\(\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx\) [614]
Optimal result
Integrand size = 29, antiderivative size = 75 \[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=-\frac {4 c^2 \tan (2 a+2 b x)}{3 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{3 b}
\]
[Out]
-4/3*c^2*tan(2*b*x+2*a)/b/(-c+c*sec(2*b*x+2*a))^(1/2)+1/3*c*(-c+c*sec(2*b*x+2*a))^(1/2)*tan(2*b*x+2*a)/b
Rubi [A] (verified)
Time = 0.13 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of
steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4482, 3878, 3877}
\[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\frac {c \tan (2 a+2 b x) \sqrt {c \sec (2 a+2 b x)-c}}{3 b}-\frac {4 c^2 \tan (2 a+2 b x)}{3 b \sqrt {c \sec (2 a+2 b x)-c}}
\]
[In]
Int[Sec[2*(a + b*x)]*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]
[Out]
(-4*c^2*Tan[2*a + 2*b*x])/(3*b*Sqrt[-c + c*Sec[2*a + 2*b*x]]) + (c*Sqrt[-c + c*Sec[2*a + 2*b*x]]*Tan[2*a + 2*b
*x])/(3*b)
Rule 3877
Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*b*(Cot[e + f*x]/(
f*Sqrt[a + b*Csc[e + f*x]])), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]
Rule 3878
Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(-b)*Cot[e + f*x]*(
(a + b*Csc[e + f*x])^(m - 1)/(f*m)), x] + Dist[a*((2*m - 1)/m), Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1),
x], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, 1/2] && IntegerQ[2*m]
Rule 4482
Int[u_, x_Symbol] :> Int[TrigSimplify[u], x] /; TrigSimplifyQ[u]
Rubi steps \begin{align*}
\text {integral}& = \int \sec (2 a+2 b x) (-c+c \sec (2 a+2 b x))^{3/2} \, dx \\ & = \frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{3 b}-\frac {1}{3} (4 c) \int \sec (2 a+2 b x) \sqrt {-c+c \sec (2 a+2 b x)} \, dx \\ & = -\frac {4 c^2 \tan (2 a+2 b x)}{3 b \sqrt {-c+c \sec (2 a+2 b x)}}+\frac {c \sqrt {-c+c \sec (2 a+2 b x)} \tan (2 a+2 b x)}{3 b} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.48 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.68
\[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=-\frac {\cot (a+b x) (-1+4 \cot (a+b x) \cot (2 (a+b x))) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2}}{3 b}
\]
[In]
Integrate[Sec[2*(a + b*x)]*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2),x]
[Out]
-1/3*(Cot[a + b*x]*(-1 + 4*Cot[a + b*x]*Cot[2*(a + b*x)])*(c*Tan[a + b*x]*Tan[2*(a + b*x)])^(3/2))/b
Maple [A] (verified)
Time = 5.74 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.95
| | |
| method | result | size |
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| default |
\(-\frac {\sqrt {2}\, \cot \left (x b +a \right ) c \left (5 \cos \left (x b +a \right )^{2}-3\right ) \sqrt {\frac {c \sin \left (x b +a \right )^{2}}{2 \cos \left (x b +a \right )^{2}-1}}\, \sqrt {4}}{3 b \left (2 \cos \left (x b +a \right )^{2}-1\right )}\) |
\(71\) |
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[In]
int(sec(2*b*x+2*a)*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x,method=_RETURNVERBOSE)
[Out]
-1/3*2^(1/2)/b*cot(b*x+a)*c*(5*cos(b*x+a)^2-3)*(c*sin(b*x+a)^2/(2*cos(b*x+a)^2-1))^(1/2)/(2*cos(b*x+a)^2-1)*4^
(1/2)
Fricas [A] (verification not implemented)
none
Time = 0.25 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89
\[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=-\frac {2 \, \sqrt {2} {\left (3 \, c \tan \left (b x + a\right )^{2} - 2 \, c\right )} \sqrt {-\frac {c \tan \left (b x + a\right )^{2}}{\tan \left (b x + a\right )^{2} - 1}}}{3 \, {\left (b \tan \left (b x + a\right )^{3} - b \tan \left (b x + a\right )\right )}}
\]
[In]
integrate(sec(2*b*x+2*a)*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x, algorithm="fricas")
[Out]
-2/3*sqrt(2)*(3*c*tan(b*x + a)^2 - 2*c)*sqrt(-c*tan(b*x + a)^2/(tan(b*x + a)^2 - 1))/(b*tan(b*x + a)^3 - b*tan
(b*x + a))
Sympy [F(-1)]
Timed out. \[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out}
\]
[In]
integrate(sec(2*b*x+2*a)*(c*tan(b*x+a)*tan(2*b*x+2*a))**(3/2),x)
[Out]
Timed out
Maxima [F]
\[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\int { \left (c \tan \left (2 \, b x + 2 \, a\right ) \tan \left (b x + a\right )\right )^{\frac {3}{2}} \sec \left (2 \, b x + 2 \, a\right ) \,d x }
\]
[In]
integrate(sec(2*b*x+2*a)*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x, algorithm="maxima")
[Out]
1/3*(6*(3*b*c*integrate(-(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(1/4)*(((cos(8*b*x
+ 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*cos(3/
2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) + (cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(8*b*x + 8*a)*si
n(4*b*x + 4*a))*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*cos(3/2*arctan2(sin(4*b*x + 4*a), c
os(4*b*x + 4*a))) + ((cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*cos(3/2*arctan2(s
in(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)) - (cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(8*b*x
+ 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*sin(
3/2*arctan2(sin(4*b*x + 4*a), cos(4*b*x + 4*a))))/((cos(4*b*x + 4*a)^4 + sin(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a
