\(\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [111]

Optimal result

Integrand size = 33, antiderivative size = 228 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d} \]

[Out]

1/4*a^4*(14*A+11*C)*arctanh(sin(d*x+c))/d+16/35*a^4*(14*A+11*C)*tan(d*x+c)/d+27/140*a^4*(14*A+11*C)*sec(d*x+c)
*tan(d*x+c)/d+1/70*a^4*(14*A+11*C)*sec(d*x+c)^3*tan(d*x+c)/d+1/105*(21*A+4*C)*(a+a*sec(d*x+c))^4*tan(d*x+c)/d+
1/7*C*sec(d*x+c)^2*(a+a*sec(d*x+c))^4*tan(d*x+c)/d+2/21*C*(a+a*sec(d*x+c))^5*tan(d*x+c)/a/d+8/105*a^4*(14*A+11
*C)*tan(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4174, 4095, 4086, 3876, 3855, 3852, 8, 3853} \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \tan (c+d x) \sec ^3(c+d x)}{70 d}+\frac {27 a^4 (14 A+11 C) \tan (c+d x) \sec (c+d x)}{140 d}+\frac {(21 A+4 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{105 d}+\frac {C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 d}+\frac {2 C \tan (c+d x) (a \sec (c+d x)+a)^5}{21 a d} \]

[In]

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

[Out]

(a^4*(14*A + 11*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (16*a^4*(14*A + 11*C)*Tan[c + d*x])/(35*d) + (27*a^4*(14*A +
11*C)*Sec[c + d*x]*Tan[c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(70*d) + ((21*A +
4*C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(105*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(7*
d) + (2*C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(21*a*d) + (8*a^4*(14*A + 11*C)*Tan[c + d*x]^3)/(105*d)

Rule 8

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

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]

Rule 3876

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Int[Expand
Trig[(a + b*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]
&& IGtQ[m, 0] && RationalQ[n]

Rule 4086

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

Rule 4095

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

Rule 4174

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b
_.) + (a_))^(m_), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*(m + n + 1
))), x] + Dist[1/(b*(m + n + 1)), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n*Simp[A*b*(m + n + 1) + b*C*n +
a*C*m*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, C, m, n}, x] && EqQ[a^2 - b^2, 0] &&  !LtQ[m, -2^(
-1)] &&  !LtQ[n, -2^(-1)] && NeQ[m + n + 1, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (a (7 A+2 C)+4 a C \sec (c+d x)) \, dx}{7 a} \\ & = \frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {\int \sec (c+d x) (a+a \sec (c+d x))^4 \left (20 a^2 C+2 a^2 (21 A+4 C) \sec (c+d x)\right ) \, dx}{42 a^2} \\ & = \frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 C)) \int \sec (c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = \frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} (2 (14 A+11 C)) \int \left (a^4 \sec (c+d x)+4 a^4 \sec ^2(c+d x)+6 a^4 \sec ^3(c+d x)+4 a^4 \sec ^4(c+d x)+a^4 \sec ^5(c+d x)\right ) \, dx \\ & = \frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx+\frac {1}{35} \left (2 a^4 (14 A+11 C)\right ) \int \sec ^5(c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \sec ^2(c+d x) \, dx+\frac {1}{35} \left (8 a^4 (14 A+11 C)\right ) \int \sec ^4(c+d x) \, dx+\frac {1}{35} \left (12 a^4 (14 A+11 C)\right ) \int \sec ^3(c+d x) \, dx \\ & = \frac {2 a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{35 d}+\frac {6 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{35 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {1}{70} \left (3 a^4 (14 A+11 C)\right ) \int \sec ^3(c+d x) \, dx+\frac {1}{35} \left (6 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx-\frac {\left (8 a^4 (14 A+11 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{35 d}-\frac {\left (8 a^4 (14 A+11 C)\right ) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{35 d} \\ & = \frac {8 a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{35 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}+\frac {1}{140} \left (3 a^4 (14 A+11 C)\right ) \int \sec (c+d x) \, dx \\ & = \frac {a^4 (14 A+11 C) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac {27 a^4 (14 A+11 C) \sec (c+d x) \tan (c+d x)}{140 d}+\frac {a^4 (14 A+11 C) \sec ^3(c+d x) \tan (c+d x)}{70 d}+\frac {(21 A+4 C) (a+a \sec (c+d x))^4 \tan (c+d x)}{105 d}+\frac {C \sec ^2(c+d x) (a+a \sec (c+d x))^4 \tan (c+d x)}{7 d}+\frac {2 C (a+a \sec (c+d x))^5 \tan (c+d x)}{21 a d}+\frac {8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.85 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.57 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 \left (105 (14 A+11 C) \text {arctanh}(\sin (c+d x))+\left (4 (581 A+454 C)+105 (14 A+11 C) \sec (c+d x)+(952 A+908 C) \sec ^2(c+d x)+70 (6 A+11 C) \sec ^3(c+d x)+12 (7 A+48 C) \sec ^4(c+d x)+280 C \sec ^5(c+d x)+60 C \sec ^6(c+d x)\right ) \tan (c+d x)\right )}{420 d} \]

