3.527 \(\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x)+C \sec ^2(c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx\)

Optimal. Leaf size=277 \[ \frac{(45 A-85 B+157 C) \tan (c+d x) \sec ^2(c+d x)}{80 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(75 A-163 B+283 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(195 A-475 B+787 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{240 a^3 d}+\frac{(465 A-985 B+1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(5 A-13 B+21 C) \tan (c+d x) \sec ^3(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}} \]

[Out]

-((75*A - 163*B + 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2
)*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A - 13*B + 21*C)*Sec[c
+ d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((465*A - 985*B + 1729*C)*Tan[c + d*x])/(120*a^2
*d*Sqrt[a + a*Sec[c + d*x]]) + ((45*A - 85*B + 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c
+ d*x]]) - ((195*A - 475*B + 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)

________________________________________________________________________________________

Rubi [A]  time = 0.901622, antiderivative size = 277, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4084, 4019, 4021, 4010, 4001, 3795, 203} \[ \frac{(45 A-85 B+157 C) \tan (c+d x) \sec ^2(c+d x)}{80 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(75 A-163 B+283 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(195 A-475 B+787 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{240 a^3 d}+\frac{(465 A-985 B+1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(5 A-13 B+21 C) \tan (c+d x) \sec ^3(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((75*A - 163*B + 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2
)*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A - 13*B + 21*C)*Sec[c
+ d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((465*A - 985*B + 1729*C)*Tan[c + d*x])/(120*a^2
*d*Sqrt[a + a*Sec[c + d*x]]) + ((45*A - 85*B + 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c
+ d*x]]) - ((195*A - 475*B + 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)

Rule 4084

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*A - b*B + a*C)*Cot[e + f*x]*(a + b*Cs
c[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[a*B*n - b*C*n - A*b*(2*m + n + 1) - (b*B*(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, B, 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 4021

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

Rule 4010

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

Rule 4001

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

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
]

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])

Rubi steps

\begin{align*} \int \frac{\sec ^4(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{5/2}} \, dx &=-\frac{(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}+\frac{\int \frac{\sec ^4(c+d x) \left (4 a (B-C)+\frac{1}{2} a (5 A-5 B+13 C) \sec (c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac{(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(5 A-13 B+21 C) \sec ^3(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac{\int \frac{\sec ^3(c+d x) \left (-\frac{3}{2} a^2 (5 A-13 B+21 C)+\frac{1}{4} a^2 (45 A-85 B+157 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac{(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(5 A-13 B+21 C) \sec ^3(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac{(45 A-85 B+157 C) \sec ^2(c+d x) \tan (c+d x)}{80 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{\int \frac{\sec ^2(c+d x) \left (\frac{1}{2} a^3 (45 A-85 B+157 C)-\frac{1}{8} a^3 (195 A-475 B+787 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{20 a^5}\\ &=-\frac{(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(5 A-13 B+21 C) \sec ^3(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac{(45 A-85 B+157 C) \sec ^2(c+d x) \tan (c+d x)}{80 a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{(195 A-475 B+787 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{240 a^3 d}+\frac{\int \frac{\sec (c+d x) \left (-\frac{1}{16} a^4 (195 A-475 B+787 C)+\frac{1}{8} a^4 (465 A-985 B+1729 C) \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{30 a^6}\\ &=-\frac{(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(5 A-13 B+21 C) \sec ^3(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac{(465 A-985 B+1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{(45 A-85 B+157 C) \sec ^2(c+d x) \tan (c+d x)}{80 a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{(195 A-475 B+787 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{240 a^3 d}-\frac{(75 A-163 B+283 C) \int \frac{\sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx}{32 a^2}\\ &=-\frac{(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(5 A-13 B+21 C) \sec ^3(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac{(465 A-985 B+1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{(45 A-85 B+157 C) \sec ^2(c+d x) \tan (c+d x)}{80 a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{(195 A-475 B+787 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{240 a^3 d}+\frac{(75 A-163 B+283 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 )}{16 a^2 d}\\ &=-\frac{(75 A-163 B+283 C) \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B+C) \sec ^4(c+d x) \tan (c+d x)}{4 d (a+a \sec (c+d x))^{5/2}}-\frac{(5 A-13 B+21 C) \sec ^3(c+d x) \tan (c+d x)}{16 a d (a+a \sec (c+d x))^{3/2}}+\frac{(465 A-985 B+1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a+a \sec (c+d x)}}+\frac{(45 A-85 B+157 C) \sec ^2(c+d x) \tan (c+d x)}{80 a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{(195 A-475 B+787 C) \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{240 a^3 d}\\ \end{align*}

