3.345 \(\int \sqrt{d \tan (e+f x)} (a+a \tan (e+f x))^2 \, dx\)

Optimal. Leaf size=244 \[ \frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{a^2 \sqrt{d} \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}-\frac{a^2 \sqrt{d} \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f} \]

[Out]

(Sqrt[2]*a^2*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - (Sqrt[2]*a^2*Sqrt[d]*ArcTan[1 + (
Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (a^2*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*T
an[e + f*x]]])/(Sqrt[2]*f) - (a^2*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/
(Sqrt[2]*f) + (4*a^2*Sqrt[d*Tan[e + f*x]])/f + (2*a^2*(d*Tan[e + f*x])^(3/2))/(3*d*f)

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Rubi [A]  time = 0.219088, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.48, Rules used = {3543, 12, 16, 3473, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}+1\right )}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{a^2 \sqrt{d} \log \left (\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f}-\frac{a^2 \sqrt{d} \log \left (\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}+\sqrt{d}\right )}{\sqrt{2} f} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])^2,x]

[Out]

(Sqrt[2]*a^2*Sqrt[d]*ArcTan[1 - (Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f - (Sqrt[2]*a^2*Sqrt[d]*ArcTan[1 + (
Sqrt[2]*Sqrt[d*Tan[e + f*x]])/Sqrt[d]])/f + (a^2*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] - Sqrt[2]*Sqrt[d*T
an[e + f*x]]])/(Sqrt[2]*f) - (a^2*Sqrt[d]*Log[Sqrt[d] + Sqrt[d]*Tan[e + f*x] + Sqrt[2]*Sqrt[d*Tan[e + f*x]]])/
(Sqrt[2]*f) + (4*a^2*Sqrt[d*Tan[e + f*x]])/f + (2*a^2*(d*Tan[e + f*x])^(3/2))/(3*d*f)

Rule 3543

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[
(d^2*(a + b*Tan[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e
+ f*x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] &&  !LeQ[m, -1] &&  !(EqQ[m, 2] && EqQ
[a, 0])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 16

Int[(u_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Dist[1/b^m, Int[u*(b*v)^(m + n), x], x] /; FreeQ[{b, n}, x
] && IntegerQ[m]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3476

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[b/d, Subst[Int[x^n/(b^2 + x^2), x], x, b*Tan[c + d
*x]], x] /; FreeQ[{b, c, d, n}, x] &&  !IntegerQ[n]

Rule 329

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/c^n)^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rubi steps

\begin{align*} \int \sqrt{d \tan (e+f x)} (a+a \tan (e+f x))^2 \, dx &=\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\int 2 a^2 \tan (e+f x) \sqrt{d \tan (e+f x)} \, dx\\ &=\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\left (2 a^2\right ) \int \tan (e+f x) \sqrt{d \tan (e+f x)} \, dx\\ &=\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac{\left (2 a^2\right ) \int (d \tan (e+f x))^{3/2} \, dx}{d}\\ &=\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\left (2 a^2 d\right ) \int \frac{1}{\sqrt{d \tan (e+f x)}} \, dx\\ &=\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac{\left (2 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (d^2+x^2\right )} \, dx,x,d \tan (e+f x)\right )}{f}\\ &=\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac{\left (4 a^2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac{\left (2 a^2 d\right ) \operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}-\frac{\left (2 a^2 d\right ) \operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}+\frac{\left (a^2 \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}+\frac{\left (a^2 \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}-\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}-\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \tan (e+f x)}\right )}{f}\\ &=\frac{a^2 \sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}-\frac{a^2 \sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}+\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}-\frac{\left (\sqrt{2} a^2 \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{\left (\sqrt{2} a^2 \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}\\ &=\frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}-\frac{\sqrt{2} a^2 \sqrt{d} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \tan (e+f x)}}{\sqrt{d}}\right )}{f}+\frac{a^2 \sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)-\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}-\frac{a^2 \sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \tan (e+f x)+\sqrt{2} \sqrt{d \tan (e+f x)}\right )}{\sqrt{2} f}+\frac{4 a^2 \sqrt{d \tan (e+f x)}}{f}+\frac{2 a^2 (d \tan (e+f x))^{3/2}}{3 d f}\\ \end{align*}

