Optimal. Leaf size=644 \[ \frac {\sqrt {a+b} (c-d) d \sqrt {c+d} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b (b c-a d) f}+\frac {\sqrt {c+d} (3 b c-a d) \Pi \left (\frac {b (c+d)}{(a+b) d};\sin ^{-1}\left (\frac {\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}{\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}\right )|\frac {(a-b) (c+d)}{(a+b) (c-d)}\right ) \sec (e+f x) \sqrt {-\frac {(b c-a d) (1-\sin (e+f x))}{(c+d) (a+b \sin (e+f x))}} \sqrt {\frac {(b c-a d) (1+\sin (e+f x))}{(c-d) (a+b \sin (e+f x))}} (a+b \sin (e+f x))}{b^2 \sqrt {a+b} f}-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {a+b} (a d-b (2 c+d)) F\left (\sin ^{-1}\left (\frac {\sqrt {c+d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {c+d \sin (e+f x)}}\right )|\frac {(a+b) (c-d)}{(a-b) (c+d)}\right ) \sec (e+f x) \sqrt {\frac {(b c-a d) (1-\sin (e+f x))}{(a+b) (c+d \sin (e+f x))}} \sqrt {-\frac {(b c-a d) (1+\sin (e+f x))}{(a-b) (c+d \sin (e+f x))}} (c+d \sin (e+f x))}{b^2 \sqrt {c+d} f} \]
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Rubi [A]
time = 1.01, antiderivative size = 644, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2900, 3132,
2890, 3077, 2897, 3075} \begin {gather*} -\frac {\sqrt {a+b} (a d-b (2 c+d)) \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^2 f \sqrt {c+d}}+\frac {\sqrt {c+d} (3 b c-a d) \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))}} \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 )}{b^2 f \sqrt {a+b}}+\frac {d \sqrt {a+b} (c-d) \sqrt {c+d} \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 (b c-a d)}-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2890
Rule 2897
Rule 2900
Rule 3075
Rule 3077
Rule 3132
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^{3/2}}{\sqrt {a+b \sin (e+f x)}} \, dx &=-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}+\frac {\int \frac {-\frac {1}{2} b \left (b c d-a \left (2 c^2+d^2\right )\right )+b c (b c+a d) \sin (e+f x)+\frac {1}{2} b d (3 b c-a d) \sin ^2(e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{b}\\ &=-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}+\frac {\int \frac {-\frac {1}{2} a^2 b d (3 b c-a d)-\frac {1}{2} b^3 \left (b c d-a \left (2 c^2+d^2\right )\right )+b \left (-a b d (3 b c-a d)+b^2 c (b c+a d)\right ) \sin (e+f x)}{(a+b \sin (e+f x))^{3/2} \sqrt {c+d \sin (e+f x)}} \, dx}{b^3}+\frac {(d (3 b c-a d)) \int \frac {\sqrt {a+b \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 b^2}\\ &=\frac {\sqrt {c+d} (3 b c-a d) \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))}{b^2 \sqrt {a+b} f}-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {((a+b) d (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}{2 b}--\frac {\left (-\frac {1}{2} a^2 b d (3 b c-a d)-b \left (-a b d (3 b c-a d)+b^2 c (b c+a d)\right )-\frac {1}{2} b^3 \left (b c d-a \left (2 c^2+d^2\right )\right )\right ) \int \frac {1}{\sqrt {a+b \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{(a-b) b^3}\\ &=\frac {\sqrt {a+b} (c-d) 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))}{b (b c-a d) f}+\frac {\sqrt {c+d} (3 b c-a d) \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))}{b^2 \sqrt {a+b} f}-\frac {d \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {a+b} (a d-b (2 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))}{b^2 \sqrt {c+d} f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 34.97, size = 222963, normalized size = 346.22 \begin {gather*} \text {Result too large to show} \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 = 19.26, size = 544053, normalized size = 844.80
method | result | size |
default | \(\text {Expression too large to display}\) | \(544053\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d \sin {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\sqrt {a + b \sin {\left (e + f x \right )}}}\, 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 (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}}{\sqrt {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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