\(\int \frac {(A+C \cos ^2(c+d x)) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx\) [46]

Optimal result

Integrand size = 33, antiderivative size = 133 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {(3 A+2 C) \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {(4 A+3 C) \tan (c+d x)}{a d}-\frac {(3 A+2 C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {(4 A+3 C) \tan ^3(c+d x)}{3 a d} \]

[Out]

-1/2*(3*A+2*C)*arctanh(sin(d*x+c))/a/d+(4*A+3*C)*tan(d*x+c)/a/d-1/2*(3*A+2*C)*sec(d*x+c)*tan(d*x+c)/a/d-(A+C)*
sec(d*x+c)^2*tan(d*x+c)/d/(a+a*cos(d*x+c))+1/3*(4*A+3*C)*tan(d*x+c)^3/a/d

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3121, 2827, 3852, 3853, 3855} \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {(3 A+2 C) \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {(4 A+3 C) \tan ^3(c+d x)}{3 a d}+\frac {(4 A+3 C) \tan (c+d x)}{a d}-\frac {(3 A+2 C) \tan (c+d x) \sec (c+d x)}{2 a d}-\frac {(A+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \cos (c+d x)+a)} \]

[In]

Int[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]

[Out]

-1/2*((3*A + 2*C)*ArcTanh[Sin[c + d*x]])/(a*d) + ((4*A + 3*C)*Tan[c + d*x])/(a*d) - ((3*A + 2*C)*Sec[c + d*x]*
Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Cos[c + d*x])) + ((4*A + 3*C)*Tan[c +
d*x]^3)/(3*a*d)

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 3121

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

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]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {\int (a (4 A+3 C)-a (3 A+2 C) \cos (c+d x)) \sec ^4(c+d x) \, dx}{a^2} \\ & = -\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {(3 A+2 C) \int \sec ^3(c+d x) \, dx}{a}+\frac {(4 A+3 C) \int \sec ^4(c+d x) \, dx}{a} \\ & = -\frac {(3 A+2 C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}-\frac {(3 A+2 C) \int \sec (c+d x) \, dx}{2 a}-\frac {(4 A+3 C) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d} \\ & = -\frac {(3 A+2 C) \text {arctanh}(\sin (c+d x))}{2 a d}+\frac {(4 A+3 C) \tan (c+d x)}{a d}-\frac {(3 A+2 C) \sec (c+d x) \tan (c+d x)}{2 a d}-\frac {(A+C) \sec ^2(c+d x) \tan (c+d x)}{d (a+a \cos (c+d x))}+\frac {(4 A+3 C) \tan ^3(c+d x)}{3 a d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(765\) vs. \(2(133)=266\).

Time = 7.01 (sec) , antiderivative size = 765, normalized size of antiderivative = 5.75 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {(3 A+2 C) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))}+\frac {(-3 A-2 C) \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))}+\frac {2 \cos \left (\frac {c}{2}+\frac {d x}{2}\right ) \sec \left (\frac {c}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{d (a+a \cos (c+d x))}+\frac {A \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {d x}{2}\right )}{3 d (a+a \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (-A \cos \left (\frac {c}{2}\right )+2 A \sin \left (\frac {c}{2}\right )\right )}{3 d (a+a \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {2 \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (5 A \sin \left (\frac {d x}{2}\right )+3 C \sin \left (\frac {d x}{2}\right )\right )}{3 d (a+a \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )}+\frac {A \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {d x}{2}\right )}{3 d (a+a \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^3}+\frac {\cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \cos \left (\frac {c}{2}\right )+2 A \sin \left (\frac {c}{2}\right )\right )}{3 d (a+a \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2}+\frac {2 \cos ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (5 A \sin \left (\frac {d x}{2}\right )+3 C \sin \left (\frac {d x}{2}\right )\right )}{3 d (a+a \cos (c+d x)) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )} \]

[In]

Integrate[((A + C*Cos[c + d*x]^2)*Sec[c + d*x]^4)/(a + a*Cos[c + d*x]),x]

[Out]