Mathematica [A]  time = 0.457628, size = 175, normalized size = 0.72 \[ \frac{a^2 \sqrt{d \tan (e+f x)} \left (4 \tan ^{\frac{3}{2}}(e+f x)+6 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\tan (e+f x)}\right )-6 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt{\tan (e+f x)}+1\right )+24 \sqrt{\tan (e+f x)}+3 \sqrt{2} \log \left (\tan (e+f x)-\sqrt{2} \sqrt{\tan (e+f x)}+1\right )-3 \sqrt{2} \log \left (\tan (e+f x)+\sqrt{2} \sqrt{\tan (e+f x)}+1\right )\right )}{6 f \sqrt{\tan (e+f x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[d*Tan[e + f*x]]*(a + a*Tan[e + f*x])^2,x]

[Out]

(a^2*Sqrt[d*Tan[e + f*x]]*(6*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]] - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqr
t[Tan[e + f*x]]] + 3*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 3*Sqrt[2]*Log[1 + Sqrt[2]*Sq
rt[Tan[e + f*x]] + Tan[e + f*x]] + 24*Sqrt[Tan[e + f*x]] + 4*Tan[e + f*x]^(3/2)))/(6*f*Sqrt[Tan[e + f*x]])

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Maple [A]  time = 0.018, size = 204, normalized size = 0.8 \begin{align*}{\frac{2\,{a}^{2}}{3\,df} \left ( d\tan \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}}}+4\,{\frac{{a}^{2}\sqrt{d\tan \left ( fx+e \right ) }}{f}}-{\frac{{a}^{2}\sqrt{2}}{f}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{{a}^{2}\sqrt{2}}{f}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }-{\frac{{a}^{2}\sqrt{2}}{2\,f}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*tan(f*x+e))^(1/2)*(a+a*tan(f*x+e))^2,x)

[Out]

2/3*a^2*(d*tan(f*x+e))^(3/2)/d/f+4*a^2*(d*tan(f*x+e))^(1/2)/f-1/f*a^2*(d^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(d^2)
^(1/4)*(d*tan(f*x+e))^(1/2)+1)+1/f*a^2*(d^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)
-1/2/f*a^2*(d^2)^(1/4)*2^(1/2)*ln((d*tan(f*x+e)+(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f
*x+e)-(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))^(1/2)*(a+a*tan(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.85377, size = 1616, normalized size = 6.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))^(1/2)*(a+a*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

