Optimal. Leaf size=142 \[ \frac {2 (4 A-3 B) \tan (e+f x)}{35 a c^4 f}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {2967, 2859, 2672, 3767, 8} \[ \frac {2 (4 A-3 B) \tan (e+f x)}{35 a c^4 f}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 2672
Rule 2859
Rule 2967
Rule 3767
Rubi steps
\begin {align*} \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^4} \, dx &=\frac {\int \frac {\sec ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^3} \, dx}{a c}\\ &=\frac {(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {(4 A-3 B) \int \frac {\sec ^2(e+f x)}{(c-c \sin (e+f x))^2} \, dx}{7 a c^2}\\ &=\frac {(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {(3 (4 A-3 B)) \int \frac {\sec ^2(e+f x)}{c-c \sin (e+f x)} \, dx}{35 a c^3}\\ &=\frac {(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {(2 (4 A-3 B)) \int \sec ^2(e+f x) \, dx}{35 a c^4}\\ &=\frac {(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}-\frac {(2 (4 A-3 B)) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{35 a c^4 f}\\ &=\frac {(A+B) \sec (e+f x)}{7 a c f (c-c \sin (e+f x))^3}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^2-c^2 \sin (e+f x)\right )^2}+\frac {(4 A-3 B) \sec (e+f x)}{35 a f \left (c^4-c^4 \sin (e+f x)\right )}+\frac {2 (4 A-3 B) \tan (e+f x)}{35 a c^4 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.14, size = 240, normalized size = 1.69 \[ \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) ((182 B-406 A) \cos (e+f x)+224 (4 A-3 B) \cos (2 (e+f x))+896 A \sin (e+f x)+406 A \sin (2 (e+f x))-384 A \sin (3 (e+f x))-29 A \sin (4 (e+f x))+174 A \cos (3 (e+f x))-64 A \cos (4 (e+f x))-672 B \sin (e+f x)-182 B \sin (2 (e+f x))+288 B \sin (3 (e+f x))+13 B \sin (4 (e+f x))-78 B \cos (3 (e+f x))+48 B \cos (4 (e+f x))+560 B)}{2240 a c^4 f (\sin (e+f x)-1)^4 (\sin (e+f x)+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 141, normalized size = 0.99 \[ \frac {2 \, {\left (4 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{4} - 9 \, {\left (4 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{2} + {\left (6 \, {\left (4 \, A - 3 \, B\right )} \cos \left (f x + e\right )^{2} - 20 \, A + 15 \, B\right )} \sin \left (f x + e\right ) + 15 \, A - 20 \, B}{35 \, {\left (3 \, a c^{4} f \cos \left (f x + e\right )^{3} - 4 \, a c^{4} f \cos \left (f x + e\right ) - {\left (a c^{4} f \cos \left (f x + e\right )^{3} - 4 \, a c^{4} f \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 237, normalized size = 1.67 \[ -\frac {\frac {35 \, {\left (A - B\right )}}{a c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {525 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 35 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 1960 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 280 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 4025 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 665 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 4480 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1120 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3143 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 791 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1176 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 392 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 243 \, A - 51 \, B}{a c^{4} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}}}{280 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.49, size = 189, normalized size = 1.33 \[ \frac {-\frac {2 \left (4 A +4 B \right )}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {12 A +12 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {18 A +14 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {2 \left (19 A +17 B \right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {2 \left (\frac {15 A}{16}+\frac {B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {\frac {17 A}{4}+\frac {7 B}{4}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {45 A}{4}+\frac {27 B}{4}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {2 \left (\frac {A}{16}-\frac {B}{16}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{f a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 619, normalized size = 4.36 \[ -\frac {2 \, {\left (\frac {A {\left (\frac {43 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {77 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {7 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {175 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {105 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {35 \, \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - 13\right )}}{a c^{4} - \frac {6 \, a c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {6 \, a c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}} - \frac {B {\left (\frac {6 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {21 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {56 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {105 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {70 \, \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {35 \, \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - 1\right )}}{a c^{4} - \frac {6 \, a c^{4} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {14 \, a c^{4} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {14 \, a c^{4} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {6 \, a c^{4} \sin \left (f x + e\right )^{7}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{7}} - \frac {a c^{4} \sin \left (f x + e\right )^{8}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{8}}}\right )}}{35 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 13.23, size = 239, normalized size = 1.68 \[ \frac {2\,\left (\frac {35\,B}{4}+\frac {91\,A\,\cos \left (e+f\,x\right )}{4}-\frac {7\,B\,\cos \left (e+f\,x\right )}{4}+14\,A\,\sin \left (e+f\,x\right )-\frac {21\,B\,\sin \left (e+f\,x\right )}{2}+14\,A\,\cos \left (2\,e+2\,f\,x\right )-\frac {39\,A\,\cos \left (3\,e+3\,f\,x\right )}{4}-A\,\cos \left (4\,e+4\,f\,x\right )-\frac {21\,B\,\cos \left (2\,e+2\,f\,x\right )}{2}+\frac {3\,B\,\cos \left (3\,e+3\,f\,x\right )}{4}+\frac {3\,B\,\cos \left (4\,e+4\,f\,x\right )}{4}-\frac {91\,A\,\sin \left (2\,e+2\,f\,x\right )}{4}-6\,A\,\sin \left (3\,e+3\,f\,x\right )+\frac {13\,A\,\sin \left (4\,e+4\,f\,x\right )}{8}+\frac {7\,B\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {9\,B\,\sin \left (3\,e+3\,f\,x\right )}{2}-\frac {B\,\sin \left (4\,e+4\,f\,x\right )}{8}\right )}{35\,a\,c^4\,f\,\left (\frac {7\,\cos \left (e+f\,x\right )}{2}-\frac {3\,\cos \left (3\,e+3\,f\,x\right )}{2}-\frac {7\,\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {\sin \left (4\,e+4\,f\,x\right )}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 28.58, size = 2468, normalized size = 17.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________