[In]

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

[Out]

(a^4*(105*(14*A + 11*C)*ArcTanh[Sin[c + d*x]] + (4*(581*A + 454*C) + 105*(14*A + 11*C)*Sec[c + d*x] + (952*A +
908*C)*Sec[c + d*x]^2 + 70*(6*A + 11*C)*Sec[c + d*x]^3 + 12*(7*A + 48*C)*Sec[c + d*x]^4 + 280*C*Sec[c + d*x]^
5 + 60*C*Sec[c + d*x]^6)*Tan[c + d*x]))/(420*d)

Maple [A] (verified)

Time = 0.80 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.11

method	result	size
norman	\(\frac {\frac {512 a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {283 a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{30 d}+\frac {10 a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d}-\frac {a^{4} \left (14 A +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{2 d}+\frac {14 a^{4} \left (22 A +15 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}-\frac {a^{4} \left (50 A +53 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}-\frac {a^{4} \left (5702 A +4503 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{30 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{7}}-\frac {a^{4} \left (14 A +11 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}+\frac {a^{4} \left (14 A +11 C \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d}\)	\(253\)
parallelrisch	\(-\frac {7 a^{4} \left (\left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right ) \left (A +\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-\left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right ) \left (A +\frac {11 C}{14}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-\frac {102 A}{7}-\frac {67 C}{3}\right ) \sin \left (2 d x +2 c \right )+\left (-\frac {802 A}{35}-\frac {124 C}{5}\right ) \sin \left (3 d x +3 c \right )+\left (-\frac {72 A}{7}-\frac {220 C}{21}\right ) \sin \left (4 d x +4 c \right )+\left (-\frac {1102 A}{105}-\frac {908 C}{105}\right ) \sin \left (5 d x +5 c \right )+\left (-2 A -\frac {11 C}{7}\right ) \sin \left (6 d x +6 c \right )+\left (-\frac {908 C}{735}-\frac {166 A}{105}\right ) \sin \left (7 d x +7 c \right )-14 \left (A +\frac {10 C}{7}\right ) \sin \left (d x +c \right )\right )}{2 d \left (\cos \left (7 d x +7 c \right )+7 \cos \left (5 d x +5 c \right )+21 \cos \left (3 d x +3 c \right )+35 \cos \left (d x +c \right )\right )}\)	\(284\)
parts	\(-\frac {\left (a^{4} A +6 a^{4} C \right ) \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a^{4} A +4 a^{4} C \right ) \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 a^{4} A +a^{4} C \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {a^{4} A \tan \left (d x +c \right )}{d}-\frac {a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}+\frac {2 A \sec \left (d x +c \right ) \tan \left (d x +c \right ) a^{4}}{d}+\frac {2 A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right ) a^{4}}{d}+\frac {4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}\)	\(311\)
derivativedivides	\(\frac {a^{4} A \tan \left (d x +c \right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\)	\(374\)
default	\(\frac {a^{4} A \tan \left (d x +c \right )-a^{4} C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )-6 a^{4} C \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )+4 a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 a^{4} C \left (-\left (-\frac {\sec \left (d x +c \right )^{5}}{6}-\frac {5 \sec \left (d x +c \right )^{3}}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )-a^{4} A \left (-\frac {8}{15}-\frac {\sec \left (d x +c \right )^{4}}{5}-\frac {4 \sec \left (d x +c \right )^{2}}{15}\right ) \tan \left (d x +c \right )-a^{4} C \left (-\frac {16}{35}-\frac {\sec \left (d x +c \right )^{6}}{7}-\frac {6 \sec \left (d x +c \right )^{4}}{35}-\frac {8 \sec \left (d x +c \right )^{2}}{35}\right ) \tan \left (d x +c \right )}{d}\)	\(374\)
risch	\(-\frac {i a^{4} \left (-1816 C -2324 A -30380 A \,{\mathrm e}^{8 i \left (d x +c \right )}-10710 A \,{\mathrm e}^{5 i \left (d x +c \right )}-7560 A \,{\mathrm e}^{3 i \left (d x +c \right )}-1470 A \,{\mathrm e}^{i \left (d x +c \right )}-50960 A \,{\mathrm e}^{6 i \left (d x +c \right )}-46480 C \,{\mathrm e}^{6 i \left (d x +c \right )}-41244 A \,{\mathrm e}^{4 i \left (d x +c \right )}-37296 C \,{\mathrm e}^{4 i \left (d x +c \right )}-15848 A \,{\mathrm e}^{2 i \left (d x +c \right )}-12712 C \,{\mathrm e}^{2 i \left (d x +c \right )}-16415 C \,{\mathrm e}^{5 i \left (d x +c \right )}-7700 C \,{\mathrm e}^{3 i \left (d x +c \right )}-1155 C \,{\mathrm e}^{i \left (d x +c \right )}-420 A \,{\mathrm e}^{12 i \left (d x +c \right )}+1155 C \,{\mathrm e}^{13 i \left (d x +c \right )}+1470 A \,{\mathrm e}^{13 i \left (d x +c \right )}-840 C \,{\mathrm e}^{10 i \left (d x +c \right )}+7560 A \,{\mathrm e}^{11 i \left (d x +c \right )}+7700 C \,{\mathrm e}^{11 i \left (d x +c \right )}-7560 A \,{\mathrm e}^{10 i \left (d x +c \right )}+10710 A \,{\mathrm e}^{9 i \left (d x +c \right )}+16415 C \,{\mathrm e}^{9 i \left (d x +c \right )}-17080 C \,{\mathrm e}^{8 i \left (d x +c \right )}\right )}{210 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{7}}+\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{2 d}+\frac {11 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{4 d}-\frac {7 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{2 d}-\frac {11 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{4 d}\)	\(395\)

