3.271 \(\int (a+a \tan ^2(c+d x))^{3/2} \, dx\)
Optimal. Leaf size=68 \[ \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{2 d}+\frac {a \tan (c+d x) \sqrt {a \sec ^2(c+d x)}}{2 d} \]
[Out]
1/2*a^(3/2)*arctanh(a^(1/2)*tan(d*x+c)/(a*sec(d*x+c)^2)^(1/2))/d+1/2*a*(a*sec(d*x+c)^2)^(1/2)*tan(d*x+c)/d
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Rubi [A] time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used =
{3657, 4122, 195, 217, 206} \[ \frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{2 d}+\frac {a \tan (c+d x) \sqrt {a \sec ^2(c+d x)}}{2 d} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Tan[c + d*x]^2)^(3/2),x]
[Out]
(a^(3/2)*ArcTanh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a*Sec[c + d*x]^2]])/(2*d) + (a*Sqrt[a*Sec[c + d*x]^2]*Tan[c + d*x
])/(2*d)
Rule 195
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 217
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] && !GtQ[a, 0]
Rule 3657
Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]
Rule 4122
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p
]
Rubi steps
\begin {align*} \int \left (a+a \tan ^2(c+d x)\right )^{3/2} \, dx &=\int \left (a \sec ^2(c+d x)\right )^{3/2} \, dx\\ &=\frac {a \operatorname {Subst}\left (\int \sqrt {a+a x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {a \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{2 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+a x^2}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {a \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{2 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{2 d}\\ &=\frac {a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec ^2(c+d x)}}\right )}{2 d}+\frac {a \sqrt {a \sec ^2(c+d x)} \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 43, normalized size = 0.63 \[ \frac {a \sqrt {a \sec ^2(c+d x)} \left (\tan (c+d x)+\cos (c+d x) \tanh ^{-1}(\sin (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Tan[c + d*x]^2)^(3/2),x]
[Out]
(a*Sqrt[a*Sec[c + d*x]^2]*(ArcTanh[Sin[c + d*x]]*Cos[c + d*x] + Tan[c + d*x]))/(2*d)
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fricas [A] time = 0.41, size = 72, normalized size = 1.06 \[ \frac {a^{\frac {3}{2}} \log \left (2 \, a \tan \left (d x + c\right )^{2} + 2 \, \sqrt {a \tan \left (d x + c\right )^{2} + a} \sqrt {a} \tan \left (d x + c\right ) + a\right ) + 2 \, \sqrt {a \tan \left (d x + c\right )^{2} + a} a \tan \left (d x + c\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*tan(d*x+c)^2)^(3/2),x, algorithm="fricas")
[Out]
1/4*(a^(3/2)*log(2*a*tan(d*x + c)^2 + 2*sqrt(a*tan(d*x + c)^2 + a)*sqrt(a)*tan(d*x + c) + a) + 2*sqrt(a*tan(d*
x + c)^2 + a)*a*tan(d*x + c))/d
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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*tan(d*x+c)^2)^(3/2),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)2/d*((-sqrt(a)*a*tan(d*x/2)*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^
4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)-s
qrt(a)*a*tan(d*x/2)^3*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(
1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)+2*sqrt(a)*a*sign(-4*tan(d*x/2)^3*
tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/
2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)^3+2*sqrt(a)*a*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/
2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3
+1)*tan(1/2*c)^5-2*sqrt(a)*a*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/
2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)^7-2*sqrt(a)*a*s
ign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*t
an(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)+10*sqrt(a)*a*tan(d*x/2)*sign(-4*tan(d*x/2)^3*
tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/
2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)^2-10*sqrt(a)*a*tan(d*x/2)*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x
/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*
tan(1/2*c)^3+1)*tan(1/2*c)^6+sqrt(a)*a*tan(d*x/2)*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-ta
n(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1
/2*c)^8-18*sqrt(a)*a*tan(d*x/2)^2*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan
(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)^3-18*sqrt(
a)*a*tan(d*x/2)^2*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*
c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)^5+2*sqrt(a)*a*tan(d*x/2)^2
*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3
*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)^7+2*sqrt(a)*a*tan(d*x/2)^2*sign(-4*tan(d*x/
2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan
(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)-6*sqrt(a)*a*tan(d*x/2)^3*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan
(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x
/2)*tan(1/2*c)^3+1)*tan(1/2*c)^2+6*sqrt(a)*a*tan(d*x/2)^3*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2
*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+
