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

Optimal. Leaf size=181 \[ \frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {(12 A-35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {4 a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{8} a^4 x (52 A+35 C)-\frac {(12 A-7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}-\frac {a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d} \]

[Out]

1/8*a^4*(52*A+35*C)*x+4*a^4*A*arctanh(sin(d*x+c))/d+5/8*a^4*(4*A+7*C)*sin(d*x+c)/d-1/4*a*(4*A-C)*(a+a*cos(d*x+
c))^3*sin(d*x+c)/d-1/12*(12*A-7*C)*(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d-1/24*(12*A-35*C)*(a^4+a^4*cos(d*x+c))*s
in(d*x+c)/d+A*(a+a*cos(d*x+c))^4*tan(d*x+c)/d

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Rubi [A]  time = 0.60, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3044, 2976, 2968, 3023, 2735, 3770} \[ \frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {(12 A-7 C) \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{12 d}-\frac {(12 A-35 C) \sin (c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{24 d}+\frac {4 a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {1}{8} a^4 x (52 A+35 C)-\frac {a (4 A-C) \sin (c+d x) (a \cos (c+d x)+a)^3}{4 d}+\frac {A \tan (c+d x) (a \cos (c+d x)+a)^4}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(52*A + 35*C)*x)/8 + (4*a^4*A*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(4*A + 7*C)*Sin[c + d*x])/(8*d) - (a*(4*A
- C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) - ((12*A - 7*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(12*
d) - ((12*A - 35*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(24*d) + (A*(a + a*Cos[c + d*x])^4*Tan[c + d*x])/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 2968

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

Rule 2976

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])
^(n + 1))/(d*f*(m + n + 1)), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x]
)^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*S
in[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&
NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 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 3044

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

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+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx &=\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^4 (4 a A-a (4 A-C) \cos (c+d x)) \sec (c+d x) \, dx}{a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^3 \left (16 a^2 A-a^2 (12 A-7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{4 a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x))^2 \left (48 a^3 A-a^3 (12 A-35 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{12 a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int (a+a \cos (c+d x)) \left (96 a^4 A+15 a^4 (4 A+7 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int \left (96 a^5 A+\left (96 a^5 A+15 a^5 (4 A+7 C)\right ) \cos (c+d x)+15 a^5 (4 A+7 C) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\frac {\int \left (96 a^5 A+3 a^5 (52 A+35 C) \cos (c+d x)\right ) \sec (c+d x) \, dx}{24 a}\\ &=\frac {1}{8} a^4 (52 A+35 C) x+\frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}+\left (4 a^4 A\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (52 A+35 C) x+\frac {4 a^4 A \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (4 A+7 C) \sin (c+d x)}{8 d}-\frac {a (4 A-C) (a+a \cos (c+d x))^3 \sin (c+d x)}{4 d}-\frac {(12 A-7 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{12 d}-\frac {(12 A-35 C) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {A (a+a \cos (c+d x))^4 \tan (c+d x)}{d}\\ \end {align*}

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Mathematica [A]  time = 2.17, size = 338, normalized size = 1.87 \[ \frac {a^4 (\cos (c+d x)+1)^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (\frac {96 (4 A+7 C) \sin (c) \cos (d x)}{d}+\frac {24 (A+7 C) \sin (2 c) \cos (2 d x)}{d}+\frac {96 (4 A+7 C) \cos (c) \sin (d x)}{d}+\frac {24 (A+7 C) \cos (2 c) \sin (2 d x)}{d}+\frac {96 A \sin \left (\frac {d x}{2}\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {96 A \sin \left (\frac {d x}{2}\right )}{d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {384 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {384 A \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{d}+12 x (52 A+35 C)+\frac {32 C \sin (3 c) \cos (3 d x)}{d}+\frac {3 C \sin (4 c) \cos (4 d x)}{d}+\frac {32 C \cos (3 c) \sin (3 d x)}{d}+\frac {3 C \cos (4 c) \sin (4 d x)}{d}\right )}{1536} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(12*(52*A + 35*C)*x - (384*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)
/2]])/d + (384*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (96*(4*A + 7*C)*Cos[d*x]*Sin[c])/d + (24*(A + 7
*C)*Cos[2*d*x]*Sin[2*c])/d + (32*C*Cos[3*d*x]*Sin[3*c])/d + (3*C*Cos[4*d*x]*Sin[4*c])/d + (96*(4*A + 7*C)*Cos[
c]*Sin[d*x])/d + (24*(A + 7*C)*Cos[2*c]*Sin[2*d*x])/d + (32*C*Cos[3*c]*Sin[3*d*x])/d + (3*C*Cos[4*c]*Sin[4*d*x
])/d + (96*A*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (96*A*Sin[(d*x)/2
])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/1536

