3.172 \(\int \cos ^5(c+d x) (a+a \sec (c+d x))^{3/2} (A+C \sec ^2(c+d x)) \, dx\)
Optimal. Leaf size=245 \[ \frac {a^{3/2} (133 A+176 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{128 d}+\frac {a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (67 A+80 C) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (133 A+176 C) \sin (c+d x) \cos (c+d x)}{192 d \sqrt {a \sec (c+d x)+a}}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac {3 a A \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{40 d} \]
[Out]
1/128*a^(3/2)*(133*A+176*C)*arctan(a^(1/2)*tan(d*x+c)/(a+a*sec(d*x+c))^(1/2))/d+1/5*A*cos(d*x+c)^4*(a+a*sec(d*
x+c))^(3/2)*sin(d*x+c)/d+1/128*a^2*(133*A+176*C)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/192*a^2*(133*A+176*C)*c
os(d*x+c)*sin(d*x+c)/d/(a+a*sec(d*x+c))^(1/2)+1/240*a^2*(67*A+80*C)*cos(d*x+c)^2*sin(d*x+c)/d/(a+a*sec(d*x+c))
^(1/2)+3/40*a*A*cos(d*x+c)^3*sin(d*x+c)*(a+a*sec(d*x+c))^(1/2)/d
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Rubi [A] time = 0.64, antiderivative size = 245, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used =
{4087, 4017, 4015, 3805, 3774, 203} \[ \frac {a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt {a \sec (c+d x)+a}}+\frac {a^{3/2} (133 A+176 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{128 d}+\frac {a^2 (67 A+80 C) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (133 A+176 C) \sin (c+d x) \cos (c+d x)}{192 d \sqrt {a \sec (c+d x)+a}}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac {3 a A \sin (c+d x) \cos ^3(c+d x) \sqrt {a \sec (c+d x)+a}}{40 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]
[Out]
(a^(3/2)*(133*A + 176*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(133*A + 176*
C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(133*A + 176*C)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqr
t[a + a*Sec[c + d*x]]) + (a^2*(67*A + 80*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (3
*a*A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/
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 4087
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 ^5(c+d x) (a+a \sec (c+d x))^{3/2} \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {3 a A}{2}+\frac {5}{2} a (A+2 C) \sec (c+d x)\right ) \, dx}{5 a}\\ &=\frac {3 a A \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {\int \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {1}{4} a^2 (67 A+80 C)+\frac {5}{4} a^2 (11 A+16 C) \sec (c+d x)\right ) \, dx}{20 a}\\ &=\frac {a^2 (67 A+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {3 a A \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{96} (a (133 A+176 C)) \int \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^2 (133 A+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (67 A+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {3 a A \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{128} (a (133 A+176 C)) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (133 A+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (67 A+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {3 a A \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {1}{256} (a (133 A+176 C)) \int \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (133 A+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (67 A+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {3 a A \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}-\frac {\left (a^2 (133 A+176 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^{3/2} (133 A+176 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{128 d}+\frac {a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (133 A+176 C) \cos (c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (67 A+80 C) \cos ^2(c+d x) \sin (c+d x)}{240 d \sqrt {a+a \sec (c+d x)}}+\frac {3 a A \cos ^3(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{40 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 2.20, size = 159, normalized size = 0.65 \[ \frac {a \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sec (c+d x)+1)} \left (15 \sqrt {2} (133 A+176 C) \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \sqrt {\cos (c+d x)}+\left (\sin \left (\frac {3}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) (2 (1007 A+880 C) \cos (c+d x)+4 (181 A+80 C) \cos (2 (c+d x))+228 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+2671 A+2960 C)\right )}{3840 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]
[Out]
(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(133*A + 176*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sq
rt[Cos[c + d*x]] + (2671*A + 2960*C + 2*(1007*A + 880*C)*Cos[c + d*x] + 4*(181*A + 80*C)*Cos[2*(c + d*x)] + 22
8*A*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)