1/6*(12*sqrt(2)*(a^8*d^2/f^4)^(1/4)*f*arctan(-(a^8*d^2 + sqrt(2)*(a^8*d^2/f^4)^(3/4)*a^2*f^3*sqrt(d*sin(f*x +
e)/cos(f*x + e)) - sqrt(2)*(a^8*d^2/f^4)^(3/4)*f^3*sqrt((a^4*d*sin(f*x + e) + sqrt(2)*(a^8*d^2/f^4)^(1/4)*a^2*
f*sqrt(d*sin(f*x + e)/cos(f*x + e))*cos(f*x + e) + sqrt(a^8*d^2/f^4)*f^2*cos(f*x + e))/cos(f*x + e)))/(a^8*d^2
))*cos(f*x + e) + 12*sqrt(2)*(a^8*d^2/f^4)^(1/4)*f*arctan((a^8*d^2 - sqrt(2)*(a^8*d^2/f^4)^(3/4)*a^2*f^3*sqrt(
d*sin(f*x + e)/cos(f*x + e)) + sqrt(2)*(a^8*d^2/f^4)^(3/4)*f^3*sqrt((a^4*d*sin(f*x + e) - sqrt(2)*(a^8*d^2/f^4
)^(1/4)*a^2*f*sqrt(d*sin(f*x + e)/cos(f*x + e))*cos(f*x + e) + sqrt(a^8*d^2/f^4)*f^2*cos(f*x + e))/cos(f*x + e
)))/(a^8*d^2))*cos(f*x + e) - 3*sqrt(2)*(a^8*d^2/f^4)^(1/4)*f*cos(f*x + e)*log((a^4*d*sin(f*x + e) + sqrt(2)*(
a^8*d^2/f^4)^(1/4)*a^2*f*sqrt(d*sin(f*x + e)/cos(f*x + e))*cos(f*x + e) + sqrt(a^8*d^2/f^4)*f^2*cos(f*x + e))/
cos(f*x + e)) + 3*sqrt(2)*(a^8*d^2/f^4)^(1/4)*f*cos(f*x + e)*log((a^4*d*sin(f*x + e) - sqrt(2)*(a^8*d^2/f^4)^(
1/4)*a^2*f*sqrt(d*sin(f*x + e)/cos(f*x + e))*cos(f*x + e) + sqrt(a^8*d^2/f^4)*f^2*cos(f*x + e))/cos(f*x + e))
+ 4*(6*a^2*cos(f*x + e) + a^2*sin(f*x + e))*sqrt(d*sin(f*x + e)/cos(f*x + e)))/(f*cos(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int \sqrt{d \tan{\left (e + f x \right )}}\, dx + \int 2 \sqrt{d \tan{\left (e + f x \right )}} \tan{\left (e + f x \right )}\, dx + \int \sqrt{d \tan{\left (e + f x \right )}} \tan ^{2}{\left (e + f x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))**(1/2)*(a+a*tan(f*x+e))**2,x)

[Out]

a**2*(Integral(sqrt(d*tan(e + f*x)), x) + Integral(2*sqrt(d*tan(e + f*x))*tan(e + f*x), x) + Integral(sqrt(d*t
an(e + f*x))*tan(e + f*x)**2, x))

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Giac [A]  time = 1.25471, size = 336, normalized size = 1.38 \begin{align*} -\frac{\sqrt{2} a^{2} \sqrt{{\left | d \right |}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} + 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{f} - \frac{\sqrt{2} a^{2} \sqrt{{\left | d \right |}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | d \right |}} - 2 \, \sqrt{d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt{{\left | d \right |}}}\right )}{f} - \frac{\sqrt{2} a^{2} \sqrt{{\left | d \right |}} \log \left (d \tan \left (f x + e\right ) + \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{2 \, f} + \frac{\sqrt{2} a^{2} \sqrt{{\left | d \right |}} \log \left (d \tan \left (f x + e\right ) - \sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{{\left | d \right |}} +{\left | d \right |}\right )}{2 \, f} + \frac{2 \,{\left (\sqrt{d \tan \left (f x + e\right )} a^{2} d^{3} f^{2} \tan \left (f x + e\right ) + 6 \, \sqrt{d \tan \left (f x + e\right )} a^{2} d^{3} f^{2}\right )}}{3 \, d^{3} f^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*tan(f*x+e))^(1/2)*(a+a*tan(f*x+e))^2,x, algorithm="giac")

[Out]

-sqrt(2)*a^2*sqrt(abs(d))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) + 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/f -
sqrt(2)*a^2*sqrt(abs(d))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(d)) - 2*sqrt(d*tan(f*x + e)))/sqrt(abs(d)))/f
- 1/2*sqrt(2)*a^2*sqrt(abs(d))*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/f + 1/
2*sqrt(2)*a^2*sqrt(abs(d))*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(abs(d)) + abs(d))/f + 2/3*(s
qrt(d*tan(f*x + e))*a^2*d^3*f^2*tan(f*x + e) + 6*sqrt(d*tan(f*x + e))*a^2*d^3*f^2)/(d^3*f^3)