3.138 \(\int \frac{\sec ^4(c+d x) (A+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^3} \, dx\)

Optimal. Leaf size=198 \[ -\frac{2 (11 A+76 C) \tan (c+d x)}{15 a^3 d}+\frac{(2 A+13 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(11 A+76 C) \tan (c+d x) \sec ^2(c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(2 A+13 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(A+11 C) \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2} \]

[Out]

((2*A + 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(11*A + 76*C)*Tan[c + d*x])/(15*a^3*d) + ((2*A + 13*C)*Sec
[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A +
11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((11*A + 76*C)*Sec[c + d*x]^2*Tan[c + d*
x])/(15*d*(a^3 + a^3*Sec[c + d*x]))

________________________________________________________________________________________

Rubi [A]  time = 0.489821, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules used = {4085, 4019, 3787, 3767, 8, 3768, 3770} \[ -\frac{2 (11 A+76 C) \tan (c+d x)}{15 a^3 d}+\frac{(2 A+13 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(11 A+76 C) \tan (c+d x) \sec ^2(c+d x)}{15 d \left (a^3 \sec (c+d x)+a^3\right )}+\frac{(2 A+13 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(A+11 C) \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]

[Out]

((2*A + 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(11*A + 76*C)*Tan[c + d*x])/(15*a^3*d) + ((2*A + 13*C)*Sec
[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A +
11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((11*A + 76*C)*Sec[c + d*x]^2*Tan[c + d*
x])/(15*d*(a^3 + a^3*Sec[c + d*x]))

Rule 4085

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

Rule 4019

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

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 3767

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 8

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

Rule 3768

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 3770

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

Rubi steps

\begin{align*} \int \frac{\sec ^4(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^3} \, dx &=-\frac{(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{\int \frac{\sec ^4(c+d x) (-a (A-4 C)-a (2 A+7 C) \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{\int \frac{\sec ^3(c+d x) \left (3 a^2 (A+11 C)-a^2 (8 A+43 C) \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{15 a^4}\\ &=-\frac{(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}-\frac{\int \sec ^2(c+d x) \left (2 a^3 (11 A+76 C)-15 a^3 (2 A+13 C) \sec (c+d x)\right ) \, dx}{15 a^6}\\ &=-\frac{(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{(2 A+13 C) \int \sec ^3(c+d x) \, dx}{a^3}-\frac{(2 (11 A+76 C)) \int \sec ^2(c+d x) \, dx}{15 a^3}\\ &=\frac{(2 A+13 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac{(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}+\frac{(2 A+13 C) \int \sec (c+d x) \, dx}{2 a^3}+\frac{(2 (11 A+76 C)) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{15 a^3 d}\\ &=\frac{(2 A+13 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{2 (11 A+76 C) \tan (c+d x)}{15 a^3 d}+\frac{(2 A+13 C) \sec (c+d x) \tan (c+d x)}{2 a^3 d}-\frac{(A+C) \sec ^4(c+d x) \tan (c+d x)}{5 d (a+a \sec (c+d x))^3}-\frac{(A+11 C) \sec ^3(c+d x) \tan (c+d x)}{15 a d (a+a \sec (c+d x))^2}-\frac{(11 A+76 C) \sec ^2(c+d x) \tan (c+d x)}{15 d \left (a^3+a^3 \sec (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 2.86662, size = 632, normalized size = 3.19 \[ -\frac{\cos \left (\frac{1}{2} (c+d x)\right ) \sec (c+d x) \left (A+C \sec ^2(c+d x)\right ) \left (\sec \left (\frac{c}{2}\right ) \sec (c) \sec ^2(c+d x) \left (-654 A \sin \left (c-\frac{d x}{2}\right )+654 A \sin \left (c+\frac{d x}{2}\right )-490 A \sin \left (2 c+\frac{d x}{2}\right )-350 A \sin \left (c+\frac{3 d x}{2}\right )+530 A \sin \left (2 c+\frac{3 d x}{2}\right )-350 A \sin \left (3 c+\frac{3 d x}{2}\right )+378 A \sin \left (c+\frac{5 d x}{2}\right )-150 A \sin \left (2 c+\frac{5 d x}{2}\right )+378 A \sin \left (3 c+\frac{5 d x}{2}\right )-150 A \sin \left (4 c+\frac{5 d x}{2}\right )+190 A \sin \left (2 c+\frac{7 d x}{2}\right )-30 A \sin \left (3 c+\frac{7 d x}{2}\right )+190 A \sin \left (4 c+\frac{7 d x}{2}\right )-30 A \sin \left (5 c+\frac{7 d x}{2}\right )+44 A \sin \left (3 c+\frac{9 d x}{2}\right )+44 A \sin \left (5 c+\frac{9 d x}{2}\right )-5 (98 A+247 C) \sin \left (\frac{d x}{2}\right )+5 (106 A+761 C) \sin \left (\frac{3 d x}{2}\right )-4329 C \sin \left (c-\frac{d x}{2}\right )+1989 C \sin \left (c+\frac{d x}{2}\right )-3575 C \sin \left (2 c+\frac{d x}{2}\right )-475 C \sin \left (c+\frac{3 d x}{2}\right )+2005 C \sin \left (2 c+\frac{3 d x}{2}\right )-2275 C \sin \left (3 c+\frac{3 d x}{2}\right )+2673 C \sin \left (c+\frac{5 d x}{2}\right )+105 C \sin \left (2 c+\frac{5 d x}{2}\right )+1593 C \sin \left (3 c+\frac{5 d x}{2}\right )-975 C \sin \left (4 c+\frac{5 d x}{2}\right )+1325 C \sin \left (2 c+\frac{7 d x}{2}\right )+255 C \sin \left (3 c+\frac{7 d x}{2}\right )+875 C \sin \left (4 c+\frac{7 d x}{2}\right )-195 C \sin \left (5 c+\frac{7 d x}{2}\right )+304 C \sin \left (3 c+\frac{9 d x}{2}\right )+90 C \sin \left (4 c+\frac{9 d x}{2}\right )+214 C \sin \left (5 c+\frac{9 d x}{2}\right )\right )+1920 (2 A+13 C) \cos ^5\left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )\right )}{240 a^3 d (\sec (c+d x)+1)^3 (A \cos (2 (c+d x))+A+2 C)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]

