3.384 \(\int \frac{\sec ^4(c+d x) (B \sec (c+d x)+C \sec ^2(c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx\)

Optimal. Leaf size=243 \[ \frac{2 (9 B-C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}-\frac{2 (3 B-19 C) \tan (c+d x) \sec ^2(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (B-C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 (93 B-29 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 a d}-\frac{4 (111 B-143 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}} \]

[Out]

(Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(111*B -
143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(3*B - 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sq
rt[a + a*Sec[c + d*x]]) + (2*(9*B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec
[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(93*B - 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d
*x])/(315*a*d)

________________________________________________________________________________________

Rubi [A]  time = 0.876197, antiderivative size = 243, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4072, 4021, 4010, 4001, 3795, 203} \[ \frac{2 (9 B-C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}-\frac{2 (3 B-19 C) \tan (c+d x) \sec ^2(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (B-C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 (93 B-29 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 a d}-\frac{4 (111 B-143 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(111*B -
143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(3*B - 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sq
rt[a + a*Sec[c + d*x]]) + (2*(9*B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec
[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(93*B - 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d
*x])/(315*a*d)

Rule 4072

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

Rule 4021

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

Rule 4010

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)), I
nt[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] /; Free
Q[{a, b, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] &&  !LtQ[m, -1]

Rule 4001

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 3795

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

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^4(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx &=\int \frac{\sec ^5(c+d x) (B+C \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx\\ &=\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{2 \int \frac{\sec ^4(c+d x) \left (4 a C+\frac{1}{2} a (9 B-C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{9 a}\\ &=\frac{2 (9 B-C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{4 \int \frac{\sec ^3(c+d x) \left (\frac{3}{2} a^2 (9 B-C)-\frac{3}{4} a^2 (3 B-19 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{63 a^2}\\ &=-\frac{2 (3 B-19 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (9 B-C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{8 \int \frac{\sec ^2(c+d x) \left (-\frac{3}{2} a^3 (3 B-19 C)+\frac{3}{8} a^3 (93 B-29 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{315 a^3}\\ &=-\frac{2 (3 B-19 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (9 B-C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (93 B-29 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 a d}+\frac{16 \int \frac{\sec (c+d x) \left (\frac{3}{16} a^4 (93 B-29 C)-\frac{3}{8} a^4 (111 B-143 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{945 a^4}\\ &=-\frac{4 (111 B-143 C) \tan (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}-\frac{2 (3 B-19 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (9 B-C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (93 B-29 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 a d}+(B-C) \int \frac{\sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx\\ &=-\frac{4 (111 B-143 C) \tan (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}-\frac{2 (3 B-19 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (9 B-C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (93 B-29 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 a d}-\frac{(2 (B-C)) \operatorname{Subst}\left (\int \frac{1}{2 a+x^2} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} (B-C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{\sqrt{a} d}-\frac{4 (111 B-143 C) \tan (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}-\frac{2 (3 B-19 C) \sec ^2(c+d x) \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (9 B-C) \sec ^3(c+d x) \tan (c+d x)}{63 d \sqrt{a+a \sec (c+d x)}}+\frac{2 C \sec ^4(c+d x) \tan (c+d x)}{9 d \sqrt{a+a \sec (c+d x)}}+\frac{2 (93 B-29 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{315 a d}\\ \end{align*}

Mathematica [A]  time = 1.17028, size = 183, normalized size = 0.75 \[ \frac{\tan (c+d x) \left (\frac{1}{4} \sqrt{1-\sec (c+d x)} \sec ^4(c+d x) ((918 B-214 C) \cos (c+d x)-8 (69 B-157 C) \cos (2 (c+d x))+186 B \cos (3 (c+d x))-129 B \cos (4 (c+d x))-423 B-58 C \cos (3 (c+d x))+257 C \cos (4 (c+d x))+1279 C)+315 \sqrt{2} (B-C) \tanh ^{-1}\left (\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right )\right )}{315 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((315*Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + ((-423*B + 1279*C + (918*B - 214*C)*Cos[c + d*
x] - 8*(69*B - 157*C)*Cos[2*(c + d*x)] + 186*B*Cos[3*(c + d*x)] - 58*C*Cos[3*(c + d*x)] - 129*B*Cos[4*(c + d*x
)] + 257*C*Cos[4*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^4)/4)*Tan[c + d*x])/(315*d*Sqrt[1 - Sec[c + d
*x]]*Sqrt[a*(1 + Sec[c + d*x])])

________________________________________________________________________________________

Maple [B]  time = 0.43, size = 975, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x)

[Out]

1/5040/d/a*(315*B*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c)^4*ln(((-2*cos(d*x+c)/(cos(d*x+c)+
1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))-315*C*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c
)^4*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))+1260*B*sin(d*x+c)*(-2*cos(d*
x+c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c)^3*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(
d*x+c))-1260*C*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c)^3*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))
^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))+1890*B*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c)^
2*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))-1890*C*sin(d*x+c)*(-2*cos(d*x+
c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c)^2*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*
x+c))+1260*B*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c)*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/
2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))-1260*C*sin(d*x+c)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(9/2)*cos(d*x+c)*ln((
(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))+315*B*(-2*cos(d*x+c)/(cos(d*x+c)+1))
^(9/2)*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))*sin(d*x+c)-315*C*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(9/2)*ln(((-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)-cos(d*x+c)+1)/sin(d*x+c))*sin
(d*x+c)+4128*B*cos(d*x+c)^5-8224*C*cos(d*x+c)^5-7104*B*cos(d*x+c)^4+9152*C*cos(d*x+c)^4+3264*B*cos(d*x+c)^3-27
52*C*cos(d*x+c)^3-1728*B*cos(d*x+c)^2+1984*C*cos(d*x+c)^2+1440*B*cos(d*x+c)-1280*C*cos(d*x+c)+1120*C)*(a*(cos(
d*x+c)+1)/cos(d*x+c))^(1/2)/cos(d*x+c)^4/sin(d*x+c)

