3.555 \(\int (a+b \cos (c+d x))^4 (A+C \cos ^2(c+d x)) \sec ^5(c+d x) \, dx\)

Optimal. Leaf size=246 \[ -\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{12 d}+\frac {\left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{8 d}+\frac {\left (a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+4 a b^3 C x \]

[Out]

4*a*b^3*C*x+1/8*(8*A*b^4+24*a^2*b^2*(A+2*C)+a^4*(3*A+4*C))*arctanh(sin(d*x+c))/d-1/24*b^2*(2*b^2*(13*A-12*C)+3
*a^2*(3*A+4*C))*sin(d*x+c)/d+1/12*a*b*(12*A*b^2+a^2*(23*A+36*C))*tan(d*x+c)/d+1/8*(4*A*b^2+a^2*(3*A+4*C))*(a+b
*cos(d*x+c))^2*sec(d*x+c)*tan(d*x+c)/d+1/3*A*b*(a+b*cos(d*x+c))^3*sec(d*x+c)^2*tan(d*x+c)/d+1/4*A*(a+b*cos(d*x
+c))^4*sec(d*x+c)^3*tan(d*x+c)/d

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Rubi [A]  time = 0.93, antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3048, 3047, 3031, 3023, 2735, 3770} \[ -\frac {b^2 \left (3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (a^2 (23 A+36 C)+12 A b^2\right ) \tan (c+d x)}{12 d}+\frac {\left (24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\left (a^2 (3 A+4 C)+4 A b^2\right ) \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^2}{8 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+\frac {A b \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+4 a b^3 C x \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

4*a*b^3*C*x + ((8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*ArcTanh[Sin[c + d*x]])/(8*d) - (b^2*(2*b^2*(
13*A - 12*C) + 3*a^2*(3*A + 4*C))*Sin[c + d*x])/(24*d) + (a*b*(12*A*b^2 + a^2*(23*A + 36*C))*Tan[c + d*x])/(12
*d) + ((4*A*b^2 + a^2*(3*A + 4*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (A*b*(a + b*Cos[c
+ d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)

Rule 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 3023

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

Rule 3031

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

Rule 3047

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

Rule 3048

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[((c^2*C + A*d^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*(c + d*Sin[
e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 - d^2)), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m
- 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + c*C*(b*c*m + a*d*(n + 1)) - (A*d*(a*d*(n +
2) - b*c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*(A*d^2*(m + n + 2) + C*(c^2*(
m + 1) + d^2*(n + 1)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C}, x] && NeQ[b*c - a*d, 0] &
& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]

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 (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx &=\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^3 \left (4 A b+a (3 A+4 C) \cos (c+d x)-b (A-4 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cos (c+d x))^2 \left (3 \left (4 A b^2+a^2 (3 A+4 C)\right )+2 a b (7 A+12 C) \cos (c+d x)-b^2 (7 A-12 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{24} \int (a+b \cos (c+d x)) \left (2 \left (12 A b^3+\frac {1}{2} a^2 (46 A b+72 b C)\right )+a \left (3 a^2 (3 A+4 C)+2 b^2 (13 A+36 C)\right ) \cos (c+d x)-b \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )-96 a b^3 C \cos (c+d x)+b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )-96 a b^3 C \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=4 a b^3 C x-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (-8 A b^4-24 a^2 b^2 (A+2 C)-a^4 (3 A+4 C)\right ) \int \sec (c+d x) \, dx\\ &=4 a b^3 C x+\frac {\left (8 A b^4+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac {b^2 \left (2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a b \left (12 A b^2+a^2 (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {A b (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end {align*}

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Mathematica [B]  time = 6.34, size = 612, normalized size = 2.49 \[ \frac {a^4 A}{16 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}-\frac {a^4 A}{16 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {4 \left (2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )+3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {4 \left (2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )+3 a^3 b C \sin \left (\frac {1}{2} (c+d x)\right )+3 a A b^3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 a^3 A b \sin \left (\frac {1}{2} (c+d x)\right )}{3 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {\left (-3 a^4 A-4 a^4 C-24 a^2 A b^2-48 a^2 b^2 C-8 A b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {\left (3 a^4 A+4 a^4 C+24 a^2 A b^2+48 a^2 b^2 C+8 A b^4\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}+\frac {9 a^4 A+12 a^4 C+16 a^3 A b+72 a^2 A b^2}{48 d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {-9 a^4 A-12 a^4 C-16 a^3 A b-72 a^2 A b^2}{48 d \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 a b^3 C (c+d x)}{d}+\frac {b^4 C \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Cos[c + d*x])^4*(A + C*Cos[c + d*x]^2)*Sec[c + d*x]^5,x]

