Optimal. Leaf size=296 \[ -\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {2 d (7 b c-3 a d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 b^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 d \left (6 a b c d-3 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 b^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-a d)^3 \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b^3 (a+b) f \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.72, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2872, 3138,
2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} -\frac {2 d \left (-3 a^2 d^2+6 a b c d-\left (b^2 \left (2 c^2+d^2\right )\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 b^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{b^3 f (a+b) \sqrt {c+d \sin (e+f x)}}+\frac {2 d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{3 b^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2872
Rule 2884
Rule 2886
Rule 3081
Rule 3138
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^{5/2}}{a+b \sin (e+f x)} \, dx &=-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {2 \int \frac {\frac {1}{2} \left (3 b c^3+a d^3\right )-\frac {1}{2} d \left (2 a c d-b \left (9 c^2+d^2\right )\right ) \sin (e+f x)+\frac {1}{2} d^2 (7 b c-3 a d) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 b}\\ &=-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}-\frac {2 \int \frac {\frac {1}{2} d \left (a c d (7 b c-3 a d)-b \left (3 b c^3+a d^3\right )\right )+\frac {1}{2} d^2 \left (6 a b c d-3 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{3 b^2 d}+\frac {(d (7 b c-3 a d)) \int \sqrt {c+d \sin (e+f x)} \, dx}{3 b^2}\\ &=-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {(b c-a d)^3 \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b^3}-\frac {\left (d \left (6 a b c d-3 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 b^3}+\frac {\left (d (7 b c-3 a d) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{3 b^2 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\\ &=-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {2 d (7 b c-3 a d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 b^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((b c-a d)^3 \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{b^3 \sqrt {c+d \sin (e+f x)}}-\frac {\left (d \left (6 a b c d-3 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{3 b^3 \sqrt {c+d \sin (e+f x)}}\\ &=-\frac {2 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {2 d (7 b c-3 a d) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{3 b^2 f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}-\frac {2 d \left (6 a b c d-3 a^2 d^2-b^2 \left (2 c^2+d^2\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{3 b^3 f \sqrt {c+d \sin (e+f x)}}+\frac {2 (b c-a d)^3 \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b^3 (a+b) f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 25.58, size = 606, normalized size = 2.05 \begin {gather*} \frac {\frac {4 i \left (-2 a c d+b \left (9 c^2+d^2\right )\right ) \left ((-b c+a d) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )-a d \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{b \sqrt {-\frac {1}{c+d}} (b c-a d)}+\frac {2 i (-7 b c+3 a d) \left (-2 b (c-d) (b c-a d) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+d \left (2 (a+b) (-b c+a d) F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )+\left (-2 a^2+b^2\right ) d \Pi \left (\frac {b (c+d)}{b c-a d};i \sinh ^{-1}\left (\sqrt {-\frac {1}{c+d}} \sqrt {c+d \sin (e+f x)}\right )|\frac {c+d}{c-d}\right )\right )\right ) \sec (e+f x) \sqrt {-\frac {d (-1+\sin (e+f x))}{c+d}} \sqrt {\frac {d (1+\sin (e+f x))}{-c+d}}}{b^2 \sqrt {-\frac {1}{c+d}} (b c-a d)}-4 d^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}-\frac {2 \left (6 b c^3+7 b c d^2-a d^3\right ) \Pi \left (\frac {2 b}{a+b};\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a+b) \sqrt {c+d \sin (e+f x)}}}{6 b f} \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. \(1189\) vs.
\(2(375)=750\).
time = 19.82, size = 1190, normalized size = 4.02
method | result | size |
default | \(\text {Expression too large to display}\) | \(1190\) |
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 [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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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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+b\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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