3.96 \(\int (a+a \sin (e+f x))^{3/2} \tan ^2(e+f x) \, dx\)
Optimal. Leaf size=88 \[ \frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 \sec (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f}+\frac {7 \sec (e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f} \]
[Out]
7/3*sec(f*x+e)*(a+a*sin(f*x+e))^(3/2)/f-2/3*sec(f*x+e)*(a+a*sin(f*x+e))^(5/2)/a/f+11/3*a^2*cos(f*x+e)/f/(a+a*s
in(f*x+e))^(1/2)
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Rubi [A] time = 0.20, antiderivative size = 88, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used =
{2713, 2855, 2646} \[ \frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 \sec (e+f x) (a \sin (e+f x)+a)^{5/2}}{3 a f}+\frac {7 \sec (e+f x) (a \sin (e+f x)+a)^{3/2}}{3 f} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^(3/2)*Tan[e + f*x]^2,x]
[Out]
(11*a^2*Cos[e + f*x])/(3*f*Sqrt[a + a*Sin[e + f*x]]) + (7*Sec[e + f*x]*(a + a*Sin[e + f*x])^(3/2))/(3*f) - (2*
Sec[e + f*x]*(a + a*Sin[e + f*x])^(5/2))/(3*a*f)
Rule 2646
Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(-2*b*Cos[c + d*x])/(d*Sqrt[a + b*Sin[c + d*
x]]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 2713
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> -Simp[(a + b*Sin[e + f
*x])^(m + 1)/(b*f*m*Cos[e + f*x]), x] + Dist[1/(b*m), Int[((a + b*Sin[e + f*x])^m*(b*(m + 1) + a*Sin[e + f*x])
)/Cos[e + f*x]^2, x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && !IntegerQ[m] && !LtQ[m, 0]
Rule 2855
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> -Simp[((b*c + a*d)*(g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^m)/(a*f*g*(p +
1)), x] + Dist[(b*(a*d*m + b*c*(m + p + 1)))/(a*g^2*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x]
)^(m - 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[a^2 - b^2, 0] && GtQ[m, -1] && LtQ[p, -1]
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} \tan ^2(e+f x) \, dx &=-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{5/2}}{3 a f}+\frac {2 \int \sec ^2(e+f x) (a+a \sin (e+f x))^{3/2} \left (\frac {5 a}{2}+a \sin (e+f x)\right ) \, dx}{3 a}\\ &=\frac {7 \sec (e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{5/2}}{3 a f}-\frac {1}{6} (11 a) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=\frac {11 a^2 \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}+\frac {7 \sec (e+f x) (a+a \sin (e+f x))^{3/2}}{3 f}-\frac {2 \sec (e+f x) (a+a \sin (e+f x))^{5/2}}{3 a f}\\ \end {align*}
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Mathematica [A] time = 4.15, size = 46, normalized size = 0.52 \[ \frac {a \sec (e+f x) \sqrt {a (\sin (e+f x)+1)} (-8 \sin (e+f x)+\cos (2 (e+f x))+15)}{3 f} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^(3/2)*Tan[e + f*x]^2,x]
[Out]
(a*Sec[e + f*x]*(15 + Cos[2*(e + f*x)] - 8*Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])])/(3*f)
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fricas [A] time = 0.41, size = 48, normalized size = 0.55 \[ \frac {2 \, {\left (a \cos \left (f x + e\right )^{2} - 4 \, a \sin \left (f x + e\right ) + 7 \, a\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)*tan(f*x+e)^2,x, algorithm="fricas")
[Out]
2/3*(a*cos(f*x + e)^2 - 4*a*sin(f*x + e) + 7*a)*sqrt(a*sin(f*x + e) + a)/(f*cos(f*x + e))
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)*tan(f*x+e)^2,x, algorithm="giac")
[Out]
Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check si
gn: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)sqrt(2*a)*(8448*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))+21504*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*ex
