\(\int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx\) [57]

Optimal result

Integrand size = 36, antiderivative size = 63 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {(A+B) \sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {(2 A-B) \tan (e+f x)}{3 a c^2 f} \]

[Out]

1/3*(A+B)*sec(f*x+e)/a/f/(c^2-c^2*sin(f*x+e))+1/3*(2*A-B)*tan(f*x+e)/a/c^2/f

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3046, 2938, 3852, 8} \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {(2 A-B) \tan (e+f x)}{3 a c^2 f}+\frac {(A+B) \sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )} \]

[In]

Int[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2),x]

[Out]

((A + B)*Sec[e + f*x])/(3*a*f*(c^2 - c^2*Sin[e + f*x])) + ((2*A - B)*Tan[e + f*x])/(3*a*c^2*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2938

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

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]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sec ^2(e+f x) (A+B \sin (e+f x))}{c-c \sin (e+f x)} \, dx}{a c} \\ & = \frac {(A+B) \sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {(2 A-B) \int \sec ^2(e+f x) \, dx}{3 a c^2} \\ & = \frac {(A+B) \sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}-\frac {(2 A-B) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{3 a c^2 f} \\ & = \frac {(A+B) \sec (e+f x)}{3 a f \left (c^2-c^2 \sin (e+f x)\right )}+\frac {(2 A-B) \tan (e+f x)}{3 a c^2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.71 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {\cos (e+f x) (6 B-2 (A+B) \cos (e+f x)+(4 A-2 B) \cos (2 (e+f x))+8 A \sin (e+f x)-4 B \sin (e+f x)+A \sin (2 (e+f x))+B \sin (2 (e+f x)))}{12 a c^2 f (-1+\sin (e+f x))^2 (1+\sin (e+f x))} \]

[In]

Integrate[(A + B*Sin[e + f*x])/((a + a*Sin[e + f*x])*(c - c*Sin[e + f*x])^2),x]

[Out]

(Cos[e + f*x]*(6*B - 2*(A + B)*Cos[e + f*x] + (4*A - 2*B)*Cos[2*(e + f*x)] + 8*A*Sin[e + f*x] - 4*B*Sin[e + f*
x] + A*Sin[2*(e + f*x)] + B*Sin[2*(e + f*x)]))/(12*a*c^2*f*(-1 + Sin[e + f*x])^2*(1 + Sin[e + f*x]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.71 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.37

method	result	size
risch	\(-\frac {2 i \left (4 i A \,{\mathrm e}^{i \left (f x +e \right )}-2 i B \,{\mathrm e}^{i \left (f x +e \right )}+3 B \,{\mathrm e}^{2 i \left (f x +e \right )}+2 A -B \right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right ) a \,c^{2} f}\)	\(86\)
derivativedivides	\(\frac {-\frac {2 \left (\frac {A}{4}-\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (A +B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A +B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {3 A}{4}+\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}}{a \,c^{2} f}\)	\(93\)
default	\(\frac {-\frac {2 \left (\frac {A}{4}-\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {2 \left (A +B \right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}-\frac {A +B}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {2 \left (\frac {3 A}{4}+\frac {B}{4}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}}{a \,c^{2} f}\)	\(93\)
parallelrisch	\(\frac {-6 A \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (6 A -6 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (-2 A +4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-2 A -2 B}{3 f a \,c^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\)	\(95\)
norman	\(\frac {\frac {2 A -4 B}{6 a c f}-\frac {4 \left (2 A -B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a c f}-\frac {A \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a c f}+\frac {\left (2 A -4 B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a c f}-\frac {\left (8 A -4 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a c f}+\frac {\left (14 A -16 B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{6 a c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right ) c \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\)	\(202\)

[In]

int((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

-2/3*I*(4*I*A*exp(I*(f*x+e))-2*I*B*exp(I*(f*x+e))+3*B*exp(2*I*(f*x+e))+2*A-B)/(exp(I*(f*x+e))-I)^3/(exp(I*(f*x
+e))+I)/a/c^2/f

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.16 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=-\frac {{\left (2 \, A - B\right )} \cos \left (f x + e\right )^{2} + {\left (2 \, A - B\right )} \sin \left (f x + e\right ) - A + 2 \, B}{3 \, {\left (a c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c^{2} f \cos \left (f x + e\right )\right )}} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/3*((2*A - B)*cos(f*x + e)^2 + (2*A - B)*sin(f*x + e) - A + 2*B)/(a*c^2*f*cos(f*x + e)*sin(f*x + e) - a*c^2*
f*cos(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 578 vs. \(2 (51) = 102\).

Time = 2.23 (sec) , antiderivative size = 578, normalized size of antiderivative = 9.17 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\begin {cases} - \frac {6 A \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} + \frac {6 A \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {2 A \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {2 A}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {6 B \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} + \frac {4 B \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} - \frac {2 B}{3 a c^{2} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 6 a c^{2} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 6 a c^{2} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 3 a c^{2} f} & \text {for}\: f \neq 0 \\\frac {x \left (A + B \sin {\left (e \right )}\right )}{\left (a \sin {\left (e \right )} + a\right ) \left (- c \sin {\left (e \right )} + c\right )^{2}} & \text {otherwise} \end {cases} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))**2,x)

[Out]