[In]

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

[Out]

(512/35*a^4*(14*A+11*C)/d*tan(1/2*d*x+1/2*c)^7-283/30*a^4*(14*A+11*C)/d*tan(1/2*d*x+1/2*c)^9+10/3*a^4*(14*A+11
*C)/d*tan(1/2*d*x+1/2*c)^11-1/2*a^4*(14*A+11*C)/d*tan(1/2*d*x+1/2*c)^13+14/3*a^4*(22*A+15*C)/d*tan(1/2*d*x+1/2
*c)^3-1/2*a^4*(50*A+53*C)/d*tan(1/2*d*x+1/2*c)-1/30*a^4*(5702*A+4503*C)/d*tan(1/2*d*x+1/2*c)^5)/(tan(1/2*d*x+1
/2*c)^2-1)^7-1/4*a^4*(14*A+11*C)/d*ln(tan(1/2*d*x+1/2*c)-1)+1/4*a^4*(14*A+11*C)/d*ln(tan(1/2*d*x+1/2*c)+1)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 201, normalized size of antiderivative = 0.88 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (581 \, A + 454 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 105 \, {\left (14 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 4 \, {\left (238 \, A + 227 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, A + 11 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 12 \, {\left (7 \, A + 48 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 280 \, C a^{4} \cos \left (d x + c\right ) + 60 \, C a^{4}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \]

[In]

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

[Out]

1/840*(105*(14*A + 11*C)*a^4*cos(d*x + c)^7*log(sin(d*x + c) + 1) - 105*(14*A + 11*C)*a^4*cos(d*x + c)^7*log(-
sin(d*x + c) + 1) + 2*(4*(581*A + 454*C)*a^4*cos(d*x + c)^6 + 105*(14*A + 11*C)*a^4*cos(d*x + c)^5 + 4*(238*A
+ 227*C)*a^4*cos(d*x + c)^4 + 70*(6*A + 11*C)*a^4*cos(d*x + c)^3 + 12*(7*A + 48*C)*a^4*cos(d*x + c)^2 + 280*C*
a^4*cos(d*x + c) + 60*C*a^4)*sin(d*x + c))/(d*cos(d*x + c)^7)