p(1))^2-12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^3+75264*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi)
)*tan(1/4*exp(1))^4+12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^5+21504*a*sign(cos(1/2*(f*x+exp
(1))-1/4*pi))*tan(1/4*exp(1))^6+12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^7+8448*a*sign(cos(1
/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^8+21504*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*f*x)^2-12288*a*s
ign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*f*x)^3+75264*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*f*x)^4+122
88*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*f*x)^5+21504*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*f*x)
^6+12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*f*x)^7+8448*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4
*f*x)^8-12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))-12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*
tan(1/4*f*x)+430080*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(1/4*f*x)^2+49152*a*sign(cos(1/2
*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(1/4*f*x)^3-559104*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*ex
p(1))^2*tan(1/4*f*x)^4-49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(1/4*f*x)^5+430080*a*si
gn(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(1/4*f*x)^6-49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*t
an(1/4*exp(1))^2*tan(1/4*f*x)^7+21504*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(1/4*f*x)^8+49
152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^2*tan(1/4*f*x)+49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*
pi))*tan(1/4*exp(1))^3*tan(1/4*f*x)^2-1204224*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^3*tan(1/4*f
*x)^3+122880*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^3*tan(1/4*f*x)^4+1204224*a*sign(cos(1/2*(f*x
+exp(1))-1/4*pi))*tan(1/4*exp(1))^3*tan(1/4*f*x)^5+49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^
3*tan(1/4*f*x)^6-172032*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^3*tan(1/4*f*x)^7-12288*a*sign(cos
(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^3*tan(1/4*f*x)^8+172032*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/
4*exp(1))^3*tan(1/4*f*x)-559104*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^4*tan(1/4*f*x)^2+122880*a
*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^4*tan(1/4*f*x)^3+2171904*a*sign(cos(1/2*(f*x+exp(1))-1/4*p
i))*tan(1/4*exp(1))^4*tan(1/4*f*x)^4-122880*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^4*tan(1/4*f*x
)^5-559104*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^4*tan(1/4*f*x)^6-122880*a*sign(cos(1/2*(f*x+ex
p(1))-1/4*pi))*tan(1/4*exp(1))^4*tan(1/4*f*x)^7+75264*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^4*t
an(1/4*f*x)^8+122880*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^4*tan(1/4*f*x)-49152*a*sign(cos(1/2*
(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^5*tan(1/4*f*x)^2+1204224*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*ex
p(1))^5*tan(1/4*f*x)^3-122880*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^5*tan(1/4*f*x)^4-1204224*a*
sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^5*tan(1/4*f*x)^5-49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))
*tan(1/4*exp(1))^5*tan(1/4*f*x)^6+172032*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^5*tan(1/4*f*x)^7
+12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^5*tan(1/4*f*x)^8-172032*a*sign(cos(1/2*(f*x+exp(1)
)-1/4*pi))*tan(1/4*exp(1))^5*tan(1/4*f*x)+430080*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^6*tan(1/
