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]
(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)
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Rubi [A] time = 0.0390246, antiderivative size = 68, normalized size of antiderivative = 1.,
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 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
]
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 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 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])
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*}
Mathematica [A] time = 0.0675036, 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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Maple [A] time = 0.02, size = 62, normalized size = 0.9 \begin{align*}{\frac{a\tan \left ( dx+c \right ) }{2\,d}\sqrt{a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{1}{2\,d}{a}^{{\frac{3}{2}}}\ln \left ( \sqrt{a}\tan \left ( dx+c \right ) +\sqrt{a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2}} \right ) } \end{align*}
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 = 1.9083, size = 751, normalized size = 11.04 \begin{align*} \text{result too large to display} \end{align*}
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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Fricas [A] time = 1.29946, size = 193, normalized size = 2.84 \begin{align*} \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} \end{align*}
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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \tan ^{2}{\left (c + d x \right )} + a\right )^{\frac{3}{2}}\, dx \end{align*}
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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Giac [B] time = 3.1976, size = 2869, normalized size = 42.19 \begin{align*} \text{result too large to display} \end{align*}
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]
1/2*((a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x
)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c) - a^(3
/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/
2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1))*log(abs(-tan(1/2*d*x)*tan(
1/2*c) + tan(1/2*d*x) + tan(1/2*c) + 1))/(tan(1/2*c) - 1) - (a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1
/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*
c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c) + a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*
tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*t
an(1/2*d*x)*tan(1/2*c) + 1))*log(abs(-tan(1/2*d*x)*tan(1/2*c) - tan(1/2*d*x) - tan(1/2*c) + 1))/(tan(1/2*c) +
1) - 2*(a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d
*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*
tan(1/2*c)^8 + 6*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*
tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/
2*d*x)^3*tan(1/2*c)^6 + 2*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*
x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) +
1)*tan(1/2*d*x)^2*tan(1/2*c)^7 + a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan
(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/
2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^8 - 18*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)
^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x
)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^5 - 10*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3*
tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*t
an(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^6 - 2*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*
d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^
4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^7 - 6*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2*d*x)^3
*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)^4 - 4*
tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3*tan(1/2*c)^2 - 18*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(
1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2
*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c)^3 + 2*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4
- 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3
- tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^5 - a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan
(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/
2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^3 + 2*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/2
*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c)
^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)^2*tan(1/2*c) + 10*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4
*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - ta
n(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x)*tan(1/2*c)^2 + 2*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c
)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)
^3 - tan(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c)^3 - a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*
tan(1/2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan
(1/2*c)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*d*x) - 2*a^(3/2)*sgn(tan(1/2*d*x)^4*tan(1/2*c)^4 - 4*tan(1/
2*d*x)^3*tan(1/2*c)^3 - tan(1/2*d*x)^4 - 4*tan(1/2*d*x)^3*tan(1/2*c) - 4*tan(1/2*d*x)*tan(1/2*c)^3 - tan(1/2*c
)^4 - 4*tan(1/2*d*x)*tan(1/2*c) + 1)*tan(1/2*c))/((tan(1/2*d*x)^2*tan(1/2*c)^2 - tan(1/2*d*x)^2 - 4*tan(1/2*d*
x)*tan(1/2*c) - tan(1/2*c)^2 + 1)^2*(tan(1/2*c)^4 - 2*tan(1/2*c)^2 + 1)))/d