\(\int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx\) [25]

Optimal result

Integrand size = 34, antiderivative size = 176 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}-\frac {a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}-\frac {a (A-2 B) c \cos (e+f x)}{105 f \left (c^2-c^2 \sin (e+f x)\right )^3}-\frac {2 a (A-2 B) c \cos (e+f x)}{315 f \left (c^3-c^3 \sin (e+f x)\right )^2}-\frac {2 a (A-2 B) \cos (e+f x)}{315 f \left (c^5-c^5 \sin (e+f x)\right )} \]

[Out]

2/9*a*(A+B)*cos(f*x+e)/f/(c-c*sin(f*x+e))^5-1/63*a*(A+19*B)*cos(f*x+e)/c/f/(c-c*sin(f*x+e))^4-1/105*a*(A-2*B)*
c*cos(f*x+e)/f/(c^2-c^2*sin(f*x+e))^3-2/315*a*(A-2*B)*c*cos(f*x+e)/f/(c^3-c^3*sin(f*x+e))^2-2/315*a*(A-2*B)*co
s(f*x+e)/f/(c^5-c^5*sin(f*x+e))

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3046, 2936, 2829, 2729, 2727} \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 a (A-2 B) \cos (e+f x)}{315 f \left (c^5-c^5 \sin (e+f x)\right )}-\frac {2 a c (A-2 B) \cos (e+f x)}{315 f \left (c^3-c^3 \sin (e+f x)\right )^2}-\frac {a c (A-2 B) \cos (e+f x)}{105 f \left (c^2-c^2 \sin (e+f x)\right )^3}-\frac {a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}+\frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5} \]

[In]

Int[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]

[Out]

(2*a*(A + B)*Cos[e + f*x])/(9*f*(c - c*Sin[e + f*x])^5) - (a*(A + 19*B)*Cos[e + f*x])/(63*c*f*(c - c*Sin[e + f
*x])^4) - (a*(A - 2*B)*c*Cos[e + f*x])/(105*f*(c^2 - c^2*Sin[e + f*x])^3) - (2*a*(A - 2*B)*c*Cos[e + f*x])/(31
5*f*(c^3 - c^3*Sin[e + f*x])^2) - (2*a*(A - 2*B)*Cos[e + f*x])/(315*f*(c^5 - c^5*Sin[e + f*x]))

Rule 2727

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> Simp[-Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rule 2729

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*Cos[c + d*x]*((a + b*Sin[c + d*x])^n/(a*d
*(2*n + 1))), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2829

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*
c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(a*f*(2*m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)
), Int[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 -
b^2, 0] && LtQ[m, -2^(-1)]

Rule 2936

Int[cos[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_
)]), x_Symbol] :> Simp[2*(b*c - a*d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(2*m + 3))), x] + Dist[
1/(b^3*(2*m + 3)), Int[(a + b*Sin[e + f*x])^(m + 2)*(b*c + 2*a*d*(m + 1) - b*d*(2*m + 3)*Sin[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[m, -3/2]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = (a c) \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^6} \, dx \\ & = \frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}+\frac {a \int \frac {-A c-10 B c-9 B c \sin (e+f x)}{(c-c \sin (e+f x))^4} \, dx}{9 c^2} \\ & = \frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}-\frac {a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}-\frac {(a (A-2 B)) \int \frac {1}{(c-c \sin (e+f x))^3} \, dx}{21 c^2} \\ & = \frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}-\frac {a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}-\frac {a (A-2 B) \cos (e+f x)}{105 c^2 f (c-c \sin (e+f x))^3}-\frac {(2 a (A-2 B)) \int \frac {1}{(c-c \sin (e+f x))^2} \, dx}{105 c^3} \\ & = \frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}-\frac {a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}-\frac {a (A-2 B) \cos (e+f x)}{105 c^2 f (c-c \sin (e+f x))^3}-\frac {2 a (A-2 B) \cos (e+f x)}{315 c^3 f (c-c \sin (e+f x))^2}-\frac {(2 a (A-2 B)) \int \frac {1}{c-c \sin (e+f x)} \, dx}{315 c^4} \\ & = \frac {2 a (A+B) \cos (e+f x)}{9 f (c-c \sin (e+f x))^5}-\frac {a (A+19 B) \cos (e+f x)}{63 c f (c-c \sin (e+f x))^4}-\frac {a (A-2 B) \cos (e+f x)}{105 c^2 f (c-c \sin (e+f x))^3}-\frac {2 a (A-2 B) \cos (e+f x)}{315 c^3 f (c-c \sin (e+f x))^2}-\frac {2 a (A-2 B) \cos (e+f x)}{315 f \left (c^5-c^5 \sin (e+f x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.47 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.14 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {a \left (315 A \cos \left (e+\frac {f x}{2}\right )-42 (2 A+B) \cos \left (e+\frac {3 f x}{2}\right )+9 A \cos \left (3 e+\frac {7 f x}{2}\right )-18 B \cos \left (3 e+\frac {7 f x}{2}\right )+189 A \sin \left (\frac {f x}{2}\right )+252 B \sin \left (\frac {f x}{2}\right )+210 B \sin \left (2 e+\frac {3 f x}{2}\right )+36 A \sin \left (2 e+\frac {5 f x}{2}\right )-72 B \sin \left (2 e+\frac {5 f x}{2}\right )-A \sin \left (4 e+\frac {9 f x}{2}\right )+2 B \sin \left (4 e+\frac {9 f x}{2}\right )\right )}{1260 c^5 f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]