((3*A + 2*C)*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])) + ((-
3*A - 2*C)*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]])/(d*(a + a*Cos[c + d*x])) + (2*Co
s[c/2 + (d*x)/2]*Sec[c/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(d*(a + a*Cos[c + d*x])) + (A*Cos[c/2 + (d*x)/2]^
2*Sin[(d*x)/2])/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) +
(Cos[c/2 + (d*x)/2]^2*(-(A*Cos[c/2]) + 2*A*Sin[c/2]))/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/
2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (2*Cos[c/2 + (d*x)/2]^2*(5*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(3*d*(a
+ a*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (A*Cos[c/2 + (d*x)/2]^2*
Sin[(d*x)/2])/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (
Cos[c/2 + (d*x)/2]^2*(A*Cos[c/2] + 2*A*Sin[c/2]))/(3*d*(a + a*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (
d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (2*Cos[c/2 + (d*x)/2]^2*(5*A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(3*d*(a + a*
Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.28

method	result	size
parallelrisch	\(\frac {27 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {2 C}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-27 \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) \left (A +\frac {2 C}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+44 \left (\frac {\left (4 A +3 C \right ) \cos \left (3 d x +3 c \right )}{11}+\frac {\left (\frac {7 A}{2}+3 C \right ) \cos \left (2 d x +2 c \right )}{11}+\left (A +\frac {9 C}{11}\right ) \cos \left (d x +c \right )+\frac {A}{2}+\frac {3 C}{11}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{6 a d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\)	\(170\)
derivativedivides	\(\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (-\frac {3 A}{2}-C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\frac {5 A}{2}+C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (\frac {3 A}{2}+C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {\frac {5 A}{2}+C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{d a}\)	\(172\)
default	\(\frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) C -\frac {A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\left (-\frac {3 A}{2}-C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )-\frac {\frac {5 A}{2}+C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {A}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {A}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\left (\frac {3 A}{2}+C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\frac {\frac {5 A}{2}+C}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}}{d a}\)	\(172\)
norman	\(\frac {\frac {\left (A +C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}+\frac {\left (A +3 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {3 \left (2 A +C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {\left (4 A +3 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 \left (7 A +9 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}-\frac {2 \left (10 A +3 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}+\frac {\left (3 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a d}-\frac {\left (3 A +2 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 a d}\)	\(233\)
risch	\(\frac {i \left (9 A \,{\mathrm e}^{6 i \left (d x +c \right )}+6 C \,{\mathrm e}^{6 i \left (d x +c \right )}+9 A \,{\mathrm e}^{5 i \left (d x +c \right )}+6 C \,{\mathrm e}^{5 i \left (d x +c \right )}+24 A \,{\mathrm e}^{4 i \left (d x +c \right )}+24 C \,{\mathrm e}^{4 i \left (d x +c \right )}+24 A \,{\mathrm e}^{3 i \left (d x +c \right )}+12 C \,{\mathrm e}^{3 i \left (d x +c \right )}+39 A \,{\mathrm e}^{2 i \left (d x +c \right )}+30 C \,{\mathrm e}^{2 i \left (d x +c \right )}+7 A \,{\mathrm e}^{i \left (d x +c \right )}+6 C \,{\mathrm e}^{i \left (d x +c \right )}+16 A +12 C \right )}{3 a d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}-\frac {3 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{a d}+\frac {3 A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{a d}\)	\(275\)

[In]

int((A+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+cos(d*x+c)*a),x,method=_RETURNVERBOSE)

[Out]

1/6*(27*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*(A+2/3*C)*ln(tan(1/2*d*x+1/2*c)-1)-27*(1/3*cos(3*d*x+3*c)+cos(d*x+c))*
(A+2/3*C)*ln(tan(1/2*d*x+1/2*c)+1)+44*(1/11*(4*A+3*C)*cos(3*d*x+3*c)+1/11*(7/2*A+3*C)*cos(2*d*x+2*c)+(A+9/11*C
)*cos(d*x+c)+1/2*A+3/11*C)*tan(1/2*d*x+1/2*c))/a/d/(cos(3*d*x+3*c)+3*cos(d*x+c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.29 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {3 \, {\left ({\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (4 \, {\left (4 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{2} - A \cos \left (d x + c\right ) + 2 \, A\right )} \sin \left (d x + c\right )}{12 \, {\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\right )}} \]

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+a*cos(d*x+c)),x, algorithm="fricas")

[Out]