[Out]

(4*a*b^3*C*(c + d*x))/d + ((-3*a^4*A - 24*a^2*A*b^2 - 8*A*b^4 - 4*a^4*C - 48*a^2*b^2*C)*Log[Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]])/(8*d) + ((3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 4*a^4*C + 48*a^2*b^2*C)*Log[Cos[(c + d*x)/2]
+ Sin[(c + d*x)/2]])/(8*d) + (a^4*A)/(16*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + (9*a^4*A + 16*a^3*A*b +
72*a^2*A*b^2 + 12*a^4*C)/(48*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*a^3*A*b*Sin[(c + d*x)/2])/(3*d*(C
os[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (a^4*A)/(16*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (2*a^3*A*b*S
in[(c + d*x)/2])/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-9*a^4*A - 16*a^3*A*b - 72*a^2*A*b^2 - 12*a^
4*C)/(48*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(2*a^3*A*b*Sin[(c + d*x)/2] + 3*a*A*b^3*Sin[(c + d*x)
/2] + 3*a^3*b*C*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*(2*a^3*A*b*Sin[(c + d*x)/2
] + 3*a*A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2]))/(3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (b
^4*C*Sin[c + d*x])/d

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fricas [A]  time = 0.81, size = 236, normalized size = 0.96 \[ \frac {192 \, C a b^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C b^{4} \cos \left (d x + c\right )^{4} + 32 \, A a^{3} b \cos \left (d x + c\right ) + 6 \, A a^{4} + 32 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{3} b + 3 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="fricas")

[Out]

1/48*(192*C*a*b^3*d*x*cos(d*x + c)^4 + 3*((3*A + 4*C)*a^4 + 24*(A + 2*C)*a^2*b^2 + 8*A*b^4)*cos(d*x + c)^4*log
(sin(d*x + c) + 1) - 3*((3*A + 4*C)*a^4 + 24*(A + 2*C)*a^2*b^2 + 8*A*b^4)*cos(d*x + c)^4*log(-sin(d*x + c) + 1
) + 2*(24*C*b^4*cos(d*x + c)^4 + 32*A*a^3*b*cos(d*x + c) + 6*A*a^4 + 32*((2*A + 3*C)*a^3*b + 3*A*a*b^3)*cos(d*
x + c)^3 + 3*((3*A + 4*C)*a^4 + 24*A*a^2*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^4)

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giac [B]  time = 0.60, size = 590, normalized size = 2.40 \[ \frac {96 \, {\left (d x + c\right )} C a b^{3} + \frac {48 \, C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a^{4} + 4 \, C a^{4} + 24 \, A a^{2} b^{2} + 48 \, C a^{2} b^{2} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, C a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a b^{3} \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 )}^{4}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="giac")

[Out]

1/24*(96*(d*x + c)*C*a*b^3 + 48*C*b^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 3*(3*A*a^4 + 4*C*a^4
+ 24*A*a^2*b^2 + 48*C*a^2*b^2 + 8*A*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(3*A*a^4 + 4*C*a^4 + 24*A*a^2
*b^2 + 48*C*a^2*b^2 + 8*A*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(15*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 12*C*
a^4*tan(1/2*d*x + 1/2*c)^7 - 96*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 96*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 72*A*a^2*
b^2*tan(1/2*d*x + 1/2*c)^7 - 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 9*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 12*C*a^4*tan
(1/2*d*x + 1/2*c)^5 + 160*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 72*A*a^2*b^2*t
an(1/2*d*x + 1/2*c)^5 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 12*C*a^4*tan(1/2
*d*x + 1/2*c)^3 - 160*A*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^3 - 72*A*a^2*b^2*tan(1
/2*d*x + 1/2*c)^3 - 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 15*A*a^4*tan(1/2*d*x + 1/2*c) + 12*C*a^4*tan(1/2*d*x
+ 1/2*c) + 96*A*a^3*b*tan(1/2*d*x + 1/2*c) + 96*C*a^3*b*tan(1/2*d*x + 1/2*c) + 72*A*a^2*b^2*tan(1/2*d*x + 1/2*
c) + 96*A*a*b^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 - 1)^4)/d

