3.508 \(\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=261 \[ \frac {a^{5/2} (283 A+326 B+400 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{128 d}+\frac {a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{960 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{240 d}+\frac {a (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d} \]
[Out]
1/128*a^(5/2)*(283*A+326*B+400*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+1/8*a*(A+2*B)*cos(d*x+c)
^3*(a+a*sec(d*x+c))^(3/2)*sin(d*x+c)/d+1/5*A*cos(d*x+c)^4*(a+a*sec(d*x+c))^(5/2)*sin(d*x+c)/d+1/128*a^3*(283*A
+326*B+400*C)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/960*a^3*(787*A+950*B+1040*C)*cos(d*x+c)*sin(d*x+c)/d/(a+a*
sec(d*x+c))^(1/2)+1/240*a^2*(79*A+110*B+80*C)*cos(d*x+c)^2*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d
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Rubi [A] time = 0.81, antiderivative size = 261, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.140, Rules used =
{4086, 4017, 4015, 3805, 3774, 203} \[ \frac {a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt {a \sec (c+d x)+a}}+\frac {a^{5/2} (283 A+326 B+400 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{128 d}+\frac {a^2 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{240 d}+\frac {a^3 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{960 d \sqrt {a \sec (c+d x)+a}}+\frac {a (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(a^(5/2)*(283*A + 326*B + 400*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*
A + 326*B + 400*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(787*A + 950*B + 1040*C)*Cos[c + d*x]
*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(79*A + 110*B + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c +
d*x]]*Sin[c + d*x])/(240*d) + (a*(A + 2*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A
*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)
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 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 3805
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(a*Cot[
e + f*x]*(d*Csc[e + f*x])^n)/(f*n*Sqrt[a + b*Csc[e + f*x]]), x] + Dist[(a*(2*n + 1))/(2*b*d*n), Int[Sqrt[a + b
*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, -2
^(-1)] && IntegerQ[2*n]
Rule 4015
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(
B_.) + (A_)), x_Symbol] :> Simp[(A*b^2*Cot[e + f*x]*(d*Csc[e + f*x])^n)/(a*f*n*Sqrt[a + b*Csc[e + f*x]]), x] +
Dist[(A*b*(2*n + 1) + 2*a*B*n)/(2*a*d*n), Int[Sqrt[a + b*Csc[e + f*x]]*(d*Csc[e + f*x])^(n + 1), x], x] /; Fr
eeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0] && NeQ[A*b*(2*n + 1) + 2*a*B*n, 0] &&
LtQ[n, 0]
Rule 4017
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 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])
Rubi steps
\begin {align*} \int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {5}{2} a (A+2 B)+\frac {1}{2} a (3 A+10 C) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac {a (A+2 B) \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (79 A+110 B+80 C)+\frac {1}{4} a^2 (39 A+30 B+80 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a^2 (79 A+110 B+80 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {a (A+2 B) \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {\int \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (787 A+950 B+1040 C)+\frac {3}{8} a^3 (157 A+170 B+240 C) \sec (c+d x)\right ) \, dx}{60 a}\\ &=\frac {a^3 (787 A+950 B+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (79 A+110 B+80 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {a (A+2 B) \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {1}{128} \left (a^2 (283 A+326 B+400 C)\right ) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (787 A+950 B+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (79 A+110 B+80 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {a (A+2 B) \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}+\frac {1}{256} \left (a^2 (283 A+326 B+400 C)\right ) \int \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (787 A+950 B+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (79 A+110 B+80 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {a (A+2 B) \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}-\frac {\left (a^3 (283 A+326 B+400 C)\right ) \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 )}{128 d}\\ &=\frac {a^{5/2} (283 A+326 B+400 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{128 d}+\frac {a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (787 A+950 B+1040 C) \cos (c+d x) \sin (c+d x)}{960 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (79 A+110 B+80 C) \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{240 d}+\frac {a (A+2 B) \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{8 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 2.63, size = 183, normalized size = 0.70 \[ \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sec (c+d x)+1)} \left (15 \sqrt {2} (283 A+326 B+400 C) \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cos (c+d x)}+\left (\sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) ((3874 A+3620 B+2720 C) \cos (c+d x)+4 (331 A+230 B+80 C) \cos (2 (c+d x))+348 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+5521 A+120 B \cos (3 (c+d x))+5810 B+6320 C)\right )}{3840 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