-1/12*(3*((3*A + 2*C)*cos(d*x + c)^4 + (3*A + 2*C)*cos(d*x + c)^3)*log(sin(d*x + c) + 1) - 3*((3*A + 2*C)*cos(
d*x + c)^4 + (3*A + 2*C)*cos(d*x + c)^3)*log(-sin(d*x + c) + 1) - 2*(4*(4*A + 3*C)*cos(d*x + c)^3 + (7*A + 6*C
)*cos(d*x + c)^2 - A*cos(d*x + c) + 2*A)*sin(d*x + c))/(a*d*cos(d*x + c)^4 + a*d*cos(d*x + c)^3)

Sympy [F]

\[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\int \frac {A \sec ^{4}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx + \int \frac {C \cos ^{2}{\left (c + d x \right )} \sec ^{4}{\left (c + d x \right )}}{\cos {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate((A+C*cos(d*x+c)**2)*sec(d*x+c)**4/(a+a*cos(d*x+c)),x)

[Out]

(Integral(A*sec(c + d*x)**4/(cos(c + d*x) + 1), x) + Integral(C*cos(c + d*x)**2*sec(c + d*x)**4/(cos(c + d*x)
+ 1), x))/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (127) = 254\).

Time = 0.25 (sec) , antiderivative size = 325, normalized size of antiderivative = 2.44 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {A {\left (\frac {2 \, {\left (\frac {9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a - \frac {3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac {9 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac {6 \, \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 6 \, C {\left (\frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} - \frac {\log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a - \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \]

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+a*cos(d*x+c)),x, algorithm="maxima")

[Out]

1/6*(A*(2*(9*sin(d*x + c)/(cos(d*x + c) + 1) - 16*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 15*sin(d*x + c)^5/(cos
(d*x + c) + 1)^5)/(a - 3*a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - a*s
in(d*x + c)^6/(cos(d*x + c) + 1)^6) - 9*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a + 9*log(sin(d*x + c)/(cos(d
*x + c) + 1) - 1)/a + 6*sin(d*x + c)/(a*(cos(d*x + c) + 1))) - 6*C*(log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/a
- log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a - 2*sin(d*x + c)/((a - a*sin(d*x + c)^2/(cos(d*x + c) + 1)^2)*(c
os(d*x + c) + 1)) - sin(d*x + c)/(a*(cos(d*x + c) + 1))))/d

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.39 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=-\frac {\frac {3 \, {\left (3 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac {3 \, {\left (3 \, A + 2 \, C\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac {6 \, {\left (A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{a} + \frac {2 \, {\left (15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3} a}}{6 \, d} \]

[In]

integrate((A+C*cos(d*x+c)^2)*sec(d*x+c)^4/(a+a*cos(d*x+c)),x, algorithm="giac")

[Out]

-1/6*(3*(3*A + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a - 3*(3*A + 2*C)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/a
- 6*(A*tan(1/2*d*x + 1/2*c) + C*tan(1/2*d*x + 1/2*c))/a + 2*(15*A*tan(1/2*d*x + 1/2*c)^5 + 6*C*tan(1/2*d*x + 1
/2*c)^5 - 16*A*tan(1/2*d*x + 1/2*c)^3 - 12*C*tan(1/2*d*x + 1/2*c)^3 + 9*A*tan(1/2*d*x + 1/2*c) + 6*C*tan(1/2*d
*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^3*a))/d

Mupad [B] (verification not implemented)

Time = 1.43 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.13 \[ \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x)}{a+a \cos (c+d x)} \, dx=\frac {\left (5\,A+2\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {16\,A}{3}-4\,C\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (3\,A+2\,C\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )}-\frac {2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,A}{2}+C\right )}{a\,d}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A+C\right )}{a\,d} \]

[In]

int((A + C*cos(c + d*x)^2)/(cos(c + d*x)^4*(a + a*cos(c + d*x))),x)

[Out]

(tan(c/2 + (d*x)/2)^5*(5*A + 2*C) - tan(c/2 + (d*x)/2)^3*((16*A)/3 + 4*C) + tan(c/2 + (d*x)/2)*(3*A + 2*C))/(d
*(a - 3*a*tan(c/2 + (d*x)/2)^2 + 3*a*tan(c/2 + (d*x)/2)^4 - a*tan(c/2 + (d*x)/2)^6)) - (2*atanh(tan(c/2 + (d*x
)/2))*((3*A)/2 + C))/(a*d) + (tan(c/2 + (d*x)/2)*(A + C))/(a*d)