Optimal. Leaf size=510 \[ \frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}-\frac {32 b \left (2 a^2-b^2\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{35 a^5 \sqrt {d} f (a-b) (a+b)^{3/2}}-\frac {8 \left (5 a^2-3 a b-4 b^2\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{35 a^4 \sqrt {d} f (a-b) (a+b)^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 f \left (a^2-b^2\right )^2 \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 1.91, antiderivative size = 510, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2887, 2889, 3056, 2993, 2998, 2816, 2994} \[ \frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 f \left (a^2-b^2\right )^2 \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{3/2}}-\frac {8 \left (5 a^2-3 a b-4 b^2\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{35 a^4 \sqrt {d} f (a-b) (a+b)^{3/2}}-\frac {32 b \left (2 a^2-b^2\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{35 a^5 \sqrt {d} f (a-b) (a+b)^{3/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2816
Rule 2887
Rule 2889
Rule 2993
Rule 2994
Rule 2998
Rule 3056
Rubi steps
\begin {align*} \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{9/2}} \, dx &=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {6 \int \frac {\cos ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}} \, dx}{7 a}\\ &=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {6 \int \frac {1-\sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}} \, dx}{7 a}\\ &=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {12 \int \frac {2 \left (a^2-b^2\right ) d-\left (a^2-b^2\right ) d \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}} \, dx}{35 a^2 \left (a^2-b^2\right ) d}\\ &=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {8 \int \frac {\frac {1}{2} \left (5 a^4-9 a^2 b^2+4 b^4\right ) d^2-\frac {3}{2} a b \left (a^2-b^2\right ) d^2 \sin (e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{3/2}} \, dx}{35 a^3 \left (a^2-b^2\right )^2 d^2}\\ &=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {8 \int \frac {\frac {3}{2} a^2 b \left (a^2-b^2\right ) d^2+\frac {1}{2} b \left (5 a^4-9 a^2 b^2+4 b^4\right ) d^2+\left (\frac {3}{2} a b^2 \left (a^2-b^2\right ) d^2+\frac {1}{2} a \left (5 a^4-9 a^2 b^2+4 b^4\right ) d^2\right ) \sin (e+f x)}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}} \, dx}{35 a^3 \left (a^2-b^2\right )^3 d}\\ &=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {\left (4 \left (5 a^2-3 a b-4 b^2\right )\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{35 a^3 (a-b) (a+b)^2}+\frac {\left (16 b \left (2 a^2-b^2\right ) d\right ) \int \frac {1+\sin (e+f x)}{(d \sin (e+f x))^{3/2} \sqrt {a+b \sin (e+f x)}} \, dx}{35 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {2 \cos ^3(e+f x) \sqrt {d \sin (e+f x)}}{7 a d f (a+b \sin (e+f x))^{7/2}}+\frac {12 \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^2 d f (a+b \sin (e+f x))^{5/2}}+\frac {8 \left (a^2-2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{35 a^3 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{3/2}}+\frac {32 b \left (2 a^2-b^2\right ) \cos (e+f x)}{35 a^3 \left (a^2-b^2\right )^2 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {32 b \left (2 a^2-b^2\right ) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} E\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)}{35 a^5 (a-b) (a+b)^{3/2} \sqrt {d} f}-\frac {8 \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)}{35 a^4 (a-b) (a+b)^{3/2} \sqrt {d} f}\\ \end {align*}
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Mathematica [C] time = 6.55, size = 1670, normalized size = 3.27 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right )} \cos \left (f x + e\right )^{4}}{b^{5} d \cos \left (f x + e\right )^{6} - {\left (10 \, a^{2} b^{3} + 3 \, b^{5}\right )} d \cos \left (f x + e\right )^{4} + {\left (5 \, a^{4} b + 20 \, a^{2} b^{3} + 3 \, b^{5}\right )} d \cos \left (f x + e\right )^{2} - {\left (5 \, a^{4} b + 10 \, a^{2} b^{3} + b^{5}\right )} d - {\left (5 \, a b^{4} d \cos \left (f x + e\right )^{4} - 10 \, {\left (a^{3} b^{2} + a b^{4}\right )} d \cos \left (f x + e\right )^{2} + {\left (a^{5} + 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} d\right )} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.16, size = 24365, normalized size = 47.77 \[ \text {output too large to display} \]
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 \[ \int \frac {\cos \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {9}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\cos \left (e+f\,x\right )}^4}{\sqrt {d\,\sin \left (e+f\,x\right )}\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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