4*f*x)^2+49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^6*tan(1/4*f*x)^3-559104*a*sign(cos(1/2*(f*
x+exp(1))-1/4*pi))*tan(1/4*exp(1))^6*tan(1/4*f*x)^4-49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))
^6*tan(1/4*f*x)^5+430080*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^6*tan(1/4*f*x)^6-49152*a*sign(co
s(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^6*tan(1/4*f*x)^7+21504*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/
4*exp(1))^6*tan(1/4*f*x)^8+49152*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^6*tan(1/4*f*x)-49152*a*s
ign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^7*tan(1/4*f*x)^2-172032*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))
*tan(1/4*exp(1))^7*tan(1/4*f*x)^3-122880*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^7*tan(1/4*f*x)^4
+172032*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^7*tan(1/4*f*x)^5-49152*a*sign(cos(1/2*(f*x+exp(1)
)-1/4*pi))*tan(1/4*exp(1))^7*tan(1/4*f*x)^6-24576*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^7*tan(1
/4*f*x)^7+12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^7*tan(1/4*f*x)^8+24576*a*sign(cos(1/2*(f*
x+exp(1))-1/4*pi))*tan(1/4*exp(1))^7*tan(1/4*f*x)+21504*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^8
*tan(1/4*f*x)^2-12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^8*tan(1/4*f*x)^3+75264*a*sign(cos(1
/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^8*tan(1/4*f*x)^4+12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*e
xp(1))^8*tan(1/4*f*x)^5+21504*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^8*tan(1/4*f*x)^6+12288*a*si
gn(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^8*tan(1/4*f*x)^7+8448*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*ta
n(1/4*exp(1))^8*tan(1/4*f*x)^8-12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))^8*tan(1/4*f*x)+49152
*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))*tan(1/4*f*x)^2+172032*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi
))*tan(1/4*exp(1))*tan(1/4*f*x)^3+122880*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))*tan(1/4*f*x)^4-1
72032*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))*tan(1/4*f*x)^5+49152*a*sign(cos(1/2*(f*x+exp(1))-1/
4*pi))*tan(1/4*exp(1))*tan(1/4*f*x)^6+24576*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))*tan(1/4*f*x)^
7-12288*a*sign(cos(1/2*(f*x+exp(1))-1/4*pi))*tan(1/4*exp(1))*tan(1/4*f*x)^8-24576*a*sign(cos(1/2*(f*x+exp(1))-
1/4*pi))*tan(1/4*exp(1))*tan(1/4*f*x))/(-2304*sqrt(2)*f*tan(1/4*exp(1))-2304*sqrt(2)*f*tan(1/4*f*x)-4608*sqrt(
2)*f*tan(1/4*exp(1))^2*tan(1/4*f*x)+1152*sqrt(2)*f*tan(1/4*exp(1))^8*tan(1/4*f*x)^8+2304*sqrt(2)*f*tan(1/4*exp
(1))^7*tan(1/4*f*x)^8+2304*sqrt(2)*f*tan(1/4*exp(1))^8*tan(1/4*f*x)^7+2304*sqrt(2)*f*tan(1/4*exp(1))^6*tan(1/4
*f*x)^8-4608*sqrt(2)*f*tan(1/4*exp(1))^7*tan(1/4*f*x)^7+2304*sqrt(2)*f*tan(1/4*exp(1))^8*tan(1/4*f*x)^6+6912*s
qrt(2)*f*tan(1/4*exp(1))^5*tan(1/4*f*x)^8+4608*sqrt(2)*f*tan(1/4*exp(1))^6*tan(1/4*f*x)^7+4608*sqrt(2)*f*tan(1
/4*exp(1))^7*tan(1/4*f*x)^6+6912*sqrt(2)*f*tan(1/4*exp(1))^8*tan(1/4*f*x)^5-13824*sqrt(2)*f*tan(1/4*exp(1))^5*
tan(1/4*f*x)^7+4608*sqrt(2)*f*tan(1/4*exp(1))^6*tan(1/4*f*x)^6-13824*sqrt(2)*f*tan(1/4*exp(1))^7*tan(1/4*f*x)^
5+6912*sqrt(2)*f*tan(1/4*exp(1))^3*tan(1/4*f*x)^8+13824*sqrt(2)*f*tan(1/4*exp(1))^5*tan(1/4*f*x)^6+13824*sqrt(
2)*f*tan(1/4*exp(1))^6*tan(1/4*f*x)^5+6912*sqrt(2)*f*tan(1/4*exp(1))^8*tan(1/4*f*x)^3-2304*sqrt(2)*f*tan(1/4*e
xp(1))^2*tan(1/4*f*x)^8-13824*sqrt(2)*f*tan(1/4*exp(1))^3*tan(1/4*f*x)^7-41472*sqrt(2)*f*tan(1/4*exp(1))^5*tan