1)*tan(1/2*c)^6+sqrt(a)*a*tan(d*x/2)^3*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-
4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c)^8)/(-
tan(1/2*c)^2-4*tan(d*x/2)*tan(1/2*c)+tan(d*x/2)^2*tan(1/2*c)^2-tan(d*x/2)^2+1)^2/(-2*tan(1/2*c)^4+4*tan(1/2*c)
^2-2)+(sqrt(a)*a*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c
)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)-sqrt(a)*a*sign(-4*tan(d*x/2)^3*tan(1/2
*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*
tan(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c))*ln(abs(-tan(d*x/2)*tan(1/2*c)+tan(1/2*c)+tan(d*x/2)+1))/(-4*tan(1/2*c)+
4)+(-sqrt(a)*a*sign(-4*tan(d*x/2)^3*tan(1/2*c)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-
4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*tan(d*x/2)*tan(1/2*c)^3+1)-sqrt(a)*a*sign(-4*tan(d*x/2)^3*tan(1/2*c
)+tan(d*x/2)^4*tan(1/2*c)^4-tan(1/2*c)^4-4*tan(d*x/2)*tan(1/2*c)-4*tan(d*x/2)^3*tan(1/2*c)^3-tan(d*x/2)^4-4*ta
n(d*x/2)*tan(1/2*c)^3+1)*tan(1/2*c))*ln(abs(-tan(d*x/2)*tan(1/2*c)-tan(1/2*c)-tan(d*x/2)+1))/(4*tan(1/2*c)+4))
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maple [A] time = 0.28, size = 62, normalized size = 0.91 \[ \frac {a \tan \left (d x +c \right ) \sqrt {a +a \left (\tan ^{2}\left (d x +c \right )\right )}}{2 d}+\frac {a^{\frac {3}{2}} \ln \left (\sqrt {a}\, \tan \left (d x +c \right )+\sqrt {a +a \left (\tan ^{2}\left (d x +c \right )\right )}\right )}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+a*tan(d*x+c)^2)^(3/2),x)
[Out]
1/2/d*a*tan(d*x+c)*(a+a*tan(d*x+c)^2)^(1/2)+1/2/d*a^(3/2)*ln(a^(1/2)*tan(d*x+c)+(a+a*tan(d*x+c)^2)^(1/2))
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maxima [B] time = 1.01, size = 556, normalized size = 8.18 \[ -\frac {{\left (8 \, a \cos \left (3 \, d x + 3 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) - 8 \, a \cos \left (d x + c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 8 \, a \cos \left (2 \, d x + 2 \, c\right ) \sin \left (d x + c\right ) - 4 \, {\left (a \sin \left (3 \, d x + 3 \, c\right ) - a \sin \left (d x + c\right )\right )} \cos \left (4 \, d x + 4 \, c\right ) - {\left (a \cos \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \cos \left (2 \, d x + 2 \, c\right )^{2} + a \sin \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, a \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \cos \left (4 \, d x + 4 \, c\right ) + 4 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + {\left (a \cos \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \cos \left (2 \, d x + 2 \, c\right )^{2} + a \sin \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, a \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, {\left (2 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \cos \left (4 \, d x + 4 \, c\right ) + 4 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (a \cos \left (3 \, d x + 3 \, c\right ) - a \cos \left (d x + c\right )\right )} \sin \left (4 \, d x + 4 \, c\right ) - 4 \, {\left (2 \, a \cos \left (2 \, d x + 2 \, c\right ) + a\right )} \sin \left (3 \, d x + 3 \, c\right ) + 4 \, a \sin \left (d x + c\right )\right )} \sqrt {a}}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + \cos \left (4 \, d x + 4 \, c\right )^{2} + 4 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (4 \, d x + 4 \, c\right )^{2} + 4 \, \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 4 \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 4 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*tan(d*x+c)^2)^(3/2),x, algorithm="maxima")
[Out]
-1/4*(8*a*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 8*a*cos(d*x + c)*sin(2*d*x + 2*c) + 8*a*cos(2*d*x + 2*c)*sin(d*x
+ c) - 4*(a*sin(3*d*x + 3*c) - a*sin(d*x + c))*cos(4*d*x + 4*c) - (a*cos(4*d*x + 4*c)^2 + 4*a*cos(2*d*x + 2*c
)^2 + a*sin(4*d*x + 4*c)^2 + 4*a*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 4*a*sin(2*d*x + 2*c)^2 + 2*(2*a*cos(2*d*x
+ 2*c) + a)*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c)
+ 1) + (a*cos(4*d*x + 4*c)^2 + 4*a*cos(2*d*x + 2*c)^2 + a*sin(4*d*x + 4*c)^2 + 4*a*sin(4*d*x + 4*c)*sin(2*d*x
+ 2*c) + 4*a*sin(2*d*x + 2*c)^2 + 2*(2*a*cos(2*d*x + 2*c) + a)*cos(4*d*x + 4*c) + 4*a*cos(2*d*x + 2*c) + a)*l
og(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 4*(a*cos(3*d*x + 3*c) - a*cos(d*x + c))*sin(4*d*x +
4*c) - 4*(2*a*cos(2*d*x + 2*c) + a)*sin(3*d*x + 3*c) + 4*a*sin(d*x + c))*sqrt(a)/((2*(2*cos(2*d*x + 2*c) + 1)
*cos(4*d*x + 4*c) + cos(4*d*x + 4*c)^2 + 4*cos(2*d*x + 2*c)^2 + sin(4*d*x + 4*c)^2 + 4*sin(4*d*x + 4*c)*sin(2*
d*x + 2*c) + 4*sin(2*d*x + 2*c)^2 + 4*cos(2*d*x + 2*c) + 1)*d)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + a*tan(c + d*x)^2)^(3/2),x)
[Out]
int((a + a*tan(c + d*x)^2)^(3/2), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \tan ^{2}{\left (c + d x \right )} + a\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+a*tan(d*x+c)**2)**(3/2),x)
[Out]
Integral((a*tan(c + d*x)**2 + a)**(3/2), x)
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