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fricas [A]  time = 0.69, size = 158, normalized size = 0.87 \[ \frac {3 \, {\left (52 \, A + 35 \, C\right )} a^{4} d x \cos \left (d x + c\right ) + 48 \, A a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, A a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C a^{4} \cos \left (d x + c\right )^{4} + 32 \, C a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (4 \, A + 27 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, {\left (3 \, A + 5 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, A a^{4}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(3*(52*A + 35*C)*a^4*d*x*cos(d*x + c) + 48*A*a^4*cos(d*x + c)*log(sin(d*x + c) + 1) - 48*A*a^4*cos(d*x +
c)*log(-sin(d*x + c) + 1) + (6*C*a^4*cos(d*x + c)^4 + 32*C*a^4*cos(d*x + c)^3 + 3*(4*A + 27*C)*a^4*cos(d*x + c
)^2 + 32*(3*A + 5*C)*a^4*cos(d*x + c) + 24*A*a^4)*sin(d*x + c))/(d*cos(d*x + c))

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giac [A]  time = 0.57, size = 244, normalized size = 1.35 \[ \frac {96 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 96 \, A a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {48 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} + 3 \, {\left (52 \, A a^{4} + 35 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (84 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 105 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 276 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 385 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 300 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 511 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 108 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 279 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{24 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(96*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 96*A*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 48*A*a^4*tan
(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) + 3*(52*A*a^4 + 35*C*a^4)*(d*x + c) + 2*(84*A*a^4*tan(1/2*d*x +
1/2*c)^7 + 105*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 276*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 385*C*a^4*tan(1/2*d*x + 1/2*
c)^5 + 300*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 511*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 108*A*a^4*tan(1/2*d*x + 1/2*c) +
279*C*a^4*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 = 0.39, size = 191, normalized size = 1.06 \[ \frac {A \,a^{4} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{2 d}+\frac {13 A \,a^{4} x}{2}+\frac {13 A \,a^{4} c}{2 d}+\frac {a^{4} C \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4 d}+\frac {27 a^{4} C \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {35 a^{4} C x}{8}+\frac {35 a^{4} C c}{8 d}+\frac {4 A \,a^{4} \sin \left (d x +c \right )}{d}+\frac {4 a^{4} C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )}{3 d}+\frac {20 a^{4} C \sin \left (d x +c \right )}{3 d}+\frac {4 A \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,a^{4} \tan \left (d x +c \right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/d*A*a^4*cos(d*x+c)*sin(d*x+c)+13/2*A*a^4*x+13/2/d*A*a^4*c+1/4/d*a^4*C*sin(d*x+c)*cos(d*x+c)^3+27/8/d*a^4*C
*cos(d*x+c)*sin(d*x+c)+35/8*a^4*C*x+35/8/d*a^4*C*c+4/d*A*a^4*sin(d*x+c)+4/3/d*a^4*C*sin(d*x+c)*cos(d*x+c)^2+20
/3/d*a^4*C*sin(d*x+c)+4/d*A*a^4*ln(sec(d*x+c)+tan(d*x+c))+1/d*A*a^4*tan(d*x+c)

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maxima [A]  time = 0.34, size = 194, normalized size = 1.07 \[ \frac {24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 576 \, {\left (d x + c\right )} A a^{4} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 96 \, {\left (d x + c\right )} C a^{4} + 192 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, A a^{4} \sin \left (d x + c\right ) + 384 \, C a^{4} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(24*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 + 576*(d*x + c)*A*a^4 - 128*(sin(d*x + c)^3 - 3*sin(d*x + c))*
C*a^4 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4 + 144*(2*d*x + 2*c + sin(2*d*x + 2*c))
*C*a^4 + 96*(d*x + c)*C*a^4 + 192*A*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 384*A*a^4*sin(d*x +
c) + 384*C*a^4*sin(d*x + c) + 96*A*a^4*tan(d*x + c))/d

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mupad [B]  time = 1.09, size = 234, normalized size = 1.29 \[ \frac {4\,A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {20\,C\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {13\,A\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {8\,A\,a^4\,\mathrm {atanh}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {35\,C\,a^4\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{4\,d}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {4\,C\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {C\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d}+\frac {27\,C\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*A*a^4*sin(c + d*x))/d + (20*C*a^4*sin(c + d*x))/(3*d) + (13*A*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2
)))/d + (8*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/d + (35*C*a^4*atan(sin(c/2 + (d*x)/2)/cos(c/2 +
(d*x)/2)))/(4*d) + (A*a^4*sin(c + d*x))/(d*cos(c + d*x)) + (4*C*a^4*cos(c + d*x)^2*sin(c + d*x))/(3*d) + (C*a
^4*cos(c + d*x)^3*sin(c + d*x))/(4*d) + (A*a^4*cos(c + d*x)*sin(c + d*x))/(2*d) + (27*C*a^4*cos(c + d*x)*sin(c
+ d*x))/(8*d)

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

[Out]

Timed out

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