3.610 \(\int \cos ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx\)

Optimal. Leaf size=176 \[ -\frac{5 c \sin (2 a+2 b x)}{16 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{c \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{5 c \sin (2 a+2 b x) \cos (2 a+2 b x)}{24 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right )}{16 b} \]

[Out]

(5*Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(16*b) - (5*c*Sin[2*a + 2*b*x])/
(16*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (5*c*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(24*b*Sqrt[-c + c*Sec[2*a + 2*b
*x]]) - (c*Cos[2*a + 2*b*x]^2*Sin[2*a + 2*b*x])/(6*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])

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Rubi [A]  time = 0.288151, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {4397, 3805, 3774, 207} \[ -\frac{5 c \sin (2 a+2 b x)}{16 b \sqrt{c \sec (2 a+2 b x)-c}}-\frac{c \sin (2 a+2 b x) \cos ^2(2 a+2 b x)}{6 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{5 c \sin (2 a+2 b x) \cos (2 a+2 b x)}{24 b \sqrt{c \sec (2 a+2 b x)-c}}+\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{c \sec (2 a+2 b x)-c}}\right )}{16 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]

[Out]

(5*Sqrt[c]*ArcTanh[(Sqrt[c]*Tan[2*a + 2*b*x])/Sqrt[-c + c*Sec[2*a + 2*b*x]]])/(16*b) - (5*c*Sin[2*a + 2*b*x])/
(16*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (5*c*Cos[2*a + 2*b*x]*Sin[2*a + 2*b*x])/(24*b*Sqrt[-c + c*Sec[2*a + 2*b
*x]]) - (c*Cos[2*a + 2*b*x]^2*Sin[2*a + 2*b*x])/(6*b*Sqrt[-c + c*Sec[2*a + 2*b*x]])

Rule 4397

Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]

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 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 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \cos ^3(2 (a+b x)) \sqrt{c \tan (a+b x) \tan (2 (a+b x))} \, dx &=\int \cos ^3(2 a+2 b x) \sqrt{-c+c \sec (2 a+2 b x)} \, dx\\ &=-\frac{c \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{5}{6} \int \cos ^2(2 a+2 b x) \sqrt{-c+c \sec (2 a+2 b x)} \, dx\\ &=\frac{5 c \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{c \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt{-c+c \sec (2 a+2 b x)}}+\frac{5}{8} \int \cos (2 a+2 b x) \sqrt{-c+c \sec (2 a+2 b x)} \, dx\\ &=-\frac{5 c \sin (2 a+2 b x)}{16 b \sqrt{-c+c \sec (2 a+2 b x)}}+\frac{5 c \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{c \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{5}{16} \int \sqrt{-c+c \sec (2 a+2 b x)} \, dx\\ &=-\frac{5 c \sin (2 a+2 b x)}{16 b \sqrt{-c+c \sec (2 a+2 b x)}}+\frac{5 c \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{c \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt{-c+c \sec (2 a+2 b x)}}+\frac{(5 c) \operatorname{Subst}\left (\int \frac{1}{-c+x^2} \, dx,x,-\frac{c \tan (2 a+2 b x)}{\sqrt{-c+c \sec (2 a+2 b x)}}\right )}{16 b}\\ &=\frac{5 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \tan (2 a+2 b x)}{\sqrt{-c+c \sec (2 a+2 b x)}}\right )}{16 b}-\frac{5 c \sin (2 a+2 b x)}{16 b \sqrt{-c+c \sec (2 a+2 b x)}}+\frac{5 c \cos (2 a+2 b x) \sin (2 a+2 b x)}{24 b \sqrt{-c+c \sec (2 a+2 b x)}}-\frac{c \cos ^2(2 a+2 b x) \sin (2 a+2 b x)}{6 b \sqrt{-c+c \sec (2 a+2 b x)}}\\ \end{align*}

Mathematica [A]  time = 0.306241, size = 116, normalized size = 0.66 \[ \frac{\sqrt{c \tan (a+b x) \tan (2 (a+b x))} \left (30 \sin (2 (a+b x))-2 \sin (4 (a+b x))+4 \sin (6 (a+b x))-26 \cot (a+b x)+15 \sqrt{2} \sqrt{\cos (2 (a+b x))} \csc (a+b x) \tanh ^{-1}\left (\frac{\sqrt{2} \cos (a+b x)}{\sqrt{\cos (2 (a+b x))}}\right )\right )}{96 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[2*(a + b*x)]^3*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b*x)]],x]

[Out]

((-26*Cot[a + b*x] + 15*Sqrt[2]*ArcTanh[(Sqrt[2]*Cos[a + b*x])/Sqrt[Cos[2*(a + b*x)]]]*Sqrt[Cos[2*(a + b*x)]]*
Csc[a + b*x] + 30*Sin[2*(a + b*x)] - 2*Sin[4*(a + b*x)] + 4*Sin[6*(a + b*x)])*Sqrt[c*Tan[a + b*x]*Tan[2*(a + b
*x)]])/(96*b)

