Integrand size = 25, antiderivative size = 136 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {(A+C) \sin (c+d x)}{7 d (a+a \cos (c+d x))^4}+\frac {(3 A-11 C) \sin (c+d x)}{35 a d (a+a \cos (c+d x))^3}+\frac {(6 A+13 C) \sin (c+d x)}{105 d \left (a^2+a^2 \cos (c+d x)\right )^2}+\frac {(6 A+13 C) \sin (c+d x)}{105 d \left (a^4+a^4 \cos (c+d x)\right )} \]
1/7*(A+C)*sin(d*x+c)/d/(a+a*cos(d*x+c))^4+1/35*(3*A-11*C)*sin(d*x+c)/a/d/( a+a*cos(d*x+c))^3+1/105*(6*A+13*C)*sin(d*x+c)/d/(a^2+a^2*cos(d*x+c))^2+1/1 05*(6*A+13*C)*sin(d*x+c)/d/(a^4+a^4*cos(d*x+c))
Time = 2.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.17 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\cos \left (\frac {1}{2} (c+d x)\right ) \sec \left (\frac {c}{2}\right ) \left (70 (3 A+4 C) \sin \left (\frac {d x}{2}\right )-175 C \sin \left (c+\frac {d x}{2}\right )+126 A \sin \left (c+\frac {3 d x}{2}\right )+168 C \sin \left (c+\frac {3 d x}{2}\right )-105 C \sin \left (2 c+\frac {3 d x}{2}\right )+42 A \sin \left (2 c+\frac {5 d x}{2}\right )+91 C \sin \left (2 c+\frac {5 d x}{2}\right )+6 A \sin \left (3 c+\frac {7 d x}{2}\right )+13 C \sin \left (3 c+\frac {7 d x}{2}\right )\right )}{420 a^4 d (1+\cos (c+d x))^4} \]
(Cos[(c + d*x)/2]*Sec[c/2]*(70*(3*A + 4*C)*Sin[(d*x)/2] - 175*C*Sin[c + (d *x)/2] + 126*A*Sin[c + (3*d*x)/2] + 168*C*Sin[c + (3*d*x)/2] - 105*C*Sin[2 *c + (3*d*x)/2] + 42*A*Sin[2*c + (5*d*x)/2] + 91*C*Sin[2*c + (5*d*x)/2] + 6*A*Sin[3*c + (7*d*x)/2] + 13*C*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos [c + d*x])^4)
Time = 0.58 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 3499, 25, 3042, 3229, 3042, 3129, 3042, 3127}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+C \cos ^2(c+d x)}{(a \cos (c+d x)+a)^4} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4}dx\) |
\(\Big \downarrow \) 3499 |
\(\displaystyle \frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac {\int -\frac {a (3 A-4 C)+7 a C \cos (c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {a (3 A-4 C)+7 a C \cos (c+d x)}{(\cos (c+d x) a+a)^3}dx}{7 a^2}+\frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {a (3 A-4 C)+7 a C \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3}dx}{7 a^2}+\frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3229 |
\(\displaystyle \frac {\frac {1}{5} (6 A+13 C) \int \frac {1}{(\cos (c+d x) a+a)^2}dx+\frac {a (3 A-11 C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} (6 A+13 C) \int \frac {1}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx+\frac {a (3 A-11 C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3129 |
\(\displaystyle \frac {\frac {1}{5} (6 A+13 C) \left (\frac {\int \frac {1}{\cos (c+d x) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )+\frac {a (3 A-11 C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {1}{5} (6 A+13 C) \left (\frac {\int \frac {1}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )+\frac {a (3 A-11 C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}}{7 a^2}+\frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
\(\Big \downarrow \) 3127 |
\(\displaystyle \frac {\frac {a (3 A-11 C) \sin (c+d x)}{5 d (a \cos (c+d x)+a)^3}+\frac {1}{5} (6 A+13 C) \left (\frac {\sin (c+d x)}{3 a d (a \cos (c+d x)+a)}+\frac {\sin (c+d x)}{3 d (a \cos (c+d x)+a)^2}\right )}{7 a^2}+\frac {(A+C) \sin (c+d x)}{7 d (a \cos (c+d x)+a)^4}\) |
((A + C)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((a*(3*A - 11*C)*Sin [c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((6*A + 13*C)*(Sin[c + d*x]/(3*d *(a + a*Cos[c + d*x])^2) + Sin[c + d*x]/(3*a*d*(a + a*Cos[c + d*x]))))/5)/ (7*a^2)
3.1.69.3.1 Defintions of rubi rules used
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]
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] + Simp[(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]
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] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1)) I nt[(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)]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[b*(A + C)*Cos[e + f*x]*((a + b*Sin[e + f* x])^m/(a*f*(2*m + 1))), x] + Simp[1/(a^2*(2*m + 1)) Int[(a + b*Sin[e + f* x])^(m + 1)*Simp[a*A*(m + 1) - a*C*m + b*C*(2*m + 1)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && LtQ[m, -1] && EqQ[a^2 - b^2, 0]
Time = 1.78 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.57
method | result | size |
parallelrisch | \(\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\left (A +C \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {7 \left (3 A -C \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+7 \left (A -\frac {C}{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7 A +7 C \right )}{56 a^{4} d}\) | \(78\) |
derivativedivides | \(\frac {\frac {\left (A +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (3 A -C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A -C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{8 d \,a^{4}}\) | \(88\) |
