\(\int (a+a \sec (c+d x))^m (B-C+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [636]

Optimal result

Integrand size = 36, antiderivative size = 171 \[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} (B-C) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},1,\frac {5}{2}+m,\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) (a+a \sec (c+d x))^m \tan (c+d x)}{d (3+2 m) \sqrt {1-\sec (c+d x)}}+\frac {2^{\frac {3}{2}+m} C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right ) (1+\sec (c+d x))^{-\frac {1}{2}-m} (a+a \sec (c+d x))^m \tan (c+d x)}{d} \]

[Out]

2^(3/2+m)*C*hypergeom([1/2, -1/2-m],[3/2],1/2-1/2*sec(d*x+c))*(1+sec(d*x+c))^(-1/2-m)*(a+a*sec(d*x+c))^m*tan(d
*x+c)/d+(B-C)*AppellF1(3/2+m,1,1/2,5/2+m,1+sec(d*x+c),1/2+1/2*sec(d*x+c))*(1+sec(d*x+c))*(a+a*sec(d*x+c))^m*2^
(1/2)*tan(d*x+c)/d/(3+2*m)/(1-sec(d*x+c))^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4126, 4009, 3864, 3863, 141, 3913, 3912, 71} \[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sqrt {2} (B-C) \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^m \operatorname {AppellF1}\left (m+\frac {3}{2},\frac {1}{2},1,m+\frac {5}{2},\frac {1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right )}{d (2 m+3) \sqrt {1-\sec (c+d x)}}+\frac {C 2^{m+\frac {3}{2}} \tan (c+d x) (\sec (c+d x)+1)^{-m-\frac {1}{2}} (a \sec (c+d x)+a)^m \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-m-\frac {1}{2},\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right )}{d} \]

[In]

Int[(a + a*Sec[c + d*x])^m*(B - C + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

(Sqrt[2]*(B - C)*AppellF1[3/2 + m, 1/2, 1, 5/2 + m, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])
*(a + a*Sec[c + d*x])^m*Tan[c + d*x])/(d*(3 + 2*m)*Sqrt[1 - Sec[c + d*x]]) + (2^(3/2 + m)*C*Hypergeometric2F1[
1/2, -1/2 - m, 3/2, (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - m)*(a + a*Sec[c + d*x])^m*Tan[c + d*x])/d

Rule 71

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c
- a*d))^n))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}
, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] ||  !(Ra
tionalQ[n] && GtQ[-d/(b*c - a*d), 0]))

Rule 141

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 3863

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^n*(Cot[c + d*x]/(d*Sqrt[1 + Csc[c + d*x]]
*Sqrt[1 - Csc[c + d*x]])), Subst[Int[(1 + b*(x/a))^(n - 1/2)/(x*Sqrt[1 - b*(x/a)]), x], x, Csc[c + d*x]], x] /
; FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] && GtQ[a, 0]

Rule 3864

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[a^IntPart[n]*((a + b*Csc[c + d*x])^FracPart
[n]/(1 + (b/a)*Csc[c + d*x])^FracPart[n]), Int[(1 + (b/a)*Csc[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d, n}, x]
&& EqQ[a^2 - b^2, 0] &&  !IntegerQ[2*n] &&  !GtQ[a, 0]

Rule 3912

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[a^2*d
*(Cot[e + f*x]/(f*Sqrt[a + b*Csc[e + f*x]]*Sqrt[a - b*Csc[e + f*x]])), Subst[Int[(d*x)^(n - 1)*((a + b*x)^(m -
1/2)/Sqrt[a - b*x]), x], x, Csc[e + f*x]], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !In
tegerQ[m] && GtQ[a, 0]

Rule 3913

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[a^Int
Part[m]*((a + b*Csc[e + f*x])^FracPart[m]/(1 + (b/a)*Csc[e + f*x])^FracPart[m]), Int[(1 + (b/a)*Csc[e + f*x])^
m*(d*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !IntegerQ[m] &&  !GtQ
[a, 0]

Rule 4009

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Dist[c, I
nt[(a + b*Csc[e + f*x])^m, x], x] + Dist[d, Int[(a + b*Csc[e + f*x])^m*Csc[e + f*x], x], x] /; FreeQ[{a, b, c,
d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[2*m]

