, c^{2}+\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, d^{2}\right )}{d^{2} \left (c +d \right ) \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a \left (c +d \right ) d}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(393\) |
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. 519 vs.
\(2 (155) = 310\).
time = 0.45, size = 1368, normalized size = 8.24 \begin {gather*} \left [\frac {{\left (3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} - {\left (3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, a^{2} c^{3} + 2 \, a^{2} c^{2} d - 5 \, a^{2} c d^{2}\right )} \cos \left (f x + e\right ) + {\left (3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} + {\left (3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {a}{c d + d^{2}}} \log \left (\frac {a d^{2} \cos \left (f x + e\right )^{3} - a c^{2} - 2 \, a c d - a d^{2} - {\left (6 \, a c d + 7 \, a d^{2}\right )} \cos \left (f x + e\right )^{2} + 4 \, {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} - {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (c^{2} d + 3 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right ) - {\left (c^{2} d + 4 \, c d^{2} + 3 \, d^{3} + {\left (c d^{2} + d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {\frac {a}{c d + d^{2}}} - {\left (a c^{2} + 8 \, a c d + 9 \, a d^{2}\right )} \cos \left (f x + e\right ) + {\left (a d^{2} \cos \left (f x + e\right )^{2} - a c^{2} - 2 \, a c d - a d^{2} + 2 \, {\left (3 \, a c d + 4 \, a d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{d^{2} \cos \left (f x + e\right )^{3} + {\left (2 \, c d + d^{2}\right )} \cos \left (f x + e\right )^{2} - c^{2} - 2 \, c d - d^{2} - {\left (c^{2} + d^{2}\right )} \cos \left (f x + e\right ) + {\left (d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \cos \left (f x + e\right ) - c^{2} - 2 \, c d - d^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} + 2 \, {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, a^{2} c^{2} + a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} - 2 \, {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{4 \, {\left ({\left (c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{2} d^{2} + c d^{3}\right )} f \cos \left (f x + e\right ) - {\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f - {\left ({\left (c d^{3} + d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f\right )} \sin \left (f x + e\right )\right )}}, -\frac {{\left (3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} - {\left (3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, a^{2} c^{3} + 2 \, a^{2} c^{2} d - 5 \, a^{2} c d^{2}\right )} \cos \left (f x + e\right ) + {\left (3 \, a^{2} c^{3} + 5 \, a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 5 \, a^{2} d^{3} + {\left (3 \, a^{2} c^{2} d + 2 \, a^{2} c d^{2} - 5 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {-\frac {a}{c d + d^{2}}} \arctan \left (\frac {\sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) - c - 2 \, d\right )} \sqrt {-\frac {a}{c d + d^{2}}}}{2 \, a \cos \left (f x + e\right )}\right ) - 2 \, {\left (3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} + 2 \, {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )^{2} + {\left (3 \, a^{2} c^{2} + a^{2} d^{2}\right )} \cos \left (f x + e\right ) - {\left (3 \, a^{2} c^{2} - 2 \, a^{2} c d - a^{2} d^{2} - 2 \, {\left (a^{2} c d + a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{2 \, {\left ({\left (c d^{3} + d^{4}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{2} d^{2} + c d^{3}\right )} f \cos \left (f x + e\right ) - {\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f - {\left ({\left (c d^{3} + d^{4}\right )} f \cos \left (f x + e\right ) + {\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} f\right )} \sin \left (f x + e\right )\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 314 vs.
\(2 (155) = 310\).
time = 0.56, size = 314, normalized size = 1.89 \begin {gather*} \frac {\sqrt {2} {\left (\frac {4 \, a^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{d^{2}} + \frac {\sqrt {2} {\left (3 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 5 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \arctan \left (\frac {\sqrt {2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c d - d^{2}}}\right )}{{\left (c d^{2} + d^{3}\right )} \sqrt {-c d - d^{2}}} - \frac {2 \, {\left (a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (c d^{2} + d^{3}\right )} {\left (2 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}}\right )} \sqrt {a}}{2 \, 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 )}^{5/2}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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