[Out]

-(Cos[(c + d*x)/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(1920*(2*A + 13*C)*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x
)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-5*(98*
A + 247*C)*Sin[(d*x)/2] + 5*(106*A + 761*C)*Sin[(3*d*x)/2] - 654*A*Sin[c - (d*x)/2] - 4329*C*Sin[c - (d*x)/2]
+ 654*A*Sin[c + (d*x)/2] + 1989*C*Sin[c + (d*x)/2] - 490*A*Sin[2*c + (d*x)/2] - 3575*C*Sin[2*c + (d*x)/2] - 35
0*A*Sin[c + (3*d*x)/2] - 475*C*Sin[c + (3*d*x)/2] + 530*A*Sin[2*c + (3*d*x)/2] + 2005*C*Sin[2*c + (3*d*x)/2] -
350*A*Sin[3*c + (3*d*x)/2] - 2275*C*Sin[3*c + (3*d*x)/2] + 378*A*Sin[c + (5*d*x)/2] + 2673*C*Sin[c + (5*d*x)/
2] - 150*A*Sin[2*c + (5*d*x)/2] + 105*C*Sin[2*c + (5*d*x)/2] + 378*A*Sin[3*c + (5*d*x)/2] + 1593*C*Sin[3*c + (
5*d*x)/2] - 150*A*Sin[4*c + (5*d*x)/2] - 975*C*Sin[4*c + (5*d*x)/2] + 190*A*Sin[2*c + (7*d*x)/2] + 1325*C*Sin[
2*c + (7*d*x)/2] - 30*A*Sin[3*c + (7*d*x)/2] + 255*C*Sin[3*c + (7*d*x)/2] + 190*A*Sin[4*c + (7*d*x)/2] + 875*C
*Sin[4*c + (7*d*x)/2] - 30*A*Sin[5*c + (7*d*x)/2] - 195*C*Sin[5*c + (7*d*x)/2] + 44*A*Sin[3*c + (9*d*x)/2] + 3
04*C*Sin[3*c + (9*d*x)/2] + 90*C*Sin[4*c + (9*d*x)/2] + 44*A*Sin[5*c + (9*d*x)/2] + 214*C*Sin[5*c + (9*d*x)/2]
)))/(240*a^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^3)

________________________________________________________________________________________

Maple [A]  time = 0.074, size = 289, normalized size = 1.5 \begin{align*} -{\frac{A}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{C}{20\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5}}-{\frac{A}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{2\,C}{3\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{7\,A}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{31\,C}{4\,d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{A}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{13\,C}{2\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{C}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{7\,C}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{A}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{13\,C}{2\,d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }+{\frac{C}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{7\,C}{2\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x)

[Out]

-1/20/d/a^3*tan(1/2*d*x+1/2*c)^5*A-1/20/d/a^3*C*tan(1/2*d*x+1/2*c)^5-1/3/d/a^3*tan(1/2*d*x+1/2*c)^3*A-2/3/d/a^
3*C*tan(1/2*d*x+1/2*c)^3-7/4/d/a^3*A*tan(1/2*d*x+1/2*c)-31/4/d/a^3*C*tan(1/2*d*x+1/2*c)+1/d/a^3*ln(tan(1/2*d*x
+1/2*c)+1)*A+13/2/d/a^3*ln(tan(1/2*d*x+1/2*c)+1)*C-1/2/d/a^3*C/(tan(1/2*d*x+1/2*c)+1)^2+7/2/d/a^3*C/(tan(1/2*d
*x+1/2*c)+1)-1/d/a^3*ln(tan(1/2*d*x+1/2*c)-1)*A-13/2/d/a^3*ln(tan(1/2*d*x+1/2*c)-1)*C+1/2/d/a^3*C/(tan(1/2*d*x
+1/2*c)-1)^2+7/2/d/a^3*C/(tan(1/2*d*x+1/2*c)-1)