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

Fricas [A]  time = 0.661966, size = 1211, normalized size = 4.98 \begin{align*} \left [-\frac{315 \, \sqrt{2}{\left ({\left (B - C\right )} a \cos \left (d x + c\right )^{5} +{\left (B - C\right )} a \cos \left (d x + c\right )^{4}\right )} \sqrt{-\frac{1}{a}} \log \left (\frac{2 \, \sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{-\frac{1}{a}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 4 \,{\left ({\left (129 \, B - 257 \, C\right )} \cos \left (d x + c\right )^{4} -{\left (93 \, B - 29 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \,{\left (9 \, B - C\right )} \cos \left (d x + c\right ) - 35 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{630 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}}, -\frac{2 \,{\left ({\left (129 \, B - 257 \, C\right )} \cos \left (d x + c\right )^{4} -{\left (93 \, B - 29 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \,{\left (3 \, B - 19 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \,{\left (9 \, B - C\right )} \cos \left (d x + c\right ) - 35 \, C\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) + \frac{315 \, \sqrt{2}{\left ({\left (B - C\right )} a \cos \left (d x + c\right )^{5} +{\left (B - C\right )} a \cos \left (d x + c\right )^{4}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right )}{\sqrt{a}}}{315 \,{\left (a d \cos \left (d x + c\right )^{5} + a d \cos \left (d x + c\right )^{4}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[-1/630*(315*sqrt(2)*((B - C)*a*cos(d*x + c)^5 + (B - C)*a*cos(d*x + c)^4)*sqrt(-1/a)*log((2*sqrt(2)*sqrt((a*c
os(d*x + c) + a)/cos(d*x + c))*sqrt(-1/a)*cos(d*x + c)*sin(d*x + c) + 3*cos(d*x + c)^2 + 2*cos(d*x + c) - 1)/(
cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 4*((129*B - 257*C)*cos(d*x + c)^4 - (93*B - 29*C)*cos(d*x + c)^3 + 3*(
3*B - 19*C)*cos(d*x + c)^2 - 5*(9*B - C)*cos(d*x + c) - 35*C)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x
+ c))/(a*d*cos(d*x + c)^5 + a*d*cos(d*x + c)^4), -1/315*(2*((129*B - 257*C)*cos(d*x + c)^4 - (93*B - 29*C)*cos
(d*x + c)^3 + 3*(3*B - 19*C)*cos(d*x + c)^2 - 5*(9*B - C)*cos(d*x + c) - 35*C)*sqrt((a*cos(d*x + c) + a)/cos(d
*x + c))*sin(d*x + c) + 315*sqrt(2)*((B - C)*a*cos(d*x + c)^5 + (B - C)*a*cos(d*x + c)^4)*arctan(sqrt(2)*sqrt(
(a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c)))/sqrt(a))/(a*d*cos(d*x + c)^5 + a*d*cos
(d*x + c)^4)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (B + C \sec{\left (c + d x \right )}\right ) \sec ^{5}{\left (c + d x \right )}}{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**(1/2),x)

[Out]

Integral((B + C*sec(c + d*x))*sec(c + d*x)**5/sqrt(a*(sec(c + d*x) + 1)), x)

________________________________________________________________________________________

Giac [A]  time = 9.22317, size = 528, normalized size = 2.17 \begin{align*} \frac{\frac{315 \,{\left (\sqrt{2} B - \sqrt{2} C\right )} \log \left ({\left | -\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} \right |}\right )}{\sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{2 \,{\left (\frac{315 \, \sqrt{2} C a^{4}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} +{\left (420 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) - 840 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) -{\left (756 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) - 1638 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) -{\left (612 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) - 936 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) -{\left (276 \, \sqrt{2} B a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right ) - 383 \, \sqrt{2} C a^{4} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}}{315 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/315*(315*(sqrt(2)*B - sqrt(2)*C)*log(abs(-sqrt(-a)*tan(1/2*d*x + 1/2*c) + sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a
)))/(sqrt(-a)*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)) - 2*(315*sqrt(2)*C*a^4/sgn(tan(1/2*d*x + 1/2*c)^2 - 1) + (420*s
qrt(2)*B*a^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - 840*sqrt(2)*C*a^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - (756*sqrt(2
)*B*a^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - 1638*sqrt(2)*C*a^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - (612*sqrt(2)*B*
a^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - 936*sqrt(2)*C*a^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - (276*sqrt(2)*B*a^4*s
gn(tan(1/2*d*x + 1/2*c)^2 - 1) - 383*sqrt(2)*C*a^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1))*tan(1/2*d*x + 1/2*c)^2)*ta
n(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)^2)*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1
/2*c)^2 - a)^4*sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)))/d