3.518 \(\int \frac{\cos ^4(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx\)
Optimal. Leaf size=259 \[ -\frac{(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{(107 A-72 B+112 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{64 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{(43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-8 B) \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}} \]
[Out]
((107*A - 72*B + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A
- B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((21*A - 56*B + 16*C
)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + ((43*A - 8*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a
+ a*Sec[c + d*x]]) - ((A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x
]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])
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Rubi [A] time = 0.776892, antiderivative size = 259, normalized size of antiderivative = 1.,
number of steps used = 9, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used =
{4086, 4022, 3920, 3774, 203, 3795} \[ -\frac{(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{(107 A-72 B+112 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{64 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{(43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-8 B) \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
Int[(Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]
[Out]
((107*A - 72*B + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A
- B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((21*A - 56*B + 16*C
)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + ((43*A - 8*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a
+ a*Sec[c + d*x]]) - ((A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x
]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])
Rule 4086
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + 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] - Dist[1/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m -
b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 -
b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Rule 4022
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*
(B_.) + (A_)), x_Symbol] :> Simp[(A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^n)/(f*n), x] - Dist[1
/(b*d*n), Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - A*b*(m + n + 1)*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] && LtQ[n, 0]
Rule 3920
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c/a,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Rule 3774
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(a + x^2), x], x, (b*C
ot[c + d*x])/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, 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])
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
]
Rubi steps
\begin{align*} \int \frac{\cos ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx &=\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{\int \frac{\cos ^3(c+d x) \left (-\frac{1}{2} a (A-8 B)+\frac{1}{2} a (7 A+8 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{4 a}\\ &=-\frac{(A-8 B) \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{\int \frac{\cos ^2(c+d x) \left (\frac{1}{4} a^2 (43 A-8 B+48 C)-\frac{5}{4} a^2 (A-8 B) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{12 a^2}\\ &=\frac{(43 A-8 B+48 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}-\frac{(A-8 B) \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{\int \frac{\cos (c+d x) \left (-\frac{3}{8} a^3 (21 A-56 B+16 C)+\frac{3}{8} a^3 (43 A-8 B+48 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{24 a^3}\\ &=-\frac{(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{(43 A-8 B+48 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}-\frac{(A-8 B) \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{\int \frac{\frac{3}{16} a^4 (107 A-72 B+112 C)-\frac{3}{16} a^4 (21 A-56 B+16 C) \sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx}{24 a^4}\\ &=-\frac{(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{(43 A-8 B+48 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}-\frac{(A-8 B) \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+(-A+B-C) \int \frac{\sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx+\frac{(107 A-72 B+112 C) \int \sqrt{a+a \sec (c+d x)} \, dx}{128 a}\\ &=-\frac{(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{(43 A-8 B+48 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}-\frac{(A-8 B) \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}+\frac{(2 (A-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{(107 A-72 B+112 C) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{64 d}\\ &=\frac{(107 A-72 B+112 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{64 \sqrt{a} d}-\frac{\sqrt{2} (A-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{(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt{a+a \sec (c+d x)}}+\frac{(43 A-8 B+48 C) \cos (c+d x) \sin (c+d x)}{96 d \sqrt{a+a \sec (c+d x)}}-\frac{(A-8 B) \cos ^2(c+d x) \sin (c+d x)}{24 d \sqrt{a+a \sec (c+d x)}}+\frac{A \cos ^3(c+d x) \sin (c+d x)}{4 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.969472, size = 178, normalized size = 0.69 \[ \frac{\tan (c+d x) \left (\cos (c+d x) \sqrt{1-\sec (c+d x)} \left (2 (43 A-8 B+48 C) \cos (c+d x)-8 (A-8 B) \cos ^2(c+d x)+48 A \cos ^3(c+d x)-63 A+168 B-48 C\right )+3 (107 A-72 B+112 C) \tanh ^{-1}\left (\sqrt{1-\sec (c+d x)}\right )-192 \sqrt{2} (A-B+C) \tanh ^{-1}\left (\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right )\right )}{192 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
[In]
Integrate[(Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]
[Out]
((3*(107*A - 72*B + 112*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 192*Sqrt[2]*(A - B + C)*ArcTanh[Sqrt[1 - Sec[c +
d*x]]/Sqrt[2]] + Cos[c + d*x]*(-63*A + 168*B - 48*C + 2*(43*A - 8*B + 48*C)*Cos[c + d*x] - 8*(A - 8*B)*Cos[c +
d*x]^2 + 48*A*Cos[c + d*x]^3)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(192*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 +
Sec[c + d*x])])
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Maple [B] time = 0.413, size = 2086, normalized size = 8.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x)
[Out]
1/3072/d/a*(321*A*sin(d*x+c)*cos(d*x+c)^3*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/
cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)-216*B*sin(d*x+c)*cos(d*x+c)^3*arctanh(1/2*2^(1/2)*(-2
*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)+336*C*si
n(d*x+c)*cos(d*x+c)^3*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(
d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)+321*A*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)
/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)*sin(d*x+c)-216*B*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/
(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)*sin(d*x+c)+384*A*(-2
*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*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)+384*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*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)-768*C*cos(d*x+c)^4+1008*C*sin(d*x+c)*cos(d*x+c)^2*arctanh(1/2*2^(1/2)*(
-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)+1008*C
*sin(d*x+c)*cos(d*x+c)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)-1152*B*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))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*cos(d*x+c)^2*sin(d*x+c)-1152*B*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))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*cos(d*x+c)*sin(d*x+c
)-384*B*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))*(-2*cos(d*x+c)/(cos(d*
x+c)+1))^(7/2)*cos(d*x+c)^3*sin(d*x+c)-1008*A*cos(d*x+c)^4-1504*A*cos(d*x+c)^6+2384*A*cos(d*x+c)^5-2944*B*cos(
d*x+c)^5+2304*C*cos(d*x+c)^5+2688*B*cos(d*x+c)^4-768*A*cos(d*x+c)^8+896*A*cos(d*x+c)^7-1024*B*cos(d*x+c)^7+128
0*B*cos(d*x+c)^6-384*B*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))*(-2*cos
(d*x+c)/(cos(d*x+c)+1))^(7/2)*sin(d*x+c)+963*A*sin(d*x+c)*cos(d*x+c)^2*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos
(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)-648*B*sin(d*x+c)*cos(d*x
+c)^2*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+
c)+1))^(7/2)*2^(1/2)+963*A*sin(d*x+c)*cos(d*x+c)*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(
d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)-648*B*sin(d*x+c)*cos(d*x+c)*arctanh(1/2*2^(1/2
)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*2^(1/2)+336
*C*arctanh(1/2*2^(1/2)*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)/cos(d*x+c))*2^(1/2)*(-2*cos(d*x+c)/(cos
(d*x+c)+1))^(7/2)*sin(d*x+c)-1536*C*cos(d*x+c)^6+384*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*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)*cos(d*x+c)^3+384*C*(-2*cos(d*x+c)/(co
s(d*x+c)+1))^(7/2)*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)*
cos(d*x+c)^3+1152*A*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*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)*cos(d*x+c)^2+1152*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*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)*cos(d*x+c)^2+1152*A*(-2*cos(d*x+c)