[Out]
(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(283*A + 326*B + 400*C)*ArcSin[Sqrt[2]*Sin[(c + d
*x)/2]]*Sqrt[Cos[c + d*x]] + (5521*A + 5810*B + 6320*C + (3874*A + 3620*B + 2720*C)*Cos[c + d*x] + 4*(331*A +
230*B + 80*C)*Cos[2*(c + d*x)] + 348*A*Cos[3*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Si
n[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)
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fricas [A] time = 0.75, size = 490, normalized size = 1.88 \[ \left [\frac {15 \, {\left ({\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (29 \, A + 10 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (283 \, A + 230 \, B + 80 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (283 \, A + 326 \, B + 272 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3840 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right ) + {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (384 \, A a^{2} \cos \left (d x + c\right )^{5} + 48 \, {\left (29 \, A + 10 \, B\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (283 \, A + 230 \, B + 80 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 10 \, {\left (283 \, A + 326 \, B + 272 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (283 \, A + 326 \, B + 400 \, C\right )} a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
[1/3840*(15*((283*A + 326*B + 400*C)*a^2*cos(d*x + c) + (283*A + 326*B + 400*C)*a^2)*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*(384*A*a^2*cos(d*x + c)^5 + 48*(29*A + 10*B)*a^2*cos(d*x + c)^4 + 8*(283*A + 230*B + 80
*C)*a^2*cos(d*x + c)^3 + 10*(283*A + 326*B + 272*C)*a^2*cos(d*x + c)^2 + 15*(283*A + 326*B + 400*C)*a^2*cos(d*
x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) + d), -1/1920*(15*((283*A + 326*
B + 400*C)*a^2*cos(d*x + c) + (283*A + 326*B + 400*C)*a^2)*sqrt(a)*arctan(sqrt((a*cos(d*x + c) + a)/cos(d*x +
c))*cos(d*x + c)/(sqrt(a)*sin(d*x + c))) - (384*A*a^2*cos(d*x + c)^5 + 48*(29*A + 10*B)*a^2*cos(d*x + c)^4 + 8
*(283*A + 230*B + 80*C)*a^2*cos(d*x + c)^3 + 10*(283*A + 326*B + 272*C)*a^2*cos(d*x + c)^2 + 15*(283*A + 326*B
+ 400*C)*a^2*cos(d*x + c))*sqrt((a*cos(d*x + c) + a)/cos(d*x + c))*sin(d*x + c))/(d*cos(d*x + c) + d)]
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giac [B] time = 4.34, size = 1965, normalized size = 7.53 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac")
[Out]
-1/3840*(15*(283*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 326*B*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 400*C*sqrt(-a)*a^2*
sgn(cos(d*x + 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))) - 15*(283*A*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 326*B*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 400*C*sqrt(-a)*
a^2*sgn(cos(d*x + 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))) + 4*(4245*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^18*A*sq
rt(-a)*a^3*sgn(cos(d*x + c)) + 4890*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 +
a))^18*B*sqrt(-a)*a^3*sgn(cos(d*x + c)) + 6000*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x +
1/2*c)^2 + a))^18*C*sqrt(-a)*a^3*sgn(cos(d*x + c)) - 114615*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*t
an(1/2*d*x + 1/2*c)^2 + a))^16*A*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 132030*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c
) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^16*B*sqrt(-a)*a^4*sgn(cos(d*x + c)) - 162000*sqrt(2)*(sqrt(-a)*tan(1/
2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^16*C*sqrt(-a)*a^4*sgn(cos(d*x + c)) + 1298820*sqrt(2)*(s
qrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^14*A*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 1319
880*sqrt(2)*(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)*a^5*sgn(cos(d*
x + c)) + 1801920*sqrt(2)*(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)*
a^5*sgn(cos(d*x + c)) - 6176700*sqrt(2)*(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^6*sgn(cos(d*x + c)) - 6888120*sqrt(2)*(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^6*sgn(cos(d*x + c)) - 9764160*sqrt(2)*(sqrt(-a)*tan(1/2*d*x + 1/2*c) - sqrt(-a*t
an(1/2*d*x + 1/2*c)^2 + a))^12*C*sqrt(-a)*a^6*sgn(cos(d*x + c)) + 16394598*sqrt(2)*(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^7*sgn(cos(d*x + c)) + 18352620*sqrt(2)*(sqrt(-a)*ta