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fricas [A] time = 0.56, size = 420, normalized size = 1.71 \[ \left [\frac {15 \, {\left ({\left (133 \, A + 176 \, C\right )} a \cos \left (d x + c\right ) + {\left (133 \, A + 176 \, C\right )} a\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (384 \, A a \cos \left (d x + c\right )^{5} + 912 \, A a \cos \left (d x + c\right )^{4} + 8 \, {\left (133 \, A + 80 \, C\right )} a \cos \left (d x + c\right )^{3} + 10 \, {\left (133 \, A + 176 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \, {\left (133 \, A + 176 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3840 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {15 \, {\left ({\left (133 \, A + 176 \, C\right )} a \cos \left (d x + c\right ) + {\left (133 \, A + 176 \, C\right )} a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (384 \, A a \cos \left (d x + c\right )^{5} + 912 \, A a \cos \left (d x + c\right )^{4} + 8 \, {\left (133 \, A + 80 \, C\right )} a \cos \left (d x + c\right )^{3} + 10 \, {\left (133 \, A + 176 \, C\right )} a \cos \left (d x + c\right )^{2} + 15 \, {\left (133 \, A + 176 \, C\right )} a \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{1920 \, {\left (d \cos \left (d x + c\right ) + d\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="fricas")
[Out]
[1/3840*(15*((133*A + 176*C)*a*cos(d*x + c) + (133*A + 176*C)*a)*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*cos(d*x + c)^5 + 912*A*a*cos(d*x + c)^4 + 8*(133*A + 80*C)*a*cos(d*x + c)^3 + 10*(133*A + 176*C)*a
*cos(d*x + c)^2 + 15*(133*A + 176*C)*a*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*((133*A + 176*C)*a*cos(d*x + c) + (133*A + 176*C)*a)*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*cos(d*x + c)^5 + 912*A*a*cos(d*x
+ c)^4 + 8*(133*A + 80*C)*a*cos(d*x + c)^3 + 10*(133*A + 176*C)*a*cos(d*x + c)^2 + 15*(133*A + 176*C)*a*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 = 8.44, size = 1369, normalized size = 5.59 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x, algorithm="giac")
[Out]
-1/3840*(15*(133*A*sqrt(-a)*a*sgn(cos(d*x + c)) + 176*C*sqrt(-a)*a*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*(133*A*sqrt(-a)*a*sgn(cos(d
*x + c)) + 176*C*sqrt(-a)*a*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*(1995*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*sqrt(-a)*a^2*sgn(cos(d*x + c)) + 2640*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))^18*C*sqrt(-a)*a^2*sgn(cos(d*x + c)) - 38505*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*A*sqrt(-a)*a^3*sgn(cos(d*x + c)) - 55920*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^3*sgn(cos(d*x + c)) + 561660*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*A*sqrt(-a)*a^4*sgn(cos(d*x + c)) + 582720*
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^4*sgn(cos(d*x +
c)) - 2684100*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^5*
sgn(cos(d*x + c)) - 3395520*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*C
*sqrt(-a)*a^5*sgn(cos(d*x + c)) + 7371738*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^6*sgn(cos(d*x + c)) + 9329760*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^6*sgn(cos(d*x + c)) - 6407470*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^7*sgn(cos(d*x + c)) - 8110880*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^7*sgn(cos(d*x + c)) + 2176620*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^8*sgn(cos(d*x + c)) + 2882880*s
qrt(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^8*sgn(cos(d*x + c)
) - 399860*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^9*sgn(
cos(d*x + c)) - 498880*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^9*sgn(cos(d*x + c)) + 34035*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^10*sgn(cos(d*x + c)) + 42960*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^10*sgn(cos(d*x + c)) - 1201*sqrt(2)*A*sqrt(-a)*a^11*sgn(cos(d*x + c)) - 1520*sqrt
(2)*C*sqrt(-a)*a^11*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 = 1.88, size = 934, normalized size = 3.81 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^5*(a+a*sec(d*x+c))^(3/2)*(A+C*sec(d*x+c)^2),x)
[Out]
-1/61440/d*(1995*A*cos(d*x+c)^4*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)+2640*C*cos(d*x+c)^4*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)+7980*A
*cos(d*x+c)^3*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)+10560*C*cos(d*x+c)^3*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*a
rctanh(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)+11970*A*cos(d*x+c)^2*si
n(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)+15840*C*cos(d*x+c)^2*sin(d*x+c)*(-2*cos(d*x+c)/(1+cos(d*x+c)))^(9/2)*arctanh(1/2*(-2*c
os(d*x+c)/(1+cos(d*x+c)))^(1/2)*sin(d*x+c)/cos(d*x+c)*2^(1/2))*2^(1/2)+7980*A*cos(d*x+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)+10560*C*cos(d*x+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)+1995*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)+2640*C*(-2*cos(d*x+c)/(1+co
s(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)+12288*A*cos(d*x+c)^10+16896*A*cos(d*x+c)^9+4864*A*cos(d*x+c)^8+20480*C*cos(d*x+c)^8+8512*A*cos(d*x+c)^7+
35840*C*cos(d*x+c)^7+21280*A*cos(d*x+c)^6+28160*C*cos(d*x+c)^6-63840*A*cos(d*x+c)^5-84480*C*cos(d*x+c)^5)*(a*(
1+cos(d*x+c))/cos(d*x+c))^(1/2)/cos(d*x+c)^4/sin(d*x+c)*a
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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))^(3/2)*(A+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 \[ \int {\cos \left (c+d\,x\right )}^5\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(c + d*x)^5*(A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(3/2),x)
[Out]
int(cos(c + d*x)^5*(A + C/cos(c + d*x)^2)*(a + a/cos(c + d*x))^(3/2), x)
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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(cos(d*x+c)**5*(a+a*sec(d*x+c))**(3/2)*(A+C*sec(d*x+c)**2),x)
[Out]
Timed out
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