[In]

Integrate[((a + a*Sin[e + f*x])*(A + B*Sin[e + f*x]))/(c - c*Sin[e + f*x])^5,x]

[Out]

(a*(315*A*Cos[e + (f*x)/2] - 42*(2*A + B)*Cos[e + (3*f*x)/2] + 9*A*Cos[3*e + (7*f*x)/2] - 18*B*Cos[3*e + (7*f*
x)/2] + 189*A*Sin[(f*x)/2] + 252*B*Sin[(f*x)/2] + 210*B*Sin[2*e + (3*f*x)/2] + 36*A*Sin[2*e + (5*f*x)/2] - 72*
B*Sin[2*e + (5*f*x)/2] - A*Sin[4*e + (9*f*x)/2] + 2*B*Sin[4*e + (9*f*x)/2]))/(1260*c^5*f*(Cos[e/2] - Sin[e/2])
*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^9)

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 1.04 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.94

method	result	size
risch	\(-\frac {4 \left (-42 i B a \,{\mathrm e}^{3 i \left (f x +e \right )}+210 B a \,{\mathrm e}^{6 i \left (f x +e \right )}+9 i A a \,{\mathrm e}^{i \left (f x +e \right )}-2 B a -18 i B a \,{\mathrm e}^{i \left (f x +e \right )}+a A -84 i A a \,{\mathrm e}^{3 i \left (f x +e \right )}-36 A a \,{\mathrm e}^{2 i \left (f x +e \right )}+72 B a \,{\mathrm e}^{2 i \left (f x +e \right )}+315 i A a \,{\mathrm e}^{5 i \left (f x +e \right )}-189 A a \,{\mathrm e}^{4 i \left (f x +e \right )}-252 B a \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{315 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{9} f \,c^{5}}\)	\(166\)
parallelrisch	\(-\frac {2 a \left (A \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-3 A +B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25 A}{3}-B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-11 A +3 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (61 A -7 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5}+\frac {\left (-107 A +29 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15}+\frac {\left (127 A -9 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{35}+\frac {\left (-23 A +11 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35}+\frac {58 A}{315}-\frac {11 B}{315}\right )}{f \,c^{5} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(176\)
derivativedivides	\(\frac {2 a \left (-\frac {128 A +128 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {46 A +18 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {296 A +248 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128 A +72 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {32 A +32 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {236 A +168 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {10 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {248 A +232 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\right )}{f \,c^{5}}\)	\(203\)
default	\(\frac {2 a \left (-\frac {128 A +128 B}{8 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{8}}-\frac {46 A +18 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {296 A +248 B}{6 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {128 A +72 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {32 A +32 B}{9 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}-\frac {236 A +168 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {10 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {248 A +232 B}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\right )}{f \,c^{5}}\)	\(203\)
norman	\(\frac {\frac {\left (34 a A -10 B a \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {116 a A -22 B a}{315 c f}-\frac {2 a A \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}+\frac {2 \left (3 a A -B a \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {2 \left (31 a A -3 B a \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c f}+\frac {\left (46 a A -22 B a \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{35 c f}+\frac {4 \left (241 a A -67 B a \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}-\frac {\left (896 a A -102 B a \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 c f}+\frac {2 \left (887 a A -269 B a \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}-\frac {2 \left (1259 a A -103 B a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 c f}-\frac {4 \left (1909 a A -213 B a \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}+\frac {\left (5444 a A -1508 B a \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{105 c f}-\frac {\left (12374 a A -1228 B a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{315 c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{9}}\)	\(377\)