Piecewise((-6*A*tan(e/2 + f*x/2)**3/(3*a*c**2*f*tan(e/2 + f*x/2)**4 - 6*a*c**2*f*tan(e/2 + f*x/2)**3 + 6*a*c**
2*f*tan(e/2 + f*x/2) - 3*a*c**2*f) + 6*A*tan(e/2 + f*x/2)**2/(3*a*c**2*f*tan(e/2 + f*x/2)**4 - 6*a*c**2*f*tan(
e/2 + f*x/2)**3 + 6*a*c**2*f*tan(e/2 + f*x/2) - 3*a*c**2*f) - 2*A*tan(e/2 + f*x/2)/(3*a*c**2*f*tan(e/2 + f*x/2
)**4 - 6*a*c**2*f*tan(e/2 + f*x/2)**3 + 6*a*c**2*f*tan(e/2 + f*x/2) - 3*a*c**2*f) - 2*A/(3*a*c**2*f*tan(e/2 +
f*x/2)**4 - 6*a*c**2*f*tan(e/2 + f*x/2)**3 + 6*a*c**2*f*tan(e/2 + f*x/2) - 3*a*c**2*f) - 6*B*tan(e/2 + f*x/2)*
*2/(3*a*c**2*f*tan(e/2 + f*x/2)**4 - 6*a*c**2*f*tan(e/2 + f*x/2)**3 + 6*a*c**2*f*tan(e/2 + f*x/2) - 3*a*c**2*f
) + 4*B*tan(e/2 + f*x/2)/(3*a*c**2*f*tan(e/2 + f*x/2)**4 - 6*a*c**2*f*tan(e/2 + f*x/2)**3 + 6*a*c**2*f*tan(e/2
+ f*x/2) - 3*a*c**2*f) - 2*B/(3*a*c**2*f*tan(e/2 + f*x/2)**4 - 6*a*c**2*f*tan(e/2 + f*x/2)**3 + 6*a*c**2*f*ta
n(e/2 + f*x/2) - 3*a*c**2*f), Ne(f, 0)), (x*(A + B*sin(e))/((a*sin(e) + a)*(-c*sin(e) + c)**2), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (60) = 120\).

Time = 0.23 (sec) , antiderivative size = 266, normalized size of antiderivative = 4.22 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=-\frac {2 \, {\left (\frac {B {\left (\frac {2 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 1\right )}}{a c^{2} - \frac {2 \, a c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a c^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} - \frac {A {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a c^{2} - \frac {2 \, a c^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a c^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {a c^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}\right )}}{3 \, f} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

-2/3*(B*(2*sin(f*x + e)/(cos(f*x + e) + 1) - 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 1)/(a*c^2 - 2*a*c^2*sin(f
*x + e)/(cos(f*x + e) + 1) + 2*a*c^2*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - a*c^2*sin(f*x + e)^4/(cos(f*x + e)
+ 1)^4) - A*(sin(f*x + e)/(cos(f*x + e) + 1) - 3*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 3*sin(f*x + e)^3/(cos(f
*x + e) + 1)^3 + 1)/(a*c^2 - 2*a*c^2*sin(f*x + e)/(cos(f*x + e) + 1) + 2*a*c^2*sin(f*x + e)^3/(cos(f*x + e) +
1)^3 - a*c^2*sin(f*x + e)^4/(cos(f*x + e) + 1)^4))/f

Giac [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.54 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=-\frac {\frac {3 \, {\left (A - B\right )}}{a c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {9 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, B \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, A \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, A + B}{a c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{3}}}{6 \, f} \]

[In]

integrate((A+B*sin(f*x+e))/(a+a*sin(f*x+e))/(c-c*sin(f*x+e))^2,x, algorithm="giac")

[Out]

-1/6*(3*(A - B)/(a*c^2*(tan(1/2*f*x + 1/2*e) + 1)) + (9*A*tan(1/2*f*x + 1/2*e)^2 + 3*B*tan(1/2*f*x + 1/2*e)^2
- 12*A*tan(1/2*f*x + 1/2*e) + 7*A + B)/(a*c^2*(tan(1/2*f*x + 1/2*e) - 1)^3))/f

Mupad [B] (verification not implemented)

Time = 12.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.87 \[ \int \frac {A+B \sin (e+f x)}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^2} \, dx=\frac {2\,\left (\frac {3\,B}{2}+A\,\cos \left (e+f\,x\right )+B\,\cos \left (e+f\,x\right )+2\,A\,\sin \left (e+f\,x\right )-B\,\sin \left (e+f\,x\right )+A\,\cos \left (2\,e+2\,f\,x\right )-\frac {B\,\cos \left (2\,e+2\,f\,x\right )}{2}-\frac {A\,\sin \left (2\,e+2\,f\,x\right )}{2}-\frac {B\,\sin \left (2\,e+2\,f\,x\right )}{2}\right )}{3\,a\,c^2\,f\,\left (2\,\cos \left (e+f\,x\right )-\sin \left (2\,e+2\,f\,x\right )\right )} \]

[In]

int((A + B*sin(e + f*x))/((a + a*sin(e + f*x))*(c - c*sin(e + f*x))^2),x)

[Out]

(2*((3*B)/2 + A*cos(e + f*x) + B*cos(e + f*x) + 2*A*sin(e + f*x) - B*sin(e + f*x) + A*cos(2*e + 2*f*x) - (B*co
s(2*e + 2*f*x))/2 - (A*sin(2*e + 2*f*x))/2 - (B*sin(2*e + 2*f*x))/2))/(3*a*c^2*f*(2*cos(e + f*x) - sin(2*e + 2
*f*x)))