)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a)^2 + 2*cos(4*b*x + 4*a)^3 + (cos(4*b*x + 4*
a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a)^2 + (2*cos(4*b*x + 4*a)^2 + 2*cos(4*b*x +
4*a) + 1)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*
a)^2 + cos(4*b*x + 4*a))*cos(8*b*x + 8*a) + cos(4*b*x + 4*a)^2 + 2*(sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 +
2*cos(4*b*x + 4*a) + 1)*sin(4*b*x + 4*a))*sin(8*b*x + 8*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*
a) - 1))^2 + (cos(4*b*x + 4*a)^4 + sin(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x
+ 4*a) + 1)*cos(8*b*x + 8*a)^2 + 2*cos(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*
x + 4*a) + 1)*sin(8*b*x + 8*a)^2 + (2*cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b*x + 4*a)^2 + 2*(cos
(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a)^2 + cos(4*b*x + 4*a))*cos(8*b*x + 8
*a) + cos(4*b*x + 4*a)^2 + 2*(sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b*x + 4
*a))*sin(8*b*x + 8*a))*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))^2), x) + b*c*integrate(-(cos(
4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(1/4)*(((cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos
(4*b*x + 4*a)^2 + sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin(4*b*x + 4*a)^2)*cos(3/2*arctan2(sin(4*b*x + 4*a), -c
os(4*b*x + 4*a) - 1)) + (cos(4*b*x + 4*a)*sin(8*b*x + 8*a) - cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*sin(3/2*arctan
2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*cos(1/2*arctan2(sin(4*b*x + 4*a), cos(4*b*x + 4*a))) + ((cos(4*b*
x + 4*a)*sin(8*b*x + 8*a) - cos(8*b*x + 8*a)*sin(4*b*x + 4*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x +
4*a) - 1)) - (cos(8*b*x + 8*a)*cos(4*b*x + 4*a) + cos(4*b*x + 4*a)^2 + sin(8*b*x + 8*a)*sin(4*b*x + 4*a) + sin
(4*b*x + 4*a)^2)*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*sin(1/2*arctan2(sin(4*b*x + 4*a),
cos(4*b*x + 4*a))))/((cos(4*b*x + 4*a)^4 + sin(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*c
os(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a)^2 + 2*cos(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*
cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a)^2 + (2*cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b*x + 4*a)^2
+ 2*(cos(4*b*x + 4*a)^3 + cos(4*b*x + 4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a)^2 + cos(4*b*x + 4*a))*cos(8
*b*x + 8*a) + cos(4*b*x + 4*a)^2 + 2*(sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4
*b*x + 4*a))*sin(8*b*x + 8*a))*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))^2 + (cos(4*b*x + 4*a)
^4 + sin(4*b*x + 4*a)^4 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*cos(8*b*x + 8*a)^
2 + 2*cos(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(8*b*x + 8*a)
^2 + (2*cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b*x + 4*a)^2 + 2*(cos(4*b*x + 4*a)^3 + cos(4*b*x +
4*a)*sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a)^2 + cos(4*b*x + 4*a))*cos(8*b*x + 8*a) + cos(4*b*x + 4*a)^2 + 2*(
sin(4*b*x + 4*a)^3 + (cos(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)*sin(4*b*x + 4*a))*sin(8*b*x + 8*a))*sin(3/2
*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))^2), x))*(cos(4*b*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*
b*x + 4*a) + 1)^(3/4)*sqrt(c) - (3*c*cos(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1))*sin(4*b*x + 4*a
) + (3*c*cos(4*b*x + 4*a) + 5*c)*sin(3/2*arctan2(sin(4*b*x + 4*a), -cos(4*b*x + 4*a) - 1)))*sqrt(c))/((cos(4*b
*x + 4*a)^2 + sin(4*b*x + 4*a)^2 + 2*cos(4*b*x + 4*a) + 1)^(3/4)*b)
Giac [F(-1)]
Timed out. \[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=\text {Timed out}
\]
[In]
integrate(sec(2*b*x+2*a)*(c*tan(b*x+a)*tan(2*b*x+2*a))^(3/2),x, algorithm="giac")
[Out]
Timed out
Mupad [B] (verification not implemented)
Time = 35.04 (sec) , antiderivative size = 158, normalized size of antiderivative = 2.11
\[
\int \sec (2 (a+b x)) (c \tan (a+b x) \tan (2 (a+b x)))^{3/2} \, dx=-\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,3{}\mathrm {i}+{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,3{}\mathrm {i}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}\,5{}\mathrm {i}+5{}\mathrm {i}\right )\,\sqrt {\frac {c\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )}{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}+1\right )\,\left ({\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+1\right )}}}{3\,b\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}+b\,x\,2{}\mathrm {i}}-{\mathrm {e}}^{a\,4{}\mathrm {i}+b\,x\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}+b\,x\,6{}\mathrm {i}}-1\right )}
\]
[In]
int((c*tan(a + b*x)*tan(2*a + 2*b*x))^(3/2)/cos(2*a + 2*b*x),x)
[Out]
-(c*(exp(a*2i + b*x*2i)*3i + exp(a*4i + b*x*4i)*3i + exp(a*6i + b*x*6i)*5i + 5i)*((c*(exp(a*2i + b*x*2i)*1i -
1i)*(exp(a*4i + b*x*4i)*1i - 1i))/((exp(a*2i + b*x*2i) + 1)*(exp(a*4i + b*x*4i) + 1)))^(1/2))/(3*b*(exp(a*2i +
b*x*2i) - exp(a*4i + b*x*4i) + exp(a*6i + b*x*6i) - 1))