Optimal. Leaf size=502 \[ -\frac {3 a b \left (-2 a^2+b^2\right ) \cos (e+f x) \sqrt {a+b \sin (e+f x)}}{5 f \sqrt {d \sin (e+f x)}}+\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}-\frac {3 a \sec ^3(e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)} \left (-a \left (7 a^2+b^2\right )+2 b \left (-7 a^2+b^2\right ) \sin (e+f x)+5 a \left (a^2-b^2\right ) \sin ^2(e+f x)+\left (8 a^2 b-4 b^3\right ) \sin ^3(e+f x)\right )}{20 d f}-\frac {3 a (a+b)^{3/2} \left (5 a^2+3 a b-4 b^2\right ) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{20 \sqrt {d} f}-\frac {3 b \left (2 a^4-3 a^2 b^2+b^4\right ) \sqrt {-\frac {a (-1+\csc (e+f x))}{a+b}} E\left (\sin ^{-1}\left (\sqrt {-\frac {b+a \csc (e+f x)}{a-b}}\right )|1-\frac {2 a}{a+b}\right ) \sqrt {d \sin (e+f x)} \sqrt {-\frac {a \csc ^2(e+f x) (1+\sin (e+f x)) (a+b \sin (e+f x))}{(a-b)^2}} \tan (e+f x)}{5 d f \sqrt {a+b \sin (e+f x)}} \]
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Rubi [F]
time = 0.23, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sec ^6(e+f x) (a+b \sin (e+f x))^{9/2}}{\sqrt {d \sin (e+f x)}} \, dx &=\frac {\sec ^5(e+f x) \sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}}{5 d f}+\frac {1}{10} (9 a) \int \frac {\sec ^4(e+f x) (a+b \sin (e+f x))^{7/2}}{\sqrt {d \sin (e+f x)}} \, dx\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 33.45, size = 1600, normalized size = 3.19 \begin {gather*} \frac {\sin (e+f x) \sqrt {a+b \sin (e+f x)} \left (\frac {1}{20} \sec (e+f x) \left (15 a^4-15 a^2 b^2+4 b^4+24 a^3 b \sin (e+f x)-12 a b^3 \sin (e+f x)\right )+\frac {1}{10} \sec ^3(e+f x) \left (3 a^4-3 a^2 b^2-4 b^4+9 a^3 b \sin (e+f x)-5 a b^3 \sin (e+f x)\right )+\frac {1}{5} \sec ^5(e+f x) \left (a^4+6 a^2 b^2+b^4+4 a^3 b \sin (e+f x)+4 a b^3 \sin (e+f x)\right )\right )}{f \sqrt {d \sin (e+f x)}}+\frac {3 a \sqrt {\sin (e+f x)} \left (\frac {4 a \left (5 a^4-9 a^2 b^2+4 b^4\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+4 a \left (-8 a^3 b+4 a b^3\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \Pi \left (-\frac {a}{b};\sin ^{-1}\left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )+2 \left (8 a^2 b^2-4 b^4\right ) \left (\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\sin (e+f x)}}+\frac {i \cos \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x) E\left (i \sinh ^{-1}\left (\frac {\sin \left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{\sqrt {\sin (e+f x)}}\right )|-\frac {2 a}{-a-b}\right ) \sqrt {a+b \sin (e+f x)}}{b \sqrt {\cos ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \csc (e+f x)} \sqrt {\frac {\csc (e+f x) (a+b \sin (e+f x))}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} F\left (\sin ^{-1}\left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{(a+b) \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right )}{-a+b}} \Pi \left (-\frac {a}{b};\sin ^{-1}\left (\frac {\sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{\sqrt {2}}\right )|-\frac {2 a}{-a+b}\right ) \sec (e+f x) \sin ^4\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sqrt {-\frac {(a+b) \csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) \sin (e+f x)}{a}} \sqrt {\frac {\csc ^2\left (\frac {1}{2} \left (-e+\frac {\pi }{2}-f x\right )\right ) (a+b \sin (e+f x))}{a}}}{b \sqrt {\sin (e+f x)} \sqrt {a+b \sin (e+f x)}}\right )}{b}\right )\right )}{40 f \sqrt {d \sin (e+f x)}} \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. \(5433\) vs.
\(2(461)=922\).
time = 0.92, size = 5434, normalized size = 10.82
method | result | size |
default | \(\text {Expression too large to display}\) | \(5434\) |
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 (a+b\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\cos \left (e+f\,x\right )}^6\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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