Rule 4126

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_.), x_Symbol] :> Dist[1/b^2, Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[b*B - a*C + b*C*Csc[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\int (a+a \sec (c+d x))^{1+m} (a (B-C)+a C \sec (c+d x)) \, dx}{a^2} \\ & = \frac {(B-C) \int (a+a \sec (c+d x))^{1+m} \, dx}{a}+\frac {C \int \sec (c+d x) (a+a \sec (c+d x))^{1+m} \, dx}{a} \\ & = \left ((B-C) (1+\sec (c+d x))^{-m} (a+a \sec (c+d x))^m\right ) \int (1+\sec (c+d x))^{1+m} \, dx+\left (C (1+\sec (c+d x))^{-m} (a+a \sec (c+d x))^m\right ) \int \sec (c+d x) (1+\sec (c+d x))^{1+m} \, dx \\ & = -\frac {\left ((B-C) (1+\sec (c+d x))^{-\frac {1}{2}-m} (a+a \sec (c+d x))^m \tan (c+d x)\right ) \text {Subst}\left (\int \frac {(1+x)^{\frac {1}{2}+m}}{\sqrt {1-x} x} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)}}-\frac {\left (C (1+\sec (c+d x))^{-\frac {1}{2}-m} (a+a \sec (c+d x))^m \tan (c+d x)\right ) \text {Subst}\left (\int \frac {(1+x)^{\frac {1}{2}+m}}{\sqrt {1-x}} \, dx,x,\sec (c+d x)\right )}{d \sqrt {1-\sec (c+d x)}} \\ & = \frac {\sqrt {2} (B-C) \operatorname {AppellF1}\left (\frac {3}{2}+m,\frac {1}{2},1,\frac {5}{2}+m,\frac {1}{2} (1+\sec (c+d x)),1+\sec (c+d x)\right ) (1+\sec (c+d x)) (a+a \sec (c+d x))^m \tan (c+d x)}{d (3+2 m) \sqrt {1-\sec (c+d x)}}+\frac {2^{\frac {3}{2}+m} C \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-\frac {1}{2}-m,\frac {3}{2},\frac {1}{2} (1-\sec (c+d x))\right ) (1+\sec (c+d x))^{-\frac {1}{2}-m} (a+a \sec (c+d x))^m \tan (c+d x)}{d} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(1817\) vs. \(2(171)=342\).

Time = 12.89 (sec) , antiderivative size = 1817, normalized size of antiderivative = 10.63 \[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\frac {C \left (1+\cos (c+d x)+4 m \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2+m,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^m\right ) (a (1+\sec (c+d x)))^{1+m} (B-C+C \sec (c+d x)) \tan \left (\frac {1}{2} (c+d x)\right )}{d (1+2 m) (C+B \cos (c+d x)-C \cos (c+d x)) (1+\sec (c+d x))}+\frac {2^{1+m} B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},1+m,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )^m \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^m (1+\sec (c+d x))^{-1-m} (a (1+\sec (c+d x)))^{1+m} (B-C+C \sec (c+d x)) \tan \left (\frac {1}{2} (c+d x)\right )}{d (C+B \cos (c+d x)-C \cos (c+d x))}+\frac {30 B \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \cos ^2(c+d x) (a (1+\sec (c+d x)))^{1+m} (B-C+C \sec (c+d x)) \sin (c+d x) \left (3 \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 \left (\operatorname {AppellF1}\left (\frac {3}{2},m,2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-m \operatorname {AppellF1}\left (\frac {3}{2},1+m,1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{d (C+B \cos (c+d x)-C \cos (c+d x)) (1+\sec (c+d x)) \left (45 \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) (1+2 m-2 m \cos (c+d x)+\cos (2 (c+d x)))+6 \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (-5 \operatorname {AppellF1}\left (\frac {3}{2},m,2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+2 m-2 (2+m) \cos (c+d x)+\cos (2 (c+d x)))+5 m \operatorname {AppellF1}\left (\frac {3}{2},1+m,1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+2 m-2 (2+m) \cos (c+d x)+\cos (2 (c+d x)))-48 \left (2 \operatorname {AppellF1}\left (\frac {5}{2},m,3,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 m \operatorname {AppellF1}\left (\frac {5}{2},1+m,2,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+m (1+m) \operatorname {AppellF1}\left (\frac {5}{2},2+m,1,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \cot (c+d x) \csc (c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )\right )+40 \left (\operatorname {AppellF1}\left (\frac {3}{2},m,2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-m \operatorname {AppellF1}\left (\frac {3}{2},1+m,1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )^2 \cos (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}-\frac {30 C \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \cos ^2(c+d x) (a (1+\sec (c+d x)))^{1+m} (B-C+C \sec (c+d x)) \sin (c+d x) \left (3 \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 \left (\operatorname {AppellF1}\left (\frac {3}{2},m,2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-m \operatorname {AppellF1}\left (\frac {3}{2},1+m,1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}{d (C+B \cos (c+d x)-C \cos (c+d x)) (1+\sec (c+d x)) \left (45 \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )^2 \cos ^2\left (\frac {1}{2} (c+d x)\right ) (1+2 m-2 m \cos (c+d x)+\cos (2 (c+d x)))+6 \operatorname {AppellF1}\left (\frac {1}{2},m,1,\frac {3}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (-5 \operatorname {AppellF1}\left (\frac {3}{2},m,2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+2 m-2 (2+m) \cos (c+d x)+\cos (2 (c+d x)))+5 m \operatorname {AppellF1}\left (\frac {3}{2},1+m,1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right ) (1+2 m-2 (2+m) \cos (c+d x)+\cos (2 (c+d x)))-48 \left (2 \operatorname {AppellF1}\left (\frac {5}{2},m,3,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-2 m \operatorname {AppellF1}\left (\frac {5}{2},1+m,2,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )+m (1+m) \operatorname {AppellF1}\left (\frac {5}{2},2+m,1,\frac {7}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right ) \cot (c+d x) \csc (c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right )\right )+40 \left (\operatorname {AppellF1}\left (\frac {3}{2},m,2,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )-m \operatorname {AppellF1}\left (\frac {3}{2},1+m,1,\frac {5}{2},\tan ^2\left (\frac {1}{2} (c+d x)\right ),-\tan ^2\left (\frac {1}{2} (c+d x)\right )\right )\right )^2 \cos (c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \tan ^2\left (\frac {1}{2} (c+d x)\right )\right )}}{a} \]