Mathematica [C]  time = 25.1746, size = 7237, normalized size = 26.13 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [B]  time = 0.421, size = 1439, normalized size = 5.2 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1920/d/a^3*(-1+cos(d*x+c))^2*(-768*C+13720*B*cos(d*x+c)^3+15072*C*cos(d*x+c)^4+1125*A*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))^(5/2)*sin(d*x+c)-244
5*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))^(5/2)*sin(d*x+c)+4245*C*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))^(5/2)*sin(d*x+c)+4320*A*cos(d*x+c)^4-23896*C*cos(d*x+c)^3-13824*C*cos(d*x+c)^2+2
048*C*cos(d*x+c)+6750*A*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*co
s(d*x+c)/(cos(d*x+c)+1))^(5/2)*cos(d*x+c)^2*sin(d*x+c)-14670*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))^(5/2)*cos(d*x+c)^2*sin(d*x+c)+25470*C*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))^(5/2)*c
os(d*x+c)^2*sin(d*x+c)+4500*A*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))^(5/2)*cos(d*x+c)*sin(d*x+c)-9780*B*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*s
in(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)*cos(d*x+c)*sin(d*x+c)+16980*C*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))^(5/2)*
cos(d*x+c)*sin(d*x+c)+5880*A*cos(d*x+c)^5-11960*B*cos(d*x+c)^5+21368*C*cos(d*x+c)^5-8160*B*cos(d*x+c)^4-6360*A
*cos(d*x+c)^3+7680*B*cos(d*x+c)^2+4500*A*cos(d*x+c)^3*sin(d*x+c)*ln(-(-(-2*cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*si
n(d*x+c)+cos(d*x+c)-1)/sin(d*x+c))*(-2*cos(d*x+c)/(cos(d*x+c)+1))^(5/2)-9780*B*cos(d*x+c)^3*sin(d*x+c)*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))^(5/2)+
16980*C*cos(d*x+c)^3*sin(d*x+c)*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))^(5/2)+4245*C*sin(d*x+c)*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))^(5/2)*cos(d*x+c)^4+1125*A*sin(d*x+c)*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))^(5/2)*cos(d*x+
c)^4-2445*B*sin(d*x+c)*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))^(5/2)*cos(d*x+c)^4-3840*A*cos(d*x+c)^2-1280*B*cos(d*x+c))*(a*(cos(d*x+c)+1)/cos(d*x+c)
)^(1/2)/sin(d*x+c)^5/cos(d*x+c)^2

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Maxima [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(sec(d*x+c)^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2)/(a+a*sec(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [A]  time = 0.698488, size = 1735, normalized size = 6.26 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/960*(15*sqrt(2)*((75*A - 163*B + 283*C)*cos(d*x + c)^5 + 3*(75*A - 163*B + 283*C)*cos(d*x + c)^4 + 3*(75*A
- 163*B + 283*C)*cos(d*x + c)^3 + (75*A - 163*B + 283*C)*cos(d*x + c)^2)*sqrt(-a)*log(-(2*sqrt(2)*sqrt(-a)*sq
rt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)*sin(d*x + c) - 3*a*cos(d*x + c)^2 - 2*a*cos(d*x + c) + a)/(
cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - 4*((735*A - 1495*B + 2671*C)*cos(d*x + c)^4 + 5*(255*A - 503*B + 911*C
)*cos(d*x + c)^3 + 32*(15*A - 25*B + 49*C)*cos(d*x + c)^2 + 160*(B - C)*cos(d*x + c) + 96*C)*sqrt((a*cos(d*x +
c) + a)/cos(d*x + 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), 1/480*(15*sqrt(2)*((75*A - 163*B + 283*C)*cos(d*x + c)^5 + 3*(75*A - 163*B + 283*C)*co
s(d*x + c)^4 + 3*(75*A - 163*B + 283*C)*cos(d*x + c)^3 + (75*A - 163*B + 283*C)*cos(d*x + c)^2)*sqrt(a)*arctan
(sqrt(2)*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) + 2*((735*A - 1495*B + 2
671*C)*cos(d*x + c)^4 + 5*(255*A - 503*B + 911*C)*cos(d*x + c)^3 + 32*(15*A - 25*B + 49*C)*cos(d*x + c)^2 + 16
0*(B - C)*cos(d*x + c) + 96*C)*sqrt((a*cos(d*x + c) + a)/cos(d*x + 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)]

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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(sec(d*x+c)**4*(A+B*sec(d*x+c)+C*sec(d*x+c)**2)/(a+a*sec(d*x+c))**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 10.4614, size = 755, normalized size = 2.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*((((15*(2*(sqrt(2)*A*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - sqrt(2)*B*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)
+ sqrt(2)*C*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1))*tan(1/2*d*x + 1/2*c)^2/a^2 + (13*sqrt(2)*A*a^2*sgn(tan(1/2*d
*x + 1/2*c)^2 - 1) - 21*sqrt(2)*B*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) + 29*sqrt(2)*C*a^2*sgn(tan(1/2*d*x + 1/2
*c)^2 - 1))/a^2)*tan(1/2*d*x + 1/2*c)^2 - (1725*sqrt(2)*A*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - 3685*sqrt(2)*B
*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) + 6733*sqrt(2)*C*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1))/a^2)*tan(1/2*d*x +
1/2*c)^2 + 5*(549*sqrt(2)*A*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - 1133*sqrt(2)*B*a^2*sgn(tan(1/2*d*x + 1/2*c)^
2 - 1) + 1973*sqrt(2)*C*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1))/a^2)*tan(1/2*d*x + 1/2*c)^2 - 15*(83*sqrt(2)*A*a^
2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) - 155*sqrt(2)*B*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1) + 291*sqrt(2)*C*a^2*sgn(
tan(1/2*d*x + 1/2*c)^2 - 1))/a^2)*tan(1/2*d*x + 1/2*c)/((a*tan(1/2*d*x + 1/2*c)^2 - a)^2*sqrt(-a*tan(1/2*d*x +
1/2*c)^2 + a)) - 15*(75*sqrt(2)*A - 163*sqrt(2)*B + 283*sqrt(2)*C)*log(abs(-sqrt(-a)*tan(1/2*d*x + 1/2*c) + s
qrt(-a*tan(1/2*d*x + 1/2*c)^2 + a)))/(sqrt(-a)*a^2*sgn(tan(1/2*d*x + 1/2*c)^2 - 1)))/d