Optimal. Leaf size=390 \[ \frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{b^2 \left (a^2-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(b c-a d) \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 d^2\right )\right ) F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{b^3 \left (a^2-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^2 \left (2 a b c+3 a^2 d-5 b^2 d\right ) \Pi \left (\frac {2 b}{a+b};\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{(a-b) b^3 (a+b)^2 f \sqrt {c+d \sin (e+f x)}} \]
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Rubi [A]
time = 0.80, antiderivative size = 390, 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 = {2871, 3138,
2734, 2732, 3081, 2742, 2740, 2886, 2884} \begin {gather*} -\frac {\left (-3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2-2 d^2\right )\right )\right ) \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 )}{b^2 f \left (a^2-b^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b f \left (a^2-b^2\right ) (a+b \sin (e+f x))}+\frac {\left (3 a^2 d^2+2 a b c d-\left (b^2 \left (c^2+4 d^2\right )\right )\right ) (b c-a d) \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 )}{b^3 f \left (a^2-b^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\left (3 a^2 d+2 a b c-5 b^2 d\right ) (b c-a d)^2 \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) (a+b)^2 \sqrt {c+d \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2871
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))^2} \, dx &=\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\int \frac {\frac {1}{2} \left (5 b^2 c^2 d+a^2 d^3-2 a b c \left (c^2+2 d^2\right )\right )-d \left (a^2 c d-3 b^2 c d+a b \left (c^2+d^2\right )\right ) \sin (e+f x)+\frac {1}{2} d \left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\int \frac {\frac {1}{2} d (b c-a d) \left (a b c^2+3 a^2 c d-5 b^2 c d+a b d^2\right )-\frac {1}{2} d (b c-a d) \left (b^2 c^2-2 a b c d-3 a^2 d^2+4 b^2 d^2\right ) \sin (e+f x)}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{b^2 \left (a^2-b^2\right ) d}-\frac {\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \int \sqrt {c+d \sin (e+f x)} \, dx}{2 b^2 \left (a^2-b^2\right )}\\ &=\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}+\frac {\left ((b c-a d)^2 \left (2 a b c+3 a^2 d-5 b^2 d\right )\right ) \int \frac {1}{(a+b \sin (e+f x)) \sqrt {c+d \sin (e+f x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}+\frac {\left ((b c-a d) \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 d^2\right )\right )\right ) \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 b^3 \left (a^2-b^2\right )}-\frac {\left (\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{2 b^2 \left (a^2-b^2\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}\\ &=\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) 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)}}{b^2 \left (a^2-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left ((b c-a d)^2 \left (2 a b c+3 a^2 d-5 b^2 d\right ) \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}{2 b^3 \left (a^2-b^2\right ) \sqrt {c+d \sin (e+f x)}}+\frac {\left ((b c-a d) \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 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}{2 b^3 \left (a^2-b^2\right ) \sqrt {c+d \sin (e+f x)}}\\ &=\frac {(b c-a d)^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{b \left (a^2-b^2\right ) f (a+b \sin (e+f x))}-\frac {\left (2 a b c d-3 a^2 d^2-b^2 \left (c^2-2 d^2\right )\right ) 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)}}{b^2 \left (a^2-b^2\right ) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {(b c-a d) \left (2 a b c d+3 a^2 d^2-b^2 \left (c^2+4 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}}}{b^3 \left (a^2-b^2\right ) f \sqrt {c+d \sin (e+f x)}}+\frac {(b c-a d)^2 \left (2 a b c+3 a^2 d-5 b^2 d\right ) \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}}}{(a-b) b^3 (a+b)^2 f \sqrt {c+d \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 27.44, size = 986, normalized size = 2.53 \begin {gather*} \frac {\left (-b^2 c^2 \cos (e+f x)+2 a b c d \cos (e+f x)-a^2 d^2 \cos (e+f x)\right ) \sqrt {c+d \sin (e+f x)}}{b \left (-a^2+b^2\right ) f (a+b \sin (e+f x))}+\frac {-\frac {2 \left (4 a b c^3-9 b^2 c^2 d+6 a b c d^2+a^2 d^3-2 b^2 d^3\right ) \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}}}{(a+b) \sqrt {c+d \sin (e+f x)}}-\frac {2 i \left (4 a b c^2 d+4 a^2 c d^2-12 b^2 c d^2+4 a b d^3\right ) \cos (e+f x) \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 ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b d^2 \sqrt {-\frac {1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}-\frac {2 i \left (-b^2 c^2 d+2 a b c d^2-3 a^2 d^3+2 b^2 d^3\right ) \cos (e+f x) \cos (2 (e+f x)) \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 ) \sqrt {\frac {d-d \sin (e+f x)}{c+d}} \sqrt {-\frac {d+d \sin (e+f x)}{c-d}} (-b c+a d+b (c+d \sin (e+f x)))}{b^2 d \sqrt {-\frac {1}{c+d}} (b c-a d) (a+b \sin (e+f x)) \sqrt {1-\sin ^2(e+f x)} \left (-2 c^2+d^2+4 c (c+d \sin (e+f x))-2 (c+d \sin (e+f x))^2\right ) \sqrt {-\frac {c^2-d^2-2 c (c+d \sin (e+f x))+(c+d \sin (e+f x))^2}{d^2}}}}{4 (a-b) b (a+b) f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1362\) vs.
\(2(476)=952\).
time = 28.56, size = 1363, normalized size = 3.49
method | result | size |
default | \(\text {Expression too large to display}\) | \(1363\) |
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]
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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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}}{{\left (a+b\,\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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