Optimal. Leaf size=630 \[ \frac {2 (A b-a B) (c-d) \sqrt {c+d} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b \sqrt {a+b} (b c-a d) f}+\frac {2 \sqrt {a+b} (A b-a B) (c-d) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b \sqrt {c+d} (b c-a d) f}+\frac {2 \sqrt {a+b} B \Pi \left (\frac {(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b^2 \sqrt {c+d} f} \]
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Rubi [A]
time = 0.57, antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3071, 2890,
2874, 2897, 3075} \begin {gather*} \frac {2 \sqrt {a+b} (c-d) (A b-a B) \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} F\left (\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{b f (a-b) \sqrt {c+d} (b c-a d)}+\frac {2 (c-d) \sqrt {c+d} (A b-a B) \sec (e+f x) (a+b \sin (e+f x)) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} E\left (\text {ArcSin}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right )}{b f (a-b) \sqrt {a+b} (b c-a d)}+\frac {2 B \sqrt {a+b} \sec (e+f x) (c+d \sin (e+f x)) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} \Pi \left (\frac {(a+b) d}{b (c+d)};\text {ArcSin}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right )}{b^2 f \sqrt {c+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2874
Rule 2890
Rule 2897
Rule 3071
Rule 3075
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) \sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx &=\frac {B \int \frac {\sqrt {c+d \sin (e+f x)}}{\sqrt {a+b \sin (e+f x)}} \, dx}{b}+\frac {(A b-a B) \int \frac {\sqrt {c+d \sin (e+f x)}}{(a+b \sin (e+f x))^{3/2}} \, dx}{b}\\ &=\frac {2 \sqrt {a+b} B \Pi \left (\frac {(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b^2 \sqrt {c+d} f}+\frac {((A b-a B) (c-d)) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{(a-b) b}-\frac {((A b-a B) (b c-a d)) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{(a-b) b}\\ &=\frac {2 (A b-a B) (c-d) \sqrt {c+d} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{(a-b) b \sqrt {a+b} (b c-a d) f}+\frac {2 \sqrt {a+b} (A b-a B) (c-d) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{(a-b) b \sqrt {c+d} (b c-a d) f}+\frac {2 \sqrt {a+b} B \Pi \left (\frac {(a+b) d}{b (c+d)};\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b^2 \sqrt {c+d} f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1901\) vs. \(2(630)=1260\).
time = 29.54, size = 1901, normalized size = 3.02 \begin {gather*} -\frac {2 (-A b \cos (e+f x)+a B \cos (e+f x)) \sqrt {c+d \sin (e+f x)}}{\left (a^2-b^2\right ) f \sqrt {a+b \sin (e+f x)}}+\frac {-\frac {4 (a A c-b B c) (-b c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-4 (-b c+a d) (A b c-a B c+a A d-b B d) \left (\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {\sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \Pi \left (\frac {-b c+a d}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) d \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )+2 (-A b d+a B d) \left (\frac {\cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {\frac {a-b}{a+b}} (a+b) \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {a-b}{a+b}} \sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\right )|\frac {2 (-b c+a d)}{(a-b) (c+d)}\right ) \sqrt {c+d \sin (e+f x)}}{b d \sqrt {\frac {(a+b) \cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{a+b \sin (e+f x)}} \sqrt {a+b \sin (e+f x)} \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \sqrt {\frac {(a+b) (c+d \sin (e+f x))}{(c+d) (a+b \sin (e+f x))}}}-\frac {2 (-b c+a d) \left (\frac {((a+b) c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) (c+d) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}-\frac {(b c+a d) \sqrt {\frac {(c+d) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-c+d}} \Pi \left (\frac {-b c+a d}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{\sqrt {2}}\right )|\frac {2 (-b c+a d)}{(a+b) (-c+d)}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {\frac {(c+d) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{-b c+a d}} \sqrt {\frac {(-a-b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (c+d \sin (e+f x))}{-b c+a d}}}{(a+b) d \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{b d}\right )}{(a-b) (a+b) f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 59.50, size = 3148347, normalized size = 4997.38
method | result | size |
default | \(\text {Expression too large to display}\) | \(3148347\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + B \sin {\left (e + f x \right )}\right ) \sqrt {c + d \sin {\left (e + f x \right )}}}{\left (a + b \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \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 (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}}{{\left (a+b\,\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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