[In]

int((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x,method=_RETURNVERBOSE)

[Out]

-4/315*(-42*I*B*a*exp(3*I*(f*x+e))+210*B*a*exp(6*I*(f*x+e))+9*I*A*a*exp(I*(f*x+e))-2*B*a-18*I*B*a*exp(I*(f*x+e
))+a*A-84*I*A*a*exp(3*I*(f*x+e))-36*A*a*exp(2*I*(f*x+e))+72*B*a*exp(2*I*(f*x+e))+315*I*A*a*exp(5*I*(f*x+e))-18
9*A*a*exp(4*I*(f*x+e))-252*B*a*exp(4*I*(f*x+e)))/(exp(I*(f*x+e))-I)^9/f/c^5

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.73 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{5} - 8 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{4} - 25 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{3} + 5 \, {\left (4 \, A + 13 \, B\right )} a \cos \left (f x + e\right )^{2} - 35 \, {\left (A + B\right )} a \cos \left (f x + e\right ) - 70 \, {\left (A + B\right )} a + {\left (2 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{4} + 10 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{3} - 15 \, {\left (A - 2 \, B\right )} a \cos \left (f x + e\right )^{2} - 35 \, {\left (A + B\right )} a \cos \left (f x + e\right ) - 70 \, {\left (A + B\right )} a\right )} \sin \left (f x + e\right )}{315 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

-1/315*(2*(A - 2*B)*a*cos(f*x + e)^5 - 8*(A - 2*B)*a*cos(f*x + e)^4 - 25*(A - 2*B)*a*cos(f*x + e)^3 + 5*(4*A +
13*B)*a*cos(f*x + e)^2 - 35*(A + B)*a*cos(f*x + e) - 70*(A + B)*a + (2*(A - 2*B)*a*cos(f*x + e)^4 + 10*(A - 2
*B)*a*cos(f*x + e)^3 - 15*(A - 2*B)*a*cos(f*x + e)^2 - 35*(A + B)*a*cos(f*x + e) - 70*(A + B)*a)*sin(f*x + e))
/(c^5*f*cos(f*x + e)^5 + 5*c^5*f*cos(f*x + e)^4 - 8*c^5*f*cos(f*x + e)^3 - 20*c^5*f*cos(f*x + e)^2 + 8*c^5*f*c
os(f*x + e) + 16*c^5*f - (c^5*f*cos(f*x + e)^4 - 4*c^5*f*cos(f*x + e)^3 - 12*c^5*f*cos(f*x + e)^2 + 8*c^5*f*co
s(f*x + e) + 16*c^5*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3232 vs. \(2 (160) = 320\).

Time = 15.82 (sec) , antiderivative size = 3232, normalized size of antiderivative = 18.36 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((-630*A*a*tan(e/2 + f*x/2)**8/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11
340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c
**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*
tan(e/2 + f*x/2) - 315*c**5*f) + 1890*A*a*tan(e/2 + f*x/2)**7/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*ta
n(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2
+ f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*
x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 5250*A*a*tan(e/2 + f*x/2)**6/(315*c**5*f*tan(e/2 + f*x/
2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6
+ 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 113
40*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 6930*A*a*tan(e/2 + f*x/2)**5/(315
*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*
f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan
(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 7686*A*a*ta
n(e/2 + f*x/2)**4/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f
*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)
**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315
*c**5*f) + 4494*A*a*tan(e/2 + f*x/2)**3/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11
340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c
**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*
tan(e/2 + f*x/2) - 315*c**5*f) - 2286*A*a*tan(e/2 + f*x/2)**2/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*ta
n(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2
+ f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*
x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 414*A*a*tan(e/2 + f*x/2)/(315*c**5*f*tan(e/2 + f*x/2)**
9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39
690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c
**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 116*A*a/(315*c**5*f*tan(e/2 + f*x/2)*
*9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 3
9690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*
c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 630*B*a*tan(e/2 + f*x/2)**7/(315*c**
5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*ta
n(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2
+ f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 630*B*a*tan(e/2
+ f*x/2)**6/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)
**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 +
26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5
*f) - 1890*B*a*tan(e/2 + f*x/2)**5/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c
**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f
*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e
/2 + f*x/2) - 315*c**5*f) + 882*B*a*tan(e/2 + f*x/2)**4/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2
+ f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x
/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**
2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 1218*B*a*tan(e/2 + f*x/2)**3/(315*c**5*f*tan(e/2 + f*x/2)**9
- 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e/2 + f*x/2)**6 + 3969
0*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**
5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) + 162*B*a*tan(e/2 + f*x/2)**2/(315*c**5*f
*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 - 26460*c**5*f*tan(e
/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460*c**5*f*tan(e/2 +
f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) - 198*B*a*tan(e/2 +
f*x/2)/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 -
26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460
*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f) +
22*B*a/(315*c**5*f*tan(e/2 + f*x/2)**9 - 2835*c**5*f*tan(e/2 + f*x/2)**8 + 11340*c**5*f*tan(e/2 + f*x/2)**7 -
26460*c**5*f*tan(e/2 + f*x/2)**6 + 39690*c**5*f*tan(e/2 + f*x/2)**5 - 39690*c**5*f*tan(e/2 + f*x/2)**4 + 26460
*c**5*f*tan(e/2 + f*x/2)**3 - 11340*c**5*f*tan(e/2 + f*x/2)**2 + 2835*c**5*f*tan(e/2 + f*x/2) - 315*c**5*f), N
e(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)/(-c*sin(e) + c)**5, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1425 vs. \(2 (171) = 342\).