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Maple [B]  time = 0.413, size = 921, 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(cos(2*b*x+2*a)^3*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x)

[Out]

1/2/b*4^(1/2)*(c*(1-cos(b*x+a)^2)/(2*cos(b*x+a)^2-1))^(1/2)*sin(b*x+a)*((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(
1/2)*arctanh(1/2*2^(1/2)*cos(b*x+a)*4^(1/2)*(-1+cos(b*x+a))/sin(b*x+a)^2/((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)
^(1/2))/(-1+cos(b*x+a))-3/8*2^(1/2)/b*4^(1/2)*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*sin(b*x+a)*(2^(1/2)*co
s(b*x+a)*((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2)*arctanh(1/2*2^(1/2)*cos(b*x+a)*4^(1/2)*(-1+cos(b*x+a))/si
n(b*x+a)^2/((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2))+2^(1/2)*((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2)*ar
ctanh(1/2*2^(1/2)*cos(b*x+a)*4^(1/2)*(-1+cos(b*x+a))/sin(b*x+a)^2/((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2))
-4*cos(b*x+a)^3+2*cos(b*x+a))/(-1+cos(b*x+a)^2)+3/32*2^(1/2)/b*4^(1/2)*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/
2)*sin(b*x+a)*(-16*cos(b*x+a)^5+3*2^(1/2)*cos(b*x+a)*((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2)*arctanh(1/2*2
^(1/2)*cos(b*x+a)*4^(1/2)*(-1+cos(b*x+a))/sin(b*x+a)^2/((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2))+3*2^(1/2)*
((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2)*arctanh(1/2*2^(1/2)*cos(b*x+a)*4^(1/2)*(-1+cos(b*x+a))/sin(b*x+a)^
2/((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2))-4*cos(b*x+a)^3+6*cos(b*x+a))/(-1+cos(b*x+a)^2)-1/192*2^(1/2)/b*
4^(1/2)*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)*sin(b*x+a)*(-128*cos(b*x+a)^7-16*cos(b*x+a)^5+15*2^(1/2)*cos
(b*x+a)*((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2)*arctanh(1/2*2^(1/2)*cos(b*x+a)*4^(1/2)*(-1+cos(b*x+a))/sin
(b*x+a)^2/((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2))+15*2^(1/2)*((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2)*
arctanh(1/2*2^(1/2)*cos(b*x+a)*4^(1/2)*(-1+cos(b*x+a))/sin(b*x+a)^2/((2*cos(b*x+a)^2-1)/(cos(b*x+a)+1)^2)^(1/2
))-20*cos(b*x+a)^3+30*cos(b*x+a))/(-1+cos(b*x+a)^2)

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Maxima [B]  time = 3.38579, size = 3150, normalized size = 17.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*b*x+2*a)^3*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x, algorithm="maxima")

[Out]

-1/384*(8*(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*
x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + 1)^(3/4)*(cos(3/2*arctan2(sin(2/3*arct
an2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))*sin(6*b*
x + 6*a) + (cos(6*b*x + 6*a) + 1)*sin(3/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2
/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1)))*sqrt(c) + 12*(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b
*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a), c
os(6*b*x + 6*a))) + 1)^(1/4)*((sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 5*sin(1/3*arctan2(sin(6*
b*x + 6*a), cos(6*b*x + 6*a))))*cos(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3
*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1)) + (cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) -
3*cos(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 4)*sin(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a),
cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1)))*sqrt(c) + 15*sqrt(c)*(log(sqr
t(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))
)^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + 1)*cos(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x +
6*a), cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))^2 + sqrt(cos(2/3*arctan2
(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*a
rctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + 1)*sin(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x +
6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))^2 + 2*(cos(2/3*arctan2(sin(6*b*x + 6*a),
cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x +
6*a), cos(6*b*x + 6*a))) + 1)^(1/4)*sin(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos
(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1)) + 1) - log(sqrt(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(
6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a)
, cos(6*b*x + 6*a))) + 1)*cos(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3*arcta
n2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))^2 + sqrt(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2
+ sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a
))) + 1)*sin(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*
a), cos(6*b*x + 6*a))) - 1))^2 - 2*(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(s
in(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + 1)^(1/4)*sin(
1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x
+ 6*a))) - 1)) + 1) + log(((cos(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(1/3*arctan2(sin(6*b*
x + 6*a), cos(6*b*x + 6*a)))^2)*cos(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3
*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))^2 + (cos(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^
2 + sin(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2)*sin(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a),
cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))^2)*sqrt(cos(2/3*arctan2(sin(6*
b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(
sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + 1) + 2*(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(2/
3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + 1)
^(1/4)*(cos(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a
), cos(6*b*x + 6*a))) - 1))*sin(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + cos(1/3*arctan2(sin(6*b*x +
6*a), cos(6*b*x + 6*a)))*sin(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3*arcta
n2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))) + 1) - log(((cos(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)
))^2 + sin(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2)*cos(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a
), cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))^2 + (cos(1/3*arctan2(sin(6*
b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2)*sin(1/2*arctan2(sin
(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))
^2)*sqrt(cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x
+ 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) + 1) - 2*(cos(2/3*arctan2(sin(6*b*x + 6*a)
, cos(6*b*x + 6*a)))^2 + sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))^2 + 2*cos(2/3*arctan2(sin(6*b*x
+ 6*a), cos(6*b*x + 6*a))) + 1)^(1/4)*(cos(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))), -
cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))*sin(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))
) + cos(1/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a)))*sin(1/2*arctan2(sin(2/3*arctan2(sin(6*b*x + 6*a), cos
(6*b*x + 6*a))), -cos(2/3*arctan2(sin(6*b*x + 6*a), cos(6*b*x + 6*a))) - 1))) + 1)))/b

