3.2.17 \(\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+C \sec ^2(c+d x)) \, dx\) [117]

Optimal. Leaf size=200 \[ \frac {1}{8} a^4 (35 A+52 C) x+\frac {4 a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d} \]

[Out]

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

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Rubi [A]
time = 0.38, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {4172, 4102, 4103, 4081, 3855} \begin {gather*} \frac {5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}-\frac {(35 A-12 C) \sin (c+d x) \left (a^4 \sec (c+d x)+a^4\right )}{24 d}+\frac {1}{8} a^4 x (35 A+52 C)+\frac {4 a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {(7 A+4 C) \sin (c+d x) \cos (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{8 d}+\frac {A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}+\frac {a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3855

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

Rule 4081

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

Rule 4102

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

Rule 4103

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

Rule 4172

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

Rubi steps

\begin {align*} \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 (4 a A-a (A-4 C) \sec (c+d x)) \, dx}{4 a}\\ &=\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \left (3 a^2 (7 A+4 C)-a^2 (7 A-12 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x))^2 \left (2 a^3 (35 A+36 C)-a^3 (35 A-12 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\frac {\int \cos (c+d x) (a+a \sec (c+d x)) \left (15 a^4 (7 A+4 C)+96 a^4 C \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}-\frac {\int \left (-3 a^5 (35 A+52 C)-96 a^5 C \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {1}{8} a^4 (35 A+52 C) x+\frac {5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}+\left (4 a^4 C\right ) \int \sec (c+d x) \, dx\\ &=\frac {1}{8} a^4 (35 A+52 C) x+\frac {4 a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac {5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}+\frac {a A \cos ^2(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{3 d}+\frac {A \cos ^3(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{4 d}+\frac {(7 A+4 C) \cos (c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{8 d}-\frac {(35 A-12 C) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{24 d}\\ \end {align*}

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Mathematica [A]
time = 2.42, size = 375, normalized size = 1.88 \begin {gather*} \frac {a^4 \cos ^2(c+d x) (1+\cos (c+d x))^4 \sec ^8\left (\frac {1}{2} (c+d x)\right ) \left (A+C \sec ^2(c+d x)\right ) \left (12 (35 A+52 C) x-\frac {384 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {384 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{d}+\frac {96 (7 A+4 C) \cos (d x) \sin (c)}{d}+\frac {24 (7 A+C) \cos (2 d x) \sin (2 c)}{d}+\frac {32 A \cos (3 d x) \sin (3 c)}{d}+\frac {3 A \cos (4 d x) \sin (4 c)}{d}+\frac {96 (7 A+4 C) \cos (c) \sin (d x)}{d}+\frac {24 (7 A+C) \cos (2 c) \sin (2 d x)}{d}+\frac {32 A \cos (3 c) \sin (3 d x)}{d}+\frac {3 A \cos (4 c) \sin (4 d x)}{d}+\frac {96 C \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 C \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 )}\right )}{768 (A+2 C+A \cos (2 (c+d x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(A + C*Sec[c + d*x]^2)*(12*(35*A + 52*C)*x - (384*
C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (384*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (96*(7*A
+ 4*C)*Cos[d*x]*Sin[c])/d + (24*(7*A + C)*Cos[2*d*x]*Sin[2*c])/d + (32*A*Cos[3*d*x]*Sin[3*c])/d + (3*A*Cos[4*d
*x]*Sin[4*c])/d + (96*(7*A + 4*C)*Cos[c]*Sin[d*x])/d + (24*(7*A + C)*Cos[2*c]*Sin[2*d*x])/d + (32*A*Cos[3*c]*S
in[3*d*x])/d + (3*A*Cos[4*c]*Sin[4*d*x])/d + (96*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] -
Sin[(c + d*x)/2])) + (96*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(76
8*(A + 2*C + A*Cos[2*(c + d*x)]))

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Maple [A]
time = 0.67, size = 197, normalized size = 0.98

method	result	size
derivativedivides	\(\frac {A \,a^{4} \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{4} \sin \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \left (d x +c \right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \sin \left (d x +c \right )+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(197\)
default	\(\frac {A \,a^{4} \left (d x +c \right )+a^{4} C \tan \left (d x +c \right )+4 A \,a^{4} \sin \left (d x +c \right )+4 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 a^{4} C \left (d x +c \right )+\frac {4 A \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+4 a^{4} C \sin \left (d x +c \right )+A \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\)	\(197\)
risch	\(\frac {35 a^{4} A x}{8}+\frac {13 a^{4} x C}{2}-\frac {7 i A \,a^{4} {\mathrm e}^{2 i \left (d x +c \right )}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} a^{4} C}{8 d}-\frac {7 i A \,a^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{d}+\frac {7 i A \,a^{4} {\mathrm e}^{-i \left (d x +c \right )}}{2 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{d}+\frac {7 i A \,a^{4} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} a^{4} C}{8 d}+\frac {2 i a^{4} C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}+\frac {4 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}+\frac {A \,a^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {A \,a^{4} \sin \left (3 d x +3 c \right )}{3 d}\)	\(271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.29, size = 194, normalized size = 0.97 \begin {gather*} -\frac {128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A 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 )} A a^{4} - 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 96 \, {\left (d x + c\right )} A a^{4} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 576 \, {\left (d x + c\right )} C a^{4} - 192 \, C 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 \, C a^{4} \tan \left (d x + c\right )}{96 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.98, size = 158, normalized size = 0.79 \begin {gather*} \frac {3 \, {\left (35 \, A + 52 \, C\right )} a^{4} d x \cos \left (d x + c\right ) + 48 \, C a^{4} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 48 \, C a^{4} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, A a^{4} \cos \left (d x + c\right )^{4} + 32 \, A a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (27 \, A + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 32 \, {\left (5 \, A + 3 \, C\right )} a^{4} \cos \left (d x + c\right ) + 24 \, C a^{4}\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

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Giac [A]
time = 0.51, size = 244, normalized size = 1.22 \begin {gather*} \frac {96 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 96 \, C a^{4} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {48 \, C 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 (35 \, A a^{4} + 52 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 385 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 276 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 511 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 300 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 279 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 108 \, 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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(96*C*a^4*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 96*C*a^4*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 48*C*a^4*tan
(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) + 3*(35*A*a^4 + 52*C*a^4)*(d*x + c) + 2*(105*A*a^4*tan(1/2*d*x
+ 1/2*c)^7 + 84*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 385*A*a^4*tan(1/2*d*x + 1/2*c)^5 + 276*C*a^4*tan(1/2*d*x + 1/2*
c)^5 + 511*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 300*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 279*A*a^4*tan(1/2*d*x + 1/2*c) +
108*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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Mupad [B]
time = 2.82, size = 234, normalized size = 1.17 \begin {gather*} \frac {20\,A\,a^4\,\sin \left (c+d\,x\right )}{3\,d}+\frac {4\,C\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {35\,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 )}{4\,d}+\frac {13\,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 )}{d}+\frac {8\,C\,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 {4\,A\,a^4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{3\,d}+\frac {A\,a^4\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {C\,a^4\,\sin \left (c+d\,x\right )}{d\,\cos \left (c+d\,x\right )}+\frac {27\,A\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{8\,d}+\frac {C\,a^4\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{2\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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