/(cos(d*x+c)+1))^(7/2)*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)*cos(d*x+c)+1152*C*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(7/2)*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)*cos(d*x+c))*(a*(cos(d*x+c)+1)/cos(d*x+c))^(1/2)/sin(d*x+c)/cos(d*x+c)^
3
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{4}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*cos(d*x + c)^4/sqrt(a*sec(d*x + c) + a), x)
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Fricas [A] time = 64.5368, size = 1639, normalized size = 6.33 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
[1/384*(192*sqrt(2)*((A - B + C)*a*cos(d*x + c) + (A - B + C)*a)*sqrt(-1/a)*log((2*sqrt(2)*sqrt((a*cos(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)) - 3*((107*A - 72*B + 112*C)*cos(d*x + c) + 107*A - 72*B + 112*C)*sqrt(-a)*log((2*a
*cos(d*x + c)^2 + 2*sqrt(-a)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) + a*cos(d*x + c
) - a)/(cos(d*x + c) + 1)) + 2*(48*A*cos(d*x + c)^4 - 8*(A - 8*B)*cos(d*x + c)^3 + 2*(43*A - 8*B + 48*C)*cos(d
*x + c)^2 - 3*(21*A - 56*B + 16*C)*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(a*d*co
s(d*x + c) + a*d), -1/192*(3*((107*A - 72*B + 112*C)*cos(d*x + c) + 107*A - 72*B + 112*C)*sqrt(a)*arctan(sqrt(
(a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - (48*A*cos(d*x + c)^4 - 8*(A - 8*B)*c
os(d*x + c)^3 + 2*(43*A - 8*B + 48*C)*cos(d*x + c)^2 - 3*(21*A - 56*B + 16*C)*cos(d*x + c))*sqrt((a*cos(d*x +
c) + a)/cos(d*x + c))*sin(d*x + c) - 192*sqrt(2)*((A - B + C)*a*cos(d*x + c) + (A - B + C)*a)*arctan(sqrt(2)*s
qrt((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) + a*d)]
________________________________________________________________________________________
Sympy [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(cos(d*x+c)**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**(1/2),x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 12.3017, size = 1887, normalized size = 7.29 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(1/2),x, algorithm="giac")
[Out]
-1/384*(192*sqrt(2)*(A - B + C)*log((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2)/(
sqrt(-a)*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)) + 3*(107*A - 72*B + 112*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))^2 - a*(2*sqrt(2) + 3)))/(sqrt(-a)*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)) - 3*(1
07*A - 72*B + 112*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))^2 + a*(2*sq
rt(2) - 3)))/(sqrt(-a)*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)) + 4*sqrt(2)*(1599*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqr
t(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*A*sqrt(-a) - 1320*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x +
1/2*c)^2 + a))^14*B*sqrt(-a) + 816*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*C*
sqrt(-a) - 18219*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^12*A*sqrt(-a)*a + 18504
*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^12*B*sqrt(-a)*a - 12528*(sqrt(-a)*tan(1
/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^12*C*sqrt(-a)*a + 91467*(sqrt(-a)*tan(1/2*d*x + 1/2*c)
- sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*A*sqrt(-a)*a^2 - 96072*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(
1/2*d*x + 1/2*c)^2 + a))^10*B*sqrt(-a)*a^2 + 64752*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*
c)^2 + a))^10*C*sqrt(-a)*a^2 - 177735*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*
A*sqrt(-a)*a^3 + 215016*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*B*sqrt(-a)*a^3
- 124848*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^8*C*sqrt(-a)*a^3 + 100413*(sqr
t(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*A*sqrt(-a)*a^4 - 136056*(sqrt(-a)*tan(1/2*
d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*B*sqrt(-a)*a^4 + 70032*(sqrt(-a)*tan(1/2*d*x + 1/2*c) -
sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^6*C*sqrt(-a)*a^4 - 26881*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2
*d*x + 1/2*c)^2 + a))^4*A*sqrt(-a)*a^5 + 36056*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2
+ a))^4*B*sqrt(-a)*a^5 - 19152*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4*C*sqrt
(-a)*a^5 + 3321*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*A*sqrt(-a)*a^6 - 4632*
(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*B*sqrt(-a)*a^6 + 2640*(sqrt(-a)*tan(1/
2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*C*sqrt(-a)*a^6 - 205*A*sqrt(-a)*a^7 + 248*B*sqrt(-a)*a
^7 - 144*C*sqrt(-a)*a^7)/(((sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^4 - 6*(sqrt(-
a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^2*a + a^2)^4*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)))/
d