n(1/2*d*x + 1/2*c) - sqrt(-a*tan(1/2*d*x + 1/2*c)^2 + a))^10*B*sqrt(-a)*a^7*sgn(cos(d*x + c)) + 24060960*sqrt(
2)*(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^7*sgn(cos(d*x + c)) -
14042770*sqrt(2)*(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^8*sgn(c
os(d*x + c)) - 15746180*sqrt(2)*(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^8*sgn(cos(d*x + c)) - 19910240*sqrt(2)*(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^8*sgn(cos(d*x + c)) + 4791060*sqrt(2)*(sqrt(-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^9*sgn(cos(d*x + c)) + 5497320*sqrt(2)*(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^9*sgn(cos(d*x + c)) + 7135680*sqrt(2)*(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^9*sgn(cos(d*x + c)) - 860300*sqrt(2)*(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^10*sgn(cos(d*x + c)) - 959320*sqrt(2)*
(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^10*sgn(cos(d*x + c)) - 12
68800*sqrt(2)*(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^10*sgn(cos(
d*x + c)) + 75885*sqrt(2)*(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
^11*sgn(cos(d*x + c)) + 84810*sqrt(2)*(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^11*sgn(cos(d*x + c)) + 111600*sqrt(2)*(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^11*sgn(cos(d*x + c)) - 2671*sqrt(2)*A*sqrt(-a)*a^12*sgn(cos(d*x + c)) - 2990*sqrt(2)
*B*sqrt(-a)*a^12*sgn(cos(d*x + c)) - 3920*sqrt(2)*C*sqrt(-a)*a^12*sgn(cos(d*x + c)))/((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)^5)/d
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maple [B] time = 2.38, size = 1381, normalized size = 5.29 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
[Out]
-1/61440/d*(16980*A*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))
^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)+104960*C*cos(d*x+c)^6-135840*A*cos(d*x+c)^5-156480*B*
cos(d*x+c)^5+43520*B*cos(d*x+c)^8+45440*B*cos(d*x+c)^7+66560*C*cos(d*x+c)^7-192000*C*cos(d*x+c)^5+19560*B*sin(
d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*
x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)+24000*C*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d
*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)+4245*A*sin(d*x+c)*(-2*cos(d*x+c)
/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)
*cos(d*x+c)^4+4890*B*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c))
)^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)^4+6000*C*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(
9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)^4+1698
0*A*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c
)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)^3+19560*B*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2
*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)^3+24000*C*sin(d*x+c)*(
-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(
1/2))*2^(1/2)*cos(d*x+c)^3+25470*A*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/
(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)^2+29340*B*sin(d*x+c)*(-2*cos(d*x+c)/(1
+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*co
s(d*x+c)^2+36000*C*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^
(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)*cos(d*x+c)^2+4245*A*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*2^(1/2)*
arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*sin(d*x+c)+4890*B*(-2*cos(d*x+
c)/(1+cos(d*x+c)))^(9/2)*2^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2
))*sin(d*x+c)+6000*C*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*2^(1/2)*arctanh(1/2*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(
1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*sin(d*x+c)+27904*A*cos(d*x+c)^8+18112*A*cos(d*x+c)^7+45280*A*cos(d*x+c)^6+
12288*A*cos(d*x+c)^10+32256*A*cos(d*x+c)^9+15360*B*cos(d*x+c)^9+20480*C*cos(d*x+c)^8+52160*B*cos(d*x+c)^6)*(a*
(1+cos(d*x+c))/cos(d*x+c))^(1/2)/cos(d*x+c)^4/sin(d*x+c)*a^2
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maxima [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(cos(d*x+c)^5*(a+a*sec(d*x+c))^(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
[Out]
Timed out
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (c+d\,x\right )}^5\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^5*(a + a/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
[Out]
int(cos(c + d*x)^5*(a + a/cos(c + d*x))^(5/2)*(A + B/cos(c + d*x) + C/cos(c + d*x)^2), x)
________________________________________________________________________________________
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(cos(d*x+c)**5*(a+a*sec(d*x+c))**(5/2)*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)
[Out]
Timed out
________________________________________________________________________________________