[In]

Integrate[(a + a*Sec[c + d*x])^m*(B - C + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]

[Out]

((C*(1 + Cos[c + d*x] + 4*m*Cos[c + d*x]*Hypergeometric2F1[1/2, 2 + m, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*
Sec[(c + d*x)/2]^2)^m)*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Tan[(c + d*x)/2])/(d*(1 + 2*m)*
(C + B*Cos[c + d*x] - C*Cos[c + d*x])*(1 + Sec[c + d*x])) + (2^(1 + m)*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 1
+ m, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^m*(1 + Se
c[c + d*x])^(-1 - m)*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Tan[(c + d*x)/2])/(d*(C + B*Cos[c
+ d*x] - C*Cos[c + d*x])) + (30*B*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c +
d*x)/2]^2*Cos[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Sin[c + d*x]*(3*AppellF1[1/2,
m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c
+ d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/
(d*(C + B*Cos[c + d*x] - C*Cos[c + d*x])*(1 + Sec[c + d*x])*(45*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -
Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 + 2*m - 2*m*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*AppellF1[1/2, m,
1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(-5*AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)
/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*m*AppellF1[3/2, 1 + m, 1
, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) - 48*(2*
AppellF1[5/2, m, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*m*AppellF1[5/2, 1 + m, 2, 7/2, Tan[(c +
d*x)/2]^2, -Tan[(c + d*x)/2]^2] + m*(1 + m)*AppellF1[5/2, 2 + m, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]
^2])*Cot[c + d*x]*Csc[c + d*x]*Sin[(c + d*x)/2]^4) + 40*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c
+ d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Sin[(c
+ d*x)/2]^2*Tan[(c + d*x)/2]^2)) - (30*C*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos
[(c + d*x)/2]^2*Cos[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Sin[c + d*x]*(3*AppellF
1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -
Tan[(c + d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2
]^2))/(d*(C + B*Cos[c + d*x] - C*Cos[c + d*x])*(1 + Sec[c + d*x])*(45*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2
]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 + 2*m - 2*m*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*AppellF1[1/
2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(-5*AppellF1[3/2, m, 2, 5/2, Tan[(c
+ d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*m*AppellF1[3/2, 1
+ m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) -
48*(2*AppellF1[5/2, m, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*m*AppellF1[5/2, 1 + m, 2, 7/2, Tan
[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + m*(1 + m)*AppellF1[5/2, 2 + m, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d
*x)/2]^2])*Cot[c + d*x]*Csc[c + d*x]*Sin[(c + d*x)/2]^4) + 40*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -T
an[(c + d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*S
in[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)))/a

Maple [F]

\[\int \left (a +a \sec \left (d x +c \right )\right )^{m} \left (B -C +B \sec \left (d x +c \right )+C \sec \left (d x +c \right )^{2}\right )d x\]

[In]

int((a+a*sec(d*x+c))^m*(B-C+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

[Out]

int((a+a*sec(d*x+c))^m*(B-C+B*sec(d*x+c)+C*sec(d*x+c)^2),x)

Fricas [F]

\[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + B - C\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^m*(B-C+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")

[Out]

integral((C*sec(d*x + c)^2 + B*sec(d*x + c) + B - C)*(a*sec(d*x + c) + a)^m, x)

Sympy [F]

\[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{m} \left (\sec {\left (c + d x \right )} + 1\right ) \left (B + C \sec {\left (c + d x \right )} - C\right )\, dx \]

[In]

integrate((a+a*sec(d*x+c))**m*(B-C+B*sec(d*x+c)+C*sec(d*x+c)**2),x)

[Out]

Integral((a*(sec(c + d*x) + 1))**m*(sec(c + d*x) + 1)*(B + C*sec(c + d*x) - C), x)

Maxima [F]

\[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + B - C\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^m*(B-C+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + B - C)*(a*sec(d*x + c) + a)^m, x)

Giac [F]

\[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + B - C\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{m} \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^m*(B-C+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")

[Out]

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + B - C)*(a*sec(d*x + c) + a)^m, x)

Mupad [F(-1)]

Timed out. \[ \int (a+a \sec (c+d x))^m \left (B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^m\,\left (B-C+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]

[In]

int((a + a/cos(c + d*x))^m*(B - C + B/cos(c + d*x) + C/cos(c + d*x)^2),x)

[Out]

int((a + a/cos(c + d*x))^m*(B - C + B/cos(c + d*x) + C/cos(c + d*x)^2), x)