Time = 0.26 (sec) , antiderivative size = 1425, normalized size of antiderivative = 8.10 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

-2/315*(A*a*(432*sin(f*x + e)/(cos(f*x + e) + 1) - 1728*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3612*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 - 5418*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 5040*sin(f*x + e)^5/(cos(f*x + e) + 1)^5
- 3360*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 1260*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 315*sin(f*x + e)^8/(c
os(f*x + e) + 1)^8 - 83)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) +
1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(
f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x
+ e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) - 5*A*a*(45
*sin(f*x + e)/(cos(f*x + e) + 1) - 117*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 273*sin(f*x + e)^3/(cos(f*x + e)
+ 1)^3 - 315*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 147*sin(f*x + e)^
6/(cos(f*x + e) + 1)^6 + 63*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 - 5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) +
1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*
x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x +
e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(
f*x + e)^9/(cos(f*x + e) + 1)^9) - 5*B*a*(45*sin(f*x + e)/(cos(f*x + e) + 1) - 117*sin(f*x + e)^2/(cos(f*x + e
) + 1)^2 + 273*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 315*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 315*sin(f*x + e
)^5/(cos(f*x + e) + 1)^5 - 147*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 + 63*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 -
5)/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x
+ e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x +
e) + 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*si
n(f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9) + 14*B*a*(9*sin(f*x + e)/(cos(f*x
+ e) + 1) - 36*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 54*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 81*sin(f*x + e)
^4/(cos(f*x + e) + 1)^4 + 45*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 30*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 1)
/(c^5 - 9*c^5*sin(f*x + e)/(cos(f*x + e) + 1) + 36*c^5*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 84*c^5*sin(f*x +
e)^3/(cos(f*x + e) + 1)^3 + 126*c^5*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 126*c^5*sin(f*x + e)^5/(cos(f*x + e)
+ 1)^5 + 84*c^5*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - 36*c^5*sin(f*x + e)^7/(cos(f*x + e) + 1)^7 + 9*c^5*sin(
f*x + e)^8/(cos(f*x + e) + 1)^8 - c^5*sin(f*x + e)^9/(cos(f*x + e) + 1)^9))/f

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.43 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=-\frac {2 \, {\left (315 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} - 945 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 315 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 2625 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 315 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 3465 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 945 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3843 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 441 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2247 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 609 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1143 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 81 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 207 \, A a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 99 \, B a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 58 \, A a - 11 \, B a\right )}}{315 \, c^{5} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{9}} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))/(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-2/315*(315*A*a*tan(1/2*f*x + 1/2*e)^8 - 945*A*a*tan(1/2*f*x + 1/2*e)^7 + 315*B*a*tan(1/2*f*x + 1/2*e)^7 + 262
5*A*a*tan(1/2*f*x + 1/2*e)^6 - 315*B*a*tan(1/2*f*x + 1/2*e)^6 - 3465*A*a*tan(1/2*f*x + 1/2*e)^5 + 945*B*a*tan(
1/2*f*x + 1/2*e)^5 + 3843*A*a*tan(1/2*f*x + 1/2*e)^4 - 441*B*a*tan(1/2*f*x + 1/2*e)^4 - 2247*A*a*tan(1/2*f*x +
1/2*e)^3 + 609*B*a*tan(1/2*f*x + 1/2*e)^3 + 1143*A*a*tan(1/2*f*x + 1/2*e)^2 - 81*B*a*tan(1/2*f*x + 1/2*e)^2 -
207*A*a*tan(1/2*f*x + 1/2*e) + 99*B*a*tan(1/2*f*x + 1/2*e) + 58*A*a - 11*B*a)/(c^5*f*(tan(1/2*f*x + 1/2*e) -
1)^9)