Sympy [F]

\[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=a^{4} \left (\int A \sec ^{2}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{3}{\left (c + d x \right )}\, dx + \int 6 A \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 A \sec ^{5}{\left (c + d x \right )}\, dx + \int A \sec ^{6}{\left (c + d x \right )}\, dx + \int C \sec ^{4}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{5}{\left (c + d x \right )}\, dx + \int 6 C \sec ^{6}{\left (c + d x \right )}\, dx + \int 4 C \sec ^{7}{\left (c + d x \right )}\, dx + \int C \sec ^{8}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

a**4*(Integral(A*sec(c + d*x)**2, x) + Integral(4*A*sec(c + d*x)**3, x) + Integral(6*A*sec(c + d*x)**4, x) + I
ntegral(4*A*sec(c + d*x)**5, x) + Integral(A*sec(c + d*x)**6, x) + Integral(C*sec(c + d*x)**4, x) + Integral(4
*C*sec(c + d*x)**5, x) + Integral(6*C*sec(c + d*x)**6, x) + Integral(4*C*sec(c + d*x)**7, x) + Integral(C*sec(
c + d*x)**8, x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 462 vs. \(2 (212) = 424\).

Time = 0.22 (sec) , antiderivative size = 462, normalized size of antiderivative = 2.03 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {56 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{4} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 24 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C a^{4} + 336 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{4} + 280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} - 35 \, C a^{4} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, C a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 840 \, A a^{4} \tan \left (d x + c\right )}{840 \, d} \]

[In]

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

[Out]

1/840*(56*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 1680*(tan(d*x + c)^3 + 3*tan(d*x +
c))*A*a^4 + 24*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*C*a^4 + 336*(3*tan
(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 280*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 - 35*C*
a^4*(2*(15*sin(d*x + c)^5 - 40*sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3*sin(d*
x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 210*A*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(
d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 210*C
*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1)
+ 3*log(sin(d*x + c) - 1)) - 840*A*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(
d*x + c) - 1)) + 840*A*a^4*tan(d*x + c))/d

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.38 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (14 \, A a^{4} + 11 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 105 \, {\left (14 \, A a^{4} + 11 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (1470 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} + 1155 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 9800 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 7700 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 27734 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 21791 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 43008 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 33792 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 39914 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 31521 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 21560 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 14700 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5250 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5565 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{7}}}{420 \, d} \]

[In]

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

[Out]

1/420*(105*(14*A*a^4 + 11*C*a^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 105*(14*A*a^4 + 11*C*a^4)*log(abs(tan(1/
2*d*x + 1/2*c) - 1)) - 2*(1470*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 1155*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 9800*A*a^4
*tan(1/2*d*x + 1/2*c)^11 - 7700*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 27734*A*a^4*tan(1/2*d*x + 1/2*c)^9 + 21791*C*a
^4*tan(1/2*d*x + 1/2*c)^9 - 43008*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 33792*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 39914*A*
a^4*tan(1/2*d*x + 1/2*c)^5 + 31521*C*a^4*tan(1/2*d*x + 1/2*c)^5 - 21560*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 14700*C
*a^4*tan(1/2*d*x + 1/2*c)^3 + 5250*A*a^4*tan(1/2*d*x + 1/2*c) + 5565*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x
+ 1/2*c)^2 - 1)^7)/d

Mupad [B] (verification not implemented)

Time = 18.20 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.32 \[ \int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (14\,A+11\,C\right )}{2\,d}-\frac {\left (7\,A\,a^4+\frac {11\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (-\frac {140\,A\,a^4}{3}-\frac {110\,C\,a^4}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1981\,A\,a^4}{15}+\frac {3113\,C\,a^4}{30}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {1024\,A\,a^4}{5}-\frac {5632\,C\,a^4}{35}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {2851\,A\,a^4}{15}+\frac {1501\,C\,a^4}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-\frac {308\,A\,a^4}{3}-70\,C\,a^4\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (25\,A\,a^4+\frac {53\,C\,a^4}{2}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]

[In]

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

[Out]

(a^4*atanh(tan(c/2 + (d*x)/2))*(14*A + 11*C))/(2*d) - (tan(c/2 + (d*x)/2)*(25*A*a^4 + (53*C*a^4)/2) + tan(c/2
+ (d*x)/2)^13*(7*A*a^4 + (11*C*a^4)/2) - tan(c/2 + (d*x)/2)^11*((140*A*a^4)/3 + (110*C*a^4)/3) - tan(c/2 + (d*
x)/2)^3*((308*A*a^4)/3 + 70*C*a^4) + tan(c/2 + (d*x)/2)^5*((2851*A*a^4)/15 + (1501*C*a^4)/10) + tan(c/2 + (d*x
)/2)^9*((1981*A*a^4)/15 + (3113*C*a^4)/30) - tan(c/2 + (d*x)/2)^7*((1024*A*a^4)/5 + (5632*C*a^4)/35))/(d*(7*ta
n(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2
+ (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 - 1))