\(\int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx\) [70]

Optimal result

Integrand size = 23, antiderivative size = 138 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \]

[Out]

1/7*(A-B)*sin(d*x+c)/d/(a+a*cos(d*x+c))^4+1/35*(3*A+4*B)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^3+2/105*(3*A+4*B)*sin
(d*x+c)/d/(a^2+a^2*cos(d*x+c))^2+2/105*(3*A+4*B)*sin(d*x+c)/d/(a^4+a^4*cos(d*x+c))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2829, 2729, 2727} \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4 \cos (c+d x)+a^4\right )}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2 \cos (c+d x)+a^2\right )^2}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a \cos (c+d x)+a)^3}+\frac {(A-B) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4} \]

[In]

Int[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4,x]

[Out]

((A - B)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^
3) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*d*(a^2 + a^2*Cos[c + d*x])^2) + (2*(3*A + 4*B)*Sin[c + d*x])/(105*d*(a^
4 + a^4*Cos[c + d*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)]

Rubi steps \begin{align*} \text {integral}& = \frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \int \frac {1}{(a+a \cos (c+d x))^3} \, dx}{7 a} \\ & = \frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(2 (3 A+4 B)) \int \frac {1}{(a+a \cos (c+d x))^2} \, dx}{35 a^2} \\ & = \frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {(2 (3 A+4 B)) \int \frac {1}{a+a \cos (c+d x)} \, dx}{105 a^3} \\ & = \frac {(A-B) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A+4 B) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {2 (3 A+4 B) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.59 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\left (36 A+13 B+13 (3 A+4 B) \cos (c+d x)+8 (3 A+4 B) \cos ^2(c+d x)+(6 A+8 B) \cos ^3(c+d x)\right ) \sin (c+d x)}{105 a^4 d (1+\cos (c+d x))^4} \]

[In]

Integrate[(A + B*Cos[c + d*x])/(a + a*Cos[c + d*x])^4,x]

[Out]

((36*A + 13*B + 13*(3*A + 4*B)*Cos[c + d*x] + 8*(3*A + 4*B)*Cos[c + d*x]^2 + (6*A + 8*B)*Cos[c + d*x]^3)*Sin[c
+ d*x])/(105*a^4*d*(1 + Cos[c + d*x])^4)

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.58

method	result	size
parallelrisch	\(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (A -B \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 \left (3 A -B \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+7 \left (A +\frac {B}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 A +7 B \right )}{56 a^{4} d}\)	\(80\)
derivativedivides	\(\frac {\frac {\left (A -B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (3 A -B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A +B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\)	\(88\)
default	\(\frac {\frac {\left (A -B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (3 A -B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A +B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{4}}\)	\(88\)
risch	\(\frac {4 i \left (70 B \,{\mathrm e}^{4 i \left (d x +c \right )}+105 A \,{\mathrm e}^{3 i \left (d x +c \right )}+70 B \,{\mathrm e}^{3 i \left (d x +c \right )}+63 A \,{\mathrm e}^{2 i \left (d x +c \right )}+84 B \,{\mathrm e}^{2 i \left (d x +c \right )}+21 A \,{\mathrm e}^{i \left (d x +c \right )}+28 B \,{\mathrm e}^{i \left (d x +c \right )}+3 A +4 B \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\)	\(114\)
norman	\(\frac {\frac {\left (A -B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\left (A +B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (3 A +2 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 a d}+\frac {\left (12 A +B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}+\frac {\left (13 A -6 B \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{140 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}\)	\(141\)

[In]

int((A+B*cos(d*x+c))/(a+cos(d*x+c)*a)^4,x,method=_RETURNVERBOSE)

[Out]

1/56*tan(1/2*d*x+1/2*c)*((A-B)*tan(1/2*d*x+1/2*c)^6+7/5*(3*A-B)*tan(1/2*d*x+1/2*c)^4+7*(A+1/3*B)*tan(1/2*d*x+1
/2*c)^2+7*A+7*B)/a^4/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.91 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\left (2 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right )^{2} + 13 \, {\left (3 \, A + 4 \, B\right )} \cos \left (d x + c\right ) + 36 \, A + 13 \, B\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))^4,x, algorithm="fricas")

[Out]

1/105*(2*(3*A + 4*B)*cos(d*x + c)^3 + 8*(3*A + 4*B)*cos(d*x + c)^2 + 13*(3*A + 4*B)*cos(d*x + c) + 36*A + 13*B
)*sin(d*x + c)/(a^4*d*cos(d*x + c)^4 + 4*a^4*d*cos(d*x + c)^3 + 6*a^4*d*cos(d*x + c)^2 + 4*a^4*d*cos(d*x + c)
+ a^4*d)

Sympy [A] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.28 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} - \frac {B \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {B \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\left (c \right )}\right )}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))**4,x)

[Out]

Piecewise((A*tan(c/2 + d*x/2)**7/(56*a**4*d) + 3*A*tan(c/2 + d*x/2)**5/(40*a**4*d) + A*tan(c/2 + d*x/2)**3/(8*
a**4*d) + A*tan(c/2 + d*x/2)/(8*a**4*d) - B*tan(c/2 + d*x/2)**7/(56*a**4*d) - B*tan(c/2 + d*x/2)**5/(40*a**4*d
) + B*tan(c/2 + d*x/2)**3/(24*a**4*d) + B*tan(c/2 + d*x/2)/(8*a**4*d), Ne(d, 0)), (x*(A + B*cos(c))/(a*cos(c)
+ a)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.27 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {B {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))^4,x, algorithm="maxima")

[Out]

1/840*(B*(105*sin(d*x + c)/(cos(d*x + c) + 1) + 35*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 21*sin(d*x + c)^5/(co
s(d*x + c) + 1)^5 - 15*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^4 + 3*A*(35*sin(d*x + c)/(cos(d*x + c) + 1) + 35
*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 21*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(cos(d*x + c)
+ 1)^7)/a^4)/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.85 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 15 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 35 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]

[In]

integrate((A+B*cos(d*x+c))/(a+a*cos(d*x+c))^4,x, algorithm="giac")

[Out]

1/840*(15*A*tan(1/2*d*x + 1/2*c)^7 - 15*B*tan(1/2*d*x + 1/2*c)^7 + 63*A*tan(1/2*d*x + 1/2*c)^5 - 21*B*tan(1/2*
d*x + 1/2*c)^5 + 105*A*tan(1/2*d*x + 1/2*c)^3 + 35*B*tan(1/2*d*x + 1/2*c)^3 + 105*A*tan(1/2*d*x + 1/2*c) + 105
*B*tan(1/2*d*x + 1/2*c))/(a^4*d)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.63 \[ \int \frac {A+B \cos (c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,A+B\right )}{24\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A-B\right )}{56\,a^4}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+B\right )}{8\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,A-B\right )}{40\,a^4}}{d} \]

[In]

int((A + B*cos(c + d*x))/(a + a*cos(c + d*x))^4,x)

[Out]

((tan(c/2 + (d*x)/2)^3*(3*A + B))/(24*a^4) + (tan(c/2 + (d*x)/2)^7*(A - B))/(56*a^4) + (tan(c/2 + (d*x)/2)*(A
+ B))/(8*a^4) + (tan(c/2 + (d*x)/2)^5*(3*A - B))/(40*a^4))/d