Mupad [B] (verification not implemented)

Time = 13.05 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.76 \[ \int \frac {(a+a \sin (e+f x)) (A+B \sin (e+f x))}{(c-c \sin (e+f x))^5} \, dx=\frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {1357\,A\,a}{4}-\frac {461\,B\,a}{16}-\frac {635\,A\,a\,\cos \left (e+f\,x\right )}{4}+\frac {5\,B\,a\,\cos \left (e+f\,x\right )}{2}-\frac {1575\,A\,a\,\sin \left (e+f\,x\right )}{4}+\frac {945\,B\,a\,\sin \left (e+f\,x\right )}{8}-\frac {625\,A\,a\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {121\,A\,a\,\cos \left (3\,e+3\,f\,x\right )}{4}+\frac {7\,A\,a\,\cos \left (4\,e+4\,f\,x\right )}{2}+\frac {95\,B\,a\,\cos \left (2\,e+2\,f\,x\right )}{4}-8\,B\,a\,\cos \left (3\,e+3\,f\,x\right )-\frac {7\,B\,a\,\cos \left (4\,e+4\,f\,x\right )}{16}+\frac {399\,A\,a\,\sin \left (2\,e+2\,f\,x\right )}{4}+\frac {141\,A\,a\,\sin \left (3\,e+3\,f\,x\right )}{4}-\frac {15\,A\,a\,\sin \left (4\,e+4\,f\,x\right )}{4}-\frac {231\,B\,a\,\sin \left (2\,e+2\,f\,x\right )}{8}-\frac {39\,B\,a\,\sin \left (3\,e+3\,f\,x\right )}{8}+\frac {15\,B\,a\,\sin \left (4\,e+4\,f\,x\right )}{16}\right )}{315\,c^5\,f\,\left (\frac {63\,\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {\pi }{4}+\frac {f\,x}{2}\right )}{8}-\frac {21\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}-\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {9\,\sqrt {2}\,\cos \left (\frac {5\,e}{2}+\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}+\frac {9\,\sqrt {2}\,\cos \left (\frac {7\,e}{2}-\frac {\pi }{4}+\frac {7\,f\,x}{2}\right )}{16}+\frac {\sqrt {2}\,\cos \left (\frac {9\,e}{2}+\frac {\pi }{4}+\frac {9\,f\,x}{2}\right )}{16}\right )} \]

[In]

int(((A + B*sin(e + f*x))*(a + a*sin(e + f*x)))/(c - c*sin(e + f*x))^5,x)

[Out]

(2*cos(e/2 + (f*x)/2)*((1357*A*a)/4 - (461*B*a)/16 - (635*A*a*cos(e + f*x))/4 + (5*B*a*cos(e + f*x))/2 - (1575
*A*a*sin(e + f*x))/4 + (945*B*a*sin(e + f*x))/8 - (625*A*a*cos(2*e + 2*f*x))/4 + (121*A*a*cos(3*e + 3*f*x))/4
+ (7*A*a*cos(4*e + 4*f*x))/2 + (95*B*a*cos(2*e + 2*f*x))/4 - 8*B*a*cos(3*e + 3*f*x) - (7*B*a*cos(4*e + 4*f*x))
/16 + (399*A*a*sin(2*e + 2*f*x))/4 + (141*A*a*sin(3*e + 3*f*x))/4 - (15*A*a*sin(4*e + 4*f*x))/4 - (231*B*a*sin
(2*e + 2*f*x))/8 - (39*B*a*sin(3*e + 3*f*x))/8 + (15*B*a*sin(4*e + 4*f*x))/16))/(315*c^5*f*((63*2^(1/2)*cos(e/
2 + pi/4 + (f*x)/2))/8 - (21*2^(1/2)*cos((3*e)/2 - pi/4 + (3*f*x)/2))/4 - (9*2^(1/2)*cos((5*e)/2 + pi/4 + (5*f
*x)/2))/4 + (9*2^(1/2)*cos((7*e)/2 - pi/4 + (7*f*x)/2))/16 + (2^(1/2)*cos((9*e)/2 + pi/4 + (9*f*x)/2))/16))