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maple [A]  time = 0.40, size = 316, normalized size = 1.28 \[ \frac {A \,a^{4} \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 A \,a^{4} \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a^{4} C \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {8 A \,a^{3} b \tan \left (d x +c \right )}{3 d}+\frac {4 A \,a^{3} b \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right )}{3 d}+\frac {4 a^{3} b C \tan \left (d x +c \right )}{d}+\frac {3 A \,a^{2} b^{2} \tan \left (d x +c \right ) \sec \left (d x +c \right )}{d}+\frac {3 A \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {4 a A \,b^{3} \tan \left (d x +c \right )}{d}+4 a \,b^{3} C x +\frac {4 C a \,b^{3} c}{d}+\frac {A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {C \,b^{4} \sin \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x)

[Out]

1/4/d*A*a^4*tan(d*x+c)*sec(d*x+c)^3+3/8/d*A*a^4*sec(d*x+c)*tan(d*x+c)+3/8/d*A*a^4*ln(sec(d*x+c)+tan(d*x+c))+1/
2/d*a^4*C*sec(d*x+c)*tan(d*x+c)+1/2/d*a^4*C*ln(sec(d*x+c)+tan(d*x+c))+8/3/d*A*a^3*b*tan(d*x+c)+4/3/d*A*a^3*b*t
an(d*x+c)*sec(d*x+c)^2+4/d*a^3*b*C*tan(d*x+c)+3/d*A*a^2*b^2*tan(d*x+c)*sec(d*x+c)+3/d*A*a^2*b^2*ln(sec(d*x+c)+
tan(d*x+c))+6/d*C*a^2*b^2*ln(sec(d*x+c)+tan(d*x+c))+4/d*a*A*b^3*tan(d*x+c)+4*a*b^3*C*x+4/d*C*a*b^3*c+1/d*A*b^4
*ln(sec(d*x+c)+tan(d*x+c))+1/d*C*b^4*sin(d*x+c)

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maxima [A]  time = 0.40, size = 306, normalized size = 1.24 \[ \frac {64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 192 \, {\left (d x + c\right )} C a b^{3} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C b^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))^4*(A+C*cos(d*x+c)^2)*sec(d*x+c)^5,x, algorithm="maxima")

[Out]

1/48*(64*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^3*b + 192*(d*x + c)*C*a*b^3 - 3*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)) - 1
2*C*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)) - 72*A*a^2*b^2*(
2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 144*C*a^2*b^2*(log(sin(
d*x + c) + 1) - log(sin(d*x + c) - 1)) + 24*A*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 48*C*b^4*s
in(d*x + c) + 192*C*a^3*b*tan(d*x + c) + 192*A*a*b^3*tan(d*x + c))/d

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mupad [B]  time = 3.90, size = 1988, normalized size = 8.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^4)/cos(c + d*x)^5,x)

[Out]