(1/4*f*x)^5-13824*sqrt(2)*f*tan(1/4*exp(1))^7*tan(1/4*f*x)^3-2304*sqrt(2)*f*tan(1/4*exp(1))^8*tan(1/4*f*x)^2-4
608*sqrt(2)*f*tan(1/4*exp(1))^2*tan(1/4*f*x)^7+13824*sqrt(2)*f*tan(1/4*exp(1))^3*tan(1/4*f*x)^6+13824*sqrt(2)*
f*tan(1/4*exp(1))^6*tan(1/4*f*x)^3-4608*sqrt(2)*f*tan(1/4*exp(1))^7*tan(1/4*f*x)^2-1152*sqrt(2)*f*tan(1/4*exp(
1))^8-1152*sqrt(2)*f*tan(1/4*f*x)^8-4608*sqrt(2)*f*tan(1/4*exp(1))^2*tan(1/4*f*x)^6-41472*sqrt(2)*f*tan(1/4*ex
p(1))^3*tan(1/4*f*x)^5-41472*sqrt(2)*f*tan(1/4*exp(1))^5*tan(1/4*f*x)^3-4608*sqrt(2)*f*tan(1/4*exp(1))^6*tan(1
/4*f*x)^2-2304*sqrt(2)*f*tan(1/4*exp(1))^7-2304*sqrt(2)*f*tan(1/4*f*x)^7-13824*sqrt(2)*f*tan(1/4*exp(1))^2*tan
(1/4*f*x)^5-13824*sqrt(2)*f*tan(1/4*exp(1))^5*tan(1/4*f*x)^2-2304*sqrt(2)*f*tan(1/4*exp(1))^6-2304*sqrt(2)*f*t
an(1/4*f*x)^6-41472*sqrt(2)*f*tan(1/4*exp(1))^3*tan(1/4*f*x)^3-6912*sqrt(2)*f*tan(1/4*exp(1))^5-6912*sqrt(2)*f
*tan(1/4*f*x)^5-13824*sqrt(2)*f*tan(1/4*exp(1))^2*tan(1/4*f*x)^3-13824*sqrt(2)*f*tan(1/4*exp(1))^3*tan(1/4*f*x
)^2+4608*sqrt(2)*f*tan(1/4*exp(1))^2*tan(1/4*f*x)^2-6912*sqrt(2)*f*tan(1/4*exp(1))^3-6912*sqrt(2)*f*tan(1/4*f*
x)^3+2304*sqrt(2)*f*tan(1/4*exp(1))^2+2304*sqrt(2)*f*tan(1/4*f*x)^2+1152*sqrt(2)*f-13824*sqrt(2)*f*tan(1/4*exp
(1))^3*tan(1/4*f*x)-13824*sqrt(2)*f*tan(1/4*exp(1))^5*tan(1/4*f*x)+4608*sqrt(2)*f*tan(1/4*exp(1))^6*tan(1/4*f*
x)-4608*sqrt(2)*f*tan(1/4*exp(1))^7*tan(1/4*f*x)+2304*sqrt(2)*f*tan(1/4*exp(1))^8*tan(1/4*f*x)-4608*sqrt(2)*f*
tan(1/4*exp(1))*tan(1/4*f*x)^2-13824*sqrt(2)*f*tan(1/4*exp(1))*tan(1/4*f*x)^3-13824*sqrt(2)*f*tan(1/4*exp(1))*
tan(1/4*f*x)^5+4608*sqrt(2)*f*tan(1/4*exp(1))*tan(1/4*f*x)^6-4608*sqrt(2)*f*tan(1/4*exp(1))*tan(1/4*f*x)^7+230
4*sqrt(2)*f*tan(1/4*exp(1))*tan(1/4*f*x)^8-4608*sqrt(2)*f*tan(1/4*exp(1))*tan(1/4*f*x))
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maple [A] time = 0.58, size = 55, normalized size = 0.62 \[ -\frac {2 a^{2} \left (1+\sin \left (f x +e \right )\right ) \left (\sin ^{2}\left (f x +e \right )+4 \sin \left (f x +e \right )-8\right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*sin(f*x+e))^(3/2)*tan(f*x+e)^2,x)
[Out]
-2/3*a^2*(1+sin(f*x+e))*(sin(f*x+e)^2+4*sin(f*x+e)-8)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f
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maxima [A] time = 0.58, size = 145, normalized size = 1.65 \[ -\frac {8 \, {\left (2 \, a^{\frac {3}{2}} - \frac {2 \, a^{\frac {3}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{\frac {3}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {2 \, a^{\frac {3}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {2 \, a^{\frac {3}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )}}{3 \, f {\left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )} {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))^(3/2)*tan(f*x+e)^2,x, algorithm="maxima")
[Out]
-8/3*(2*a^(3/2) - 2*a^(3/2)*sin(f*x + e)/(cos(f*x + e) + 1) + 3*a^(3/2)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 -
2*a^(3/2)*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 2*a^(3/2)*sin(f*x + e)^4/(cos(f*x + e) + 1)^4)/(f*(sin(f*x + e
)/(cos(f*x + e) + 1) - 1)*(sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 1)^(3/2))
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {tan}\left (e+f\,x\right )}^2\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(e + f*x)^2*(a + a*sin(e + f*x))^(3/2),x)
[Out]
int(tan(e + f*x)^2*(a + a*sin(e + f*x))^(3/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*sin(f*x+e))**(3/2)*tan(f*x+e)**2,x)
[Out]
Integral((a*(sin(e + f*x) + 1))**(3/2)*tan(e + f*x)**2, x)
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