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Fricas [A]  time = 2.50927, size = 1260, normalized size = 7.16 \begin{align*} \left [\frac{15 \,{\left (\tan \left (b x + a\right )^{7} + 3 \, \tan \left (b x + a\right )^{5} + 3 \, \tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )\right )} \sqrt{c} \log \left (-\frac{c \tan \left (b x + a\right )^{5} - 14 \, c \tan \left (b x + a\right )^{3} + 4 \, \sqrt{2}{\left (\tan \left (b x + a\right )^{4} - 4 \, \tan \left (b x + a\right )^{2} + 3\right )} \sqrt{-\frac{c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}} \sqrt{c} + 17 \, c \tan \left (b x + a\right )}{\tan \left (b x + a\right )^{5} + 2 \, \tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )}\right ) + 4 \, \sqrt{2}{\left (33 \, \tan \left (b x + a\right )^{6} - 19 \, \tan \left (b x + a\right )^{4} - \tan \left (b x + a\right )^{2} - 13\right )} \sqrt{-\frac{c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{192 \,{\left (b \tan \left (b x + a\right )^{7} + 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}}, -\frac{15 \,{\left (\tan \left (b x + a\right )^{7} + 3 \, \tan \left (b x + a\right )^{5} + 3 \, \tan \left (b x + a\right )^{3} + \tan \left (b x + a\right )\right )} \sqrt{-c} \arctan \left (\frac{2 \, \sqrt{2} \sqrt{-\frac{c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}{\left (\tan \left (b x + a\right )^{2} - 1\right )} \sqrt{-c}}{c \tan \left (b x + a\right )^{3} - 3 \, c \tan \left (b x + a\right )}\right ) - 2 \, \sqrt{2}{\left (33 \, \tan \left (b x + a\right )^{6} - 19 \, \tan \left (b x + a\right )^{4} - \tan \left (b x + a\right )^{2} - 13\right )} \sqrt{-\frac{c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{96 \,{\left (b \tan \left (b x + a\right )^{7} + 3 \, b \tan \left (b x + a\right )^{5} + 3 \, b \tan \left (b x + a\right )^{3} + b \tan \left (b x + a\right )\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(2*b*x+2*a)^3*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x, algorithm="fricas")

[Out]

[1/192*(15*(tan(b*x + a)^7 + 3*tan(b*x + a)^5 + 3*tan(b*x + a)^3 + tan(b*x + a))*sqrt(c)*log(-(c*tan(b*x + a)^
5 - 14*c*tan(b*x + a)^3 + 4*sqrt(2)*(tan(b*x + a)^4 - 4*tan(b*x + a)^2 + 3)*sqrt(-c*tan(b*x + a)^2/(tan(b*x +
a)^2 - 1))*sqrt(c) + 17*c*tan(b*x + a))/(tan(b*x + a)^5 + 2*tan(b*x + a)^3 + tan(b*x + a))) + 4*sqrt(2)*(33*ta
n(b*x + a)^6 - 19*tan(b*x + a)^4 - tan(b*x + a)^2 - 13)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1)))/(b*tan(b
*x + a)^7 + 3*b*tan(b*x + a)^5 + 3*b*tan(b*x + a)^3 + b*tan(b*x + a)), -1/96*(15*(tan(b*x + a)^7 + 3*tan(b*x +
a)^5 + 3*tan(b*x + a)^3 + tan(b*x + a))*sqrt(-c)*arctan(2*sqrt(2)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1)
)*(tan(b*x + a)^2 - 1)*sqrt(-c)/(c*tan(b*x + a)^3 - 3*c*tan(b*x + a))) - 2*sqrt(2)*(33*tan(b*x + a)^6 - 19*tan
(b*x + a)^4 - tan(b*x + a)^2 - 13)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1)))/(b*tan(b*x + a)^7 + 3*b*tan(b
*x + a)^5 + 3*b*tan(b*x + a)^3 + b*tan(b*x + a))]

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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(cos(2*b*x+2*a)**3*(c*tan(b*x+a)*tan(2*b*x+2*a))**(1/2),x)

[Out]

Timed out

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Giac [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(2*b*x+2*a)^3*(c*tan(b*x+a)*tan(2*b*x+2*a))^(1/2),x, algorithm="giac")

[Out]

Timed out