((27*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/8 + 9*A*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/
2)) + (9*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + (9*A*a^4*sin(3*c + 3*d*x))/8 + (3*C*a^4*sin(3
*c + 3*d*x))/2 + (9*C*b^4*sin(3*c + 3*d*x))/4 + (3*C*b^4*sin(5*c + 5*d*x))/4 + (33*A*a^4*sin(c + d*x))/8 + (3*
C*a^4*sin(c + d*x))/2 + (3*C*b^4*sin(c + d*x))/2 + 12*A*a*b^3*sin(2*c + 2*d*x) + 16*A*a^3*b*sin(2*c + 2*d*x) +
6*A*a*b^3*sin(4*c + 4*d*x) + 4*A*a^3*b*sin(4*c + 4*d*x) + 9*A*a^2*b^2*sin(c + d*x) + 12*C*a^3*b*sin(2*c + 2*d
*x) + 6*C*a^3*b*sin(4*c + 4*d*x) + 36*C*a*b^3*atan((9*A^2*a^8*sin(c/2 + (d*x)/2) + 64*A^2*b^8*sin(c/2 + (d*x)/
2) + 16*C^2*a^8*sin(c/2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*A^2*a^4*b^4*sin(c/2 + (d*x)/2) +
144*A^2*a^6*b^2*sin(c/2 + (d*x)/2) + 1024*C^2*a^2*b^6*sin(c/2 + (d*x)/2) + 2304*C^2*a^4*b^4*sin(c/2 + (d*x)/2
) + 384*C^2*a^6*b^2*sin(c/2 + (d*x)/2) + 24*A*C*a^8*sin(c/2 + (d*x)/2) + 768*A*C*a^2*b^6*sin(c/2 + (d*x)/2) +
2368*A*C*a^4*b^4*sin(c/2 + (d*x)/2) + 480*A*C*a^6*b^2*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(9*A^2*a^8 + 64*
A^2*b^8 + 16*C^2*a^8 + 384*A^2*a^2*b^6 + 624*A^2*a^4*b^4 + 144*A^2*a^6*b^2 + 1024*C^2*a^2*b^6 + 2304*C^2*a^4*b
^4 + 384*C^2*a^6*b^2 + 24*A*C*a^8 + 768*A*C*a^2*b^6 + 2368*A*C*a^4*b^4 + 480*A*C*a^6*b^2))) + (9*A*a^4*atanh(s
in(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x))/2 + (9*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/
2))*cos(4*c + 4*d*x))/8 + 27*A*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + 12*A*b^4*atanh(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x) + 3*A*b^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c
+ 4*d*x) + 6*C*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x) + (3*C*a^4*atanh(sin(c/2 + (d
*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d*x))/2 + 54*C*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)) + 9
*A*a^2*b^2*sin(3*c + 3*d*x) + 48*C*a*b^3*cos(2*c + 2*d*x)*atan((9*A^2*a^8*sin(c/2 + (d*x)/2) + 64*A^2*b^8*sin(
c/2 + (d*x)/2) + 16*C^2*a^8*sin(c/2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*A^2*a^4*b^4*sin(c/2
+ (d*x)/2) + 144*A^2*a^6*b^2*sin(c/2 + (d*x)/2) + 1024*C^2*a^2*b^6*sin(c/2 + (d*x)/2) + 2304*C^2*a^4*b^4*sin(c
/2 + (d*x)/2) + 384*C^2*a^6*b^2*sin(c/2 + (d*x)/2) + 24*A*C*a^8*sin(c/2 + (d*x)/2) + 768*A*C*a^2*b^6*sin(c/2 +
(d*x)/2) + 2368*A*C*a^4*b^4*sin(c/2 + (d*x)/2) + 480*A*C*a^6*b^2*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(9*A
^2*a^8 + 64*A^2*b^8 + 16*C^2*a^8 + 384*A^2*a^2*b^6 + 624*A^2*a^4*b^4 + 144*A^2*a^6*b^2 + 1024*C^2*a^2*b^6 + 23
04*C^2*a^4*b^4 + 384*C^2*a^6*b^2 + 24*A*C*a^8 + 768*A*C*a^2*b^6 + 2368*A*C*a^4*b^4 + 480*A*C*a^6*b^2))) + 12*C
*a*b^3*cos(4*c + 4*d*x)*atan((9*A^2*a^8*sin(c/2 + (d*x)/2) + 64*A^2*b^8*sin(c/2 + (d*x)/2) + 16*C^2*a^8*sin(c/
2 + (d*x)/2) + 384*A^2*a^2*b^6*sin(c/2 + (d*x)/2) + 624*A^2*a^4*b^4*sin(c/2 + (d*x)/2) + 144*A^2*a^6*b^2*sin(c
/2 + (d*x)/2) + 1024*C^2*a^2*b^6*sin(c/2 + (d*x)/2) + 2304*C^2*a^4*b^4*sin(c/2 + (d*x)/2) + 384*C^2*a^6*b^2*si
n(c/2 + (d*x)/2) + 24*A*C*a^8*sin(c/2 + (d*x)/2) + 768*A*C*a^2*b^6*sin(c/2 + (d*x)/2) + 2368*A*C*a^4*b^4*sin(c
/2 + (d*x)/2) + 480*A*C*a^6*b^2*sin(c/2 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(9*A^2*a^8 + 64*A^2*b^8 + 16*C^2*a^8 +
384*A^2*a^2*b^6 + 624*A^2*a^4*b^4 + 144*A^2*a^6*b^2 + 1024*C^2*a^2*b^6 + 2304*C^2*a^4*b^4 + 384*C^2*a^6*b^2 +
24*A*C*a^8 + 768*A*C*a^2*b^6 + 2368*A*C*a^4*b^4 + 480*A*C*a^6*b^2))) + 36*A*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/
cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x) + 9*A*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d*
x) + 72*C*a^2*b^2*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2))*cos(2*c + 2*d*x) + 18*C*a^2*b^2*atanh(sin(c/2 +
(d*x)/2)/cos(c/2 + (d*x)/2))*cos(4*c + 4*d*x))/(12*d*(cos(2*c + 2*d*x)/2 + cos(4*c + 4*d*x)/8 + 3/8))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*cos(d*x+c))**4*(A+C*cos(d*x+c)**2)*sec(d*x+c)**5,x)

[Out]

Timed out

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