Optimal. Leaf size=888 \[ \frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}-\frac {\sqrt {c+d} \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt {a+b} d f}-\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {(a+b)^{3/2} \left (3 a^2 d^2-6 a b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt {c+d} f} \]
[Out]
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Rubi [A]
time = 2.22, antiderivative size = 888, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2872, 3128,
3140, 3132, 2890, 3077, 2897, 3075} \begin {gather*} -\frac {\cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)} d^2}{3 b f}-\frac {(13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} d}{12 b f}+\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) 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 ) \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) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{24 b^2 (b c-a d) f}+\frac {(a+b)^{3/2} \left (\left (33 c^2+26 d c+16 d^2\right ) b^2-6 a d (2 c+d) b+3 a^2 d^2\right ) 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 ) \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) (\sin (e+f x)+1)}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{24 b^3 \sqrt {c+d} f}-\frac {\left (\left (33 c^2+16 d^2\right ) b^2+14 a c d b-3 a^2 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {c+d} \left (-5 \left (c^3+4 d^2 c\right ) b^3-a d \left (15 c^2+4 d^2\right ) b^2+5 a^2 c d^2 b-a^3 d^3\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\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 ) \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) (\sin (e+f x)+1)}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{8 b^3 \sqrt {a+b} f d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2872
Rule 2890
Rule 2897
Rule 3075
Rule 3077
Rule 3128
Rule 3132
Rule 3140
Rubi steps
\begin {align*} \int \sqrt {a+b \sin (e+f x)} (c+d \sin (e+f x))^{5/2} \, dx &=-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {\sqrt {a+b \sin (e+f x)} \left (\frac {1}{2} \left (a d^3+3 b c \left (2 c^2+d^2\right )\right )+d \left (9 b c^2-a c d+2 b d^2\right ) \sin (e+f x)+\frac {1}{2} d^2 (13 b c-3 a d) \sin ^2(e+f x)\right )}{\sqrt {c+d \sin (e+f x)}} \, dx}{3 b}\\ &=-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {\frac {1}{4} d \left (13 b^2 c^2 d+a^2 d^3+a b \left (24 c^3+22 c d^2\right )\right )-\frac {1}{2} d \left (a^2 c d^2-a b d \left (23 c^2+7 d^2\right )-b^2 \left (12 c^3+19 c d^2\right )\right ) \sin (e+f x)+\frac {1}{4} d^2 \left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \sin ^2(e+f x)}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{6 b d}\\ &=-\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {-\frac {1}{4} d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {1}{2} d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right ) \sin (e+f x)-\frac {3}{4} d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{12 b d^2}\\ &=-\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\int \frac {-\frac {1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )+b \left (\frac {1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac {3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{12 b^3 d^2}-\frac {\left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{16 b^3}\\ &=-\frac {\sqrt {c+d} \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\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))}{8 b^3 \sqrt {a+b} d f}-\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {\left (-\frac {1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )-b \left (\frac {1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac {3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{12 (a-b) b^3 d^2}-\frac {\left (-a b \left (\frac {1}{2} b d^2 \left (13 b^3 c^2 d+a^3 c d^2+a^2 b d \left (32 c^2+15 d^2\right )+a b^2 c \left (15 c^2+44 d^2\right )\right )+\frac {3}{2} a d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )+b \left (-\frac {1}{4} b^2 d^2 \left (a^3 d^3+b^3 c \left (33 c^2+16 d^2\right )-a b^2 d \left (45 c^2+16 d^2\right )-a^2 b c \left (48 c^2+61 d^2\right )\right )+\frac {3}{4} a^2 d^2 \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right )\right )\right ) \int \frac {1+\sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{12 (a-b) b^3 d^2}\\ &=\frac {\sqrt {a+b} (c-d) \sqrt {c+d} \left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) 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))}{24 b^2 (b c-a d) f}-\frac {\sqrt {c+d} \left (5 a^2 b c d^2-a^3 d^3-a b^2 d \left (15 c^2+4 d^2\right )-5 b^3 \left (c^3+4 c d^2\right )\right ) \Pi \left (\frac {b (c+d)}{(a+b) d};\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))}{8 b^3 \sqrt {a+b} d f}-\frac {\left (14 a b c d-3 a^2 d^2+b^2 \left (33 c^2+16 d^2\right )\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{24 b f \sqrt {a+b \sin (e+f x)}}-\frac {d (13 b c-3 a d) \cos (e+f x) \sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}{12 b f}-\frac {d^2 \cos (e+f x) (a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}}{3 b f}+\frac {(a+b)^{3/2} \left (3 a^2 d^2-6 a b d (2 c+d)+b^2 \left (33 c^2+26 c d+16 d^2\right )\right ) 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))}{24 b^3 \sqrt {c+d} f}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1978\) vs. \(2(888)=1776\).
time = 7.21, size = 1978, normalized size = 2.23 \begin {gather*} -\frac {-\frac {4 (-b c+a d) \left (-48 a b c^3-59 b^2 c^2 d-58 a b c d^2+a^2 d^3-16 b^2 d^3\right ) \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) \left (-48 b^2 c^3-92 a b c^2 d+4 a^2 c d^2-76 b^2 c d^2-28 a b d^3\right ) \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 \left (33 b^2 c^2 d+14 a b c d^2-3 a^2 d^3+16 b^2 d^3\right ) \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 )}{48 b f}+\frac {\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \left (-\frac {d (13 b c+a d) \cos (e+f x)}{12 b}-\frac {1}{6} d^2 \sin (2 (e+f x))\right )}{f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 30.36, size = 409142, normalized size = 460.75
method | result | size |
default | \(\text {Expression too large to display}\) | \(409142\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 \sqrt {a+b\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\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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