Optimal. Leaf size=712 \[ -\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^7 \sqrt {d} f (a-b)^2 (a+b)^{5/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 f \left (a^2-b^2\right )^3 \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 d f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))^{3/2}}-\frac {16 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^6 \sqrt {d} f (a-b)^2 (a+b)^{5/2}}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}} \]
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Rubi [A] time = 2.64, antiderivative size = 712, 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, 2891, 3055, 2993, 2998, 2816, 2994} \[ \frac {16 b \left (-93 a^2 b^2+93 a^4+32 b^4\right ) \cos (e+f x)}{693 a^5 f \left (a^2-b^2\right )^3 \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {8 \left (-22 a^4 b^2+65 a^2 b^4+5 a^6-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 d f \left (a^2-b^2\right )^2 (a+b \sin (e+f x))^{3/2}}-\frac {4 \left (-17 a^2 b^2+5 a^4+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 d f \left (a^2-b^2\right ) (a+b \sin (e+f x))^{5/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}-\frac {16 \left (-69 a^2 b^2-48 a^3 b+45 a^4+24 a b^3+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^6 \sqrt {d} f (a-b)^2 (a+b)^{5/2}}-\frac {16 b \left (-93 a^2 b^2+93 a^4+32 b^4\right ) \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{693 a^7 \sqrt {d} f (a-b)^2 (a+b)^{5/2}}+\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2816
Rule 2887
Rule 2891
Rule 2993
Rule 2994
Rule 2998
Rule 3055
Rubi steps
\begin {align*} \int \frac {\cos ^6(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{13/2}} \, dx &=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}+\frac {10 \int \frac {\cos ^4(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{11/2}} \, dx}{11 a}\\ &=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {40 \int \frac {\frac {1}{4} \left (5 a^2-48 b^2\right )-\frac {7}{2} a b \sin (e+f x)-\frac {1}{4} \left (15 a^2-32 b^2\right ) \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{7/2}} \, dx}{693 a^3 b^2}\\ &=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {16 \int \frac {\frac {1}{4} \left (5 a^4-107 a^2 b^2+96 b^4\right ) d-\frac {5}{2} a b \left (a^2-4 b^2\right ) d \sin (e+f x)-\frac {3}{4} \left (5 a^4-17 a^2 b^2+16 b^4\right ) d \sin ^2(e+f x)}{\sqrt {d \sin (e+f x)} (a+b \sin (e+f x))^{5/2}} \, dx}{693 a^4 b^2 \left (a^2-b^2\right ) d}\\ &=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}-\frac {32 \int \frac {-\frac {3}{4} b^2 \left (45 a^4-69 a^2 b^2+32 b^4\right ) d^2+18 a b^3 \left (2 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}{2079 a^5 b^2 \left (a^2-b^2\right )^2 d^2}\\ &=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {32 \int \frac {-18 a^2 b^3 \left (2 a^2-b^2\right ) d^2-\frac {3}{4} b^3 \left (45 a^4-69 a^2 b^2+32 b^4\right ) d^2+\left (-18 a b^4 \left (2 a^2-b^2\right ) d^2-\frac {3}{4} a b^2 \left (45 a^4-69 a^2 b^2+32 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}{2079 a^5 b^2 \left (a^2-b^2\right )^3 d}\\ &=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}+\frac {\left (8 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\right )\right ) \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx}{693 a^5 (a-b)^2 (a+b)^3}+\frac {\left (8 b \left (93 a^4-93 a^2 b^2+32 b^4\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}{693 a^5 \left (a^2-b^2\right )^3}\\ &=\frac {2 \cos ^5(e+f x) \sqrt {d \sin (e+f x)}}{11 a d f (a+b \sin (e+f x))^{11/2}}-\frac {20 \left (a^2-b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{99 a^2 b^2 d f (a+b \sin (e+f x))^{9/2}}+\frac {80 \left (3 a^2+2 b^2\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^3 b^2 d f (a+b \sin (e+f x))^{7/2}}-\frac {4 \left (5 a^4-17 a^2 b^2+16 b^4\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{231 a^4 b^2 \left (a^2-b^2\right ) d f (a+b \sin (e+f x))^{5/2}}-\frac {8 \left (5 a^6-22 a^4 b^2+65 a^2 b^4-32 b^6\right ) \cos (e+f x) \sqrt {d \sin (e+f x)}}{693 a^5 b^2 \left (a^2-b^2\right )^2 d f (a+b \sin (e+f x))^{3/2}}+\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\right ) \cos (e+f x)}{693 a^5 \left (a^2-b^2\right )^3 f \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}-\frac {16 b \left (93 a^4-93 a^2 b^2+32 b^4\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)}{693 a^7 (a-b)^2 (a+b)^{5/2} \sqrt {d} f}-\frac {16 \left (45 a^4-48 a^3 b-69 a^2 b^2+24 a b^3+32 b^4\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)}{693 a^6 (a-b)^2 (a+b)^{5/2} \sqrt {d} f}\\ \end {align*}
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Mathematica [C] time = 6.98, size = 1906, normalized size = 2.68 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sin \left (f x + e\right ) + a} \sqrt {d \sin \left (f x + e\right )} \cos \left (f x + e\right )^{6}}{b^{7} d \cos \left (f x + e\right )^{8} - {\left (21 \, a^{2} b^{5} + 4 \, b^{7}\right )} d \cos \left (f x + e\right )^{6} + {\left (35 \, a^{4} b^{3} + 63 \, a^{2} b^{5} + 6 \, b^{7}\right )} d \cos \left (f x + e\right )^{4} - {\left (7 \, a^{6} b + 70 \, a^{4} b^{3} + 63 \, a^{2} b^{5} + 4 \, b^{7}\right )} d \cos \left (f x + e\right )^{2} + {\left (7 \, a^{6} b + 35 \, a^{4} b^{3} + 21 \, a^{2} b^{5} + b^{7}\right )} d - {\left (7 \, a b^{6} d \cos \left (f x + e\right )^{6} - 7 \, {\left (5 \, a^{3} b^{4} + 3 \, a b^{6}\right )} d \cos \left (f x + e\right )^{4} + 7 \, {\left (3 \, a^{5} b^{2} + 10 \, a^{3} b^{4} + 3 \, a b^{6}\right )} d \cos \left (f x + e\right )^{2} - {\left (a^{7} + 21 \, a^{5} b^{2} + 35 \, a^{3} b^{4} + 7 \, a b^{6}\right )} d\right )} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.76, size = 56846, normalized size = 79.84 \[ \text {output too large to display} \]
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 \[ \int \frac {\cos \left (f x + e\right )^{6}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{\frac {13}{2}} \sqrt {d \sin \left (f x + e\right )}}\,{d x} \]
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 \[ \int \frac {{\cos \left (e+f\,x\right )}^6}{\sqrt {d\,\sin \left (e+f\,x\right )}\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^{13/2}} \,d x \]
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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