________________________________________________________________________________________

Maxima [A]  time = 0.962566, size = 446, normalized size = 2.25 \begin{align*} -\frac{C{\left (\frac{60 \,{\left (\frac{5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{7 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a^{3} - \frac{2 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac{\frac{465 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{40 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{390 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{390 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )} + A{\left (\frac{\frac{105 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{20 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{3 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}{a^{3}} - \frac{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a^{3}} + \frac{60 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a^{3}}\right )}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/60*(C*(60*(5*sin(d*x + c)/(cos(d*x + c) + 1) - 7*sin(d*x + c)^3/(cos(d*x + c) + 1)^3)/(a^3 - 2*a^3*sin(d*x
+ c)^2/(cos(d*x + c) + 1)^2 + a^3*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (465*sin(d*x + c)/(cos(d*x + c) + 1)
+ 40*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 390*log(sin(d*x + c)/(
cos(d*x + c) + 1) + 1)/a^3 + 390*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^3) + A*((105*sin(d*x + c)/(cos(d*x
+ c) + 1) + 20*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 3*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/a^3 - 60*log(sin(
d*x + c)/(cos(d*x + c) + 1) + 1)/a^3 + 60*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/a^3))/d

________________________________________________________________________________________

Fricas [A]  time = 0.523106, size = 740, normalized size = 3.74 \begin{align*} \frac{15 \,{\left ({\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \,{\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \,{\left ({\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{5} + 3 \,{\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (2 \, A + 13 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (4 \,{\left (11 \, A + 76 \, C\right )} \cos \left (d x + c\right )^{4} + 3 \,{\left (34 \, A + 239 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (64 \, A + 479 \, C\right )} \cos \left (d x + c\right )^{2} + 45 \, C \cos \left (d x + c\right ) - 15 \, C\right )} \sin \left (d x + c\right )}{60 \,{\left (a^{3} d \cos \left (d x + c\right )^{5} + 3 \, a^{3} d \cos \left (d x + c\right )^{4} + 3 \, a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="fricas")

[Out]

1/60*(15*((2*A + 13*C)*cos(d*x + c)^5 + 3*(2*A + 13*C)*cos(d*x + c)^4 + 3*(2*A + 13*C)*cos(d*x + c)^3 + (2*A +
13*C)*cos(d*x + c)^2)*log(sin(d*x + c) + 1) - 15*((2*A + 13*C)*cos(d*x + c)^5 + 3*(2*A + 13*C)*cos(d*x + c)^4
+ 3*(2*A + 13*C)*cos(d*x + c)^3 + (2*A + 13*C)*cos(d*x + c)^2)*log(-sin(d*x + c) + 1) - 2*(4*(11*A + 76*C)*co
s(d*x + c)^4 + 3*(34*A + 239*C)*cos(d*x + c)^3 + (64*A + 479*C)*cos(d*x + c)^2 + 45*C*cos(d*x + c) - 15*C)*sin
(d*x + c))/(a^3*d*cos(d*x + c)^5 + 3*a^3*d*cos(d*x + c)^4 + 3*a^3*d*cos(d*x + c)^3 + a^3*d*cos(d*x + c)^2)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{A \sec ^{4}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx + \int \frac{C \sec ^{6}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(A+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**3,x)

[Out]

(Integral(A*sec(c + d*x)**4/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x) + Integral(C*sec(c
+ d*x)**6/(sec(c + d*x)**3 + 3*sec(c + d*x)**2 + 3*sec(c + d*x) + 1), x))/a**3

________________________________________________________________________________________

Giac [A]  time = 1.23234, size = 279, normalized size = 1.41 \begin{align*} \frac{\frac{30 \,{\left (2 \, A + 13 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{3}} - \frac{30 \,{\left (2 \, A + 13 \, C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{3}} + \frac{60 \,{\left (7 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5 \, 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 )}^{2} a^{3}} - \frac{3 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 3 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 20 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 40 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 105 \, A a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 465 \, C a^{12} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{15}}}{60 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(A+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^3,x, algorithm="giac")

[Out]

1/60*(30*(2*A + 13*C)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^3 - 30*(2*A + 13*C)*log(abs(tan(1/2*d*x + 1/2*c) -
1))/a^3 + 60*(7*C*tan(1/2*d*x + 1/2*c)^3 - 5*C*tan(1/2*d*x + 1/2*c))/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^3) - (3
*A*a^12*tan(1/2*d*x + 1/2*c)^5 + 3*C*a^12*tan(1/2*d*x + 1/2*c)^5 + 20*A*a^12*tan(1/2*d*x + 1/2*c)^3 + 40*C*a^1
2*tan(1/2*d*x + 1/2*c)^3 + 105*A*a^12*tan(1/2*d*x + 1/2*c) + 465*C*a^12*tan(1/2*d*x + 1/2*c))/a^15)/d