default | \(\frac {\frac {\left (A +C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7}+\frac {\left (3 A -C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}+\frac {\left (3 A -C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}+A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C}{8 d \,a^{4}}\) | \(88\) |
risch | \(\frac {2 i \left (105 C \,{\mathrm e}^{5 i \left (d x +c \right )}+175 C \,{\mathrm e}^{4 i \left (d x +c \right )}+210 A \,{\mathrm e}^{3 i \left (d x +c \right )}+280 C \,{\mathrm e}^{3 i \left (d x +c \right )}+126 A \,{\mathrm e}^{2 i \left (d x +c \right )}+168 C \,{\mathrm e}^{2 i \left (d x +c \right )}+42 A \,{\mathrm e}^{i \left (d x +c \right )}+91 C \,{\mathrm e}^{i \left (d x +c \right )}+6 A +13 C \right )}{105 d \,a^{4} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7}}\) | \(126\) |
norman | \(\frac {\frac {\left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a d}+\frac {\left (A +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{56 a d}+\frac {\left (9 A +5 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 a d}+\frac {\left (27 A +C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{60 a d}+\frac {\left (31 A +3 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{280 a d}+\frac {\left (123 A -31 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{420 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} a^{3}}\) | \(165\) |
1/56*tan(1/2*d*x+1/2*c)*((A+C)*tan(1/2*d*x+1/2*c)^6+7/5*(3*A-C)*tan(1/2*d* x+1/2*c)^4+7*(A-1/3*C)*tan(1/2*d*x+1/2*c)^2+7*A+7*C)/a^4/d
Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {{\left ({\left (6 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, {\left (6 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (39 \, A + 32 \, C\right )} \cos \left (d x + c\right ) + 36 \, A + 8 \, C\right )} \sin \left (d x + c\right )}{105 \, {\left (a^{4} d \cos \left (d x + c\right )^{4} + 4 \, a^{4} d \cos \left (d x + c\right )^{3} + 6 \, a^{4} d \cos \left (d x + c\right )^{2} + 4 \, a^{4} d \cos \left (d x + c\right ) + a^{4} d\right )}} \]
1/105*((6*A + 13*C)*cos(d*x + c)^3 + 4*(6*A + 13*C)*cos(d*x + c)^2 + (39*A + 32*C)*cos(d*x + c) + 36*A + 8*C)*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)
Time = 1.60 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.31 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\begin {cases} \frac {A \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} + \frac {3 A \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} + \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} + \frac {C \tan ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 a^{4} d} - \frac {C \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 a^{4} d} - \frac {C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{24 a^{4} d} + \frac {C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 a^{4} d} & \text {for}\: d \neq 0 \\\frac {x \left (A + C \cos ^{2}{\left (c \right )}\right )}{\left (a \cos {\left (c \right )} + a\right )^{4}} & \text {otherwise} \end {cases} \]
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 ) + C*tan(c/2 + d*x/2)**7/(56*a**4*d) - C*tan(c/2 + d*x/2)**5/(40*a**4*d) - C*tan(c/2 + d*x/2)**3/(24*a**4*d) + C*tan(c/2 + d*x/2)/(8*a**4*d), Ne(d, 0)), (x*(A + C*cos(c)**2)/(a*cos(c) + a)**4, True))
Time = 0.22 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.29 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {C {\left (\frac {105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {15 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}} + \frac {3 \, A {\left (\frac {35 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {35 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {21 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}\right )}}{a^{4}}}{840 \, d} \]
1/840*(C*(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/(cos(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*s in(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
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.86 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 15 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 63 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 35 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 105 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 105 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{840 \, a^{4} d} \]
1/840*(15*A*tan(1/2*d*x + 1/2*c)^7 + 15*C*tan(1/2*d*x + 1/2*c)^7 + 63*A*ta n(1/2*d*x + 1/2*c)^5 - 21*C*tan(1/2*d*x + 1/2*c)^5 + 105*A*tan(1/2*d*x + 1 /2*c)^3 - 35*C*tan(1/2*d*x + 1/2*c)^3 + 105*A*tan(1/2*d*x + 1/2*c) + 105*C *tan(1/2*d*x + 1/2*c))/(a^4*d)
Time = 1.19 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.64 \[ \int \frac {A+C \cos ^2(c+d x)}{(a+a \cos (c+d x))^4} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (A+C\right )}{56\,a^4}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+C\right )}{8\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,A-C\right )}{24\,a^4}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,A-C\right )}{40\,a^4}}{d} \]