3.22 \(\int \frac{\tan ^2(d+e x)}{(a+b \tan (d+e x)+c \tan ^2(d+e x))^{3/2}} \, dx\)

Optimal. Leaf size=638 \[ -\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tan ^{-1}\left (\frac{\left (b^2-(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )\right ) \tan (d+e x)+b \left (\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right )}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tan ^{-1}\left (\frac{\left (b^2-(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )\right ) \tan (d+e x)+b \left (-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right )}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}-\frac{2 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{e \left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \]

[Out]

-((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a
*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*
a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c
^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2
- 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a
- c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a
- c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*
Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]
^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/
((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])

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Rubi [A]  time = 3.95026, antiderivative size = 638, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {3700, 1065, 1036, 1030, 205} \[ -\frac{\sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tan ^{-1}\left (\frac{\left (b^2-(a-c) \left (-\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )\right ) \tan (d+e x)+b \left (\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right )}{\sqrt{2} \sqrt{\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{-(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}+\frac{\sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \tan ^{-1}\left (\frac{\left (b^2-(a-c) \left (\sqrt{a^2-2 a c+b^2+c^2}+a-c\right )\right ) \tan (d+e x)+b \left (-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c\right )}{\sqrt{2} \sqrt{-\sqrt{a^2-2 a c+b^2+c^2}+2 a-2 c} \sqrt{(a-c) \sqrt{a^2-2 a c+b^2+c^2}+a^2-2 a c-b^2+c^2} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{2} e \left (a^2-2 a c+b^2+c^2\right )^{3/2}}-\frac{2 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{e \left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right ) \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}} \]

Antiderivative was successfully verified.

[In]

Int[Tan[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]

[Out]

-((Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a
*c + c^2]]*ArcTan[(b*(2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a - c - Sqrt[a^2 + b^2 - 2*
a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c + Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c
^2 - (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/(Sqrt[2]*(a^2 + b^2
- 2*a*c + c^2)^(3/2)*e)) + (Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a
- c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*ArcTan[(b*(2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]) + (b^2 - (a - c)*(a
- c + Sqrt[a^2 + b^2 - 2*a*c + c^2]))*Tan[d + e*x])/(Sqrt[2]*Sqrt[2*a - 2*c - Sqrt[a^2 + b^2 - 2*a*c + c^2]]*
Sqrt[a^2 - b^2 - 2*a*c + c^2 + (a - c)*Sqrt[a^2 + b^2 - 2*a*c + c^2]]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]
^2])])/(Sqrt[2]*(a^2 + b^2 - 2*a*c + c^2)^(3/2)*e) - (2*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c)*Tan[d + e*x]))/
((b^2 + (a - c)^2)*(b^2 - 4*a*c)*e*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])

Rule 3700

Int[tan[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*((f_.)*tan[(d_.) + (e_.)*(x_)])^(n_.) + (c_.)*((f_.)*tan[(d_.
) + (e_.)*(x_)])^(n2_.))^(p_), x_Symbol] :> Dist[f/e, Subst[Int[((x/f)^m*(a + b*x^n + c*x^(2*n))^p)/(f^2 + x^2
), x], x, f*Tan[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rule 1065

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_)*((A_.) + (C_.)*(x_)^2)*((d_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Si
mp[((a + b*x + c*x^2)^(p + 1)*(d + f*x^2)^(q + 1)*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b)*(2*c^2*d + b^2*f - c
*(2*a*f)) + c*(A*(2*c^2*d + b^2*f - c*(2*a*f)) + C*(b^2*d - 2*a*(c*d - a*f)))*x))/((b^2 - 4*a*c)*(b^2*d*f + (c
*d - a*f)^2)*(p + 1)), x] + Dist[1/((b^2 - 4*a*c)*(b^2*d*f + (c*d - a*f)^2)*(p + 1)), Int[(a + b*x + c*x^2)^(p
+ 1)*(d + f*x^2)^q*Simp[(-2*A*c - 2*a*C)*((c*d - a*f)^2 - (b*d)*(-(b*f)))*(p + 1) + (b^2*(C*d + A*f) + 2*(A*c
*(c*d - a*f) - a*(c*C*d - a*C*f)))*(a*f*(p + 1) - c*d*(p + 2)) - (2*f*((A*c - a*C)*(-(b*(c*d + a*f))) + (A*b)*
(2*c^2*d + b^2*f - c*(2*a*f)))*(p + q + 2) - (b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(b*f*
(p + 1)))*x - c*f*(b^2*(C*d + A*f) + 2*(A*c*(c*d - a*f) - a*(c*C*d - a*C*f)))*(2*p + 2*q + 5)*x^2, x], x], x]
/; FreeQ[{a, b, c, d, f, A, C, q}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[b^2*d*f + (c*d - a*f)^2, 0]
&&  !( !IntegerQ[p] && ILtQ[q, -1]) &&  !IGtQ[q, 0]

Rule 1036

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[-(a*h*e) - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[-(a*c)]

Rule 1030

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 205

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

Rubi steps

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

Mathematica [C]  time = 4.81214, size = 328, normalized size = 0.51 \[ \frac{\frac{\left (-4 i a^2 c+a \left (i b^2+4 b c+4 i c^2\right )-b^2 (b+i c)\right ) \tanh ^{-1}\left (\frac{2 a+(b-2 i c) \tan (d+e x)-i b}{2 \sqrt{a-i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a-i b-c}}+\frac{i \left (4 a^2 c-a \left (b^2+4 i b c+4 c^2\right )+b^2 (c+i b)\right ) \tanh ^{-1}\left (\frac{2 a+(b+2 i c) \tan (d+e x)+i b}{2 \sqrt{a+i b-c} \sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt{a+i b-c}}-\frac{4 \left (c \left (2 a^2-2 a c+b^2\right ) \tan (d+e x)+a b (a+c)\right )}{\sqrt{a+b \tan (d+e x)+c \tan ^2(d+e x)}}}{2 e \left ((a-c)^2+b^2\right ) \left (b^2-4 a c\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Tan[d + e*x]^2/(a + b*Tan[d + e*x] + c*Tan[d + e*x]^2)^(3/2),x]

[Out]

(((-(b^2*(b + I*c)) - (4*I)*a^2*c + a*(I*b^2 + 4*b*c + (4*I)*c^2))*ArcTanh[(2*a - I*b + (b - (2*I)*c)*Tan[d +
e*x])/(2*Sqrt[a - I*b - c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a - I*b - c] + (I*(4*a^2*c + b^
2*(I*b + c) - a*(b^2 + (4*I)*b*c + 4*c^2))*ArcTanh[(2*a + I*b + (b + (2*I)*c)*Tan[d + e*x])/(2*Sqrt[a + I*b -
c]*Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])])/Sqrt[a + I*b - c] - (4*(a*b*(a + c) + c*(2*a^2 + b^2 - 2*a*c
)*Tan[d + e*x]))/Sqrt[a + b*Tan[d + e*x] + c*Tan[d + e*x]^2])/(2*(b^2 + (a - c)^2)*(b^2 - 4*a*c)*e)

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Maple [B]  time = 0.291, size = 11847956, normalized size = 18570.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x)

[Out]

result too large to display

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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(tan(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan ^{2}{\left (d + e x \right )}}{\left (a + b \tan{\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(e*x+d)**2/(a+b*tan(e*x+d)+c*tan(e*x+d)**2)**(3/2),x)

[Out]

Integral(tan(d + e*x)**2/(a + b*tan(d + e*x) + c*tan(d + e*x)**2)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (e x + d\right )^{2}}{{\left (c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(e*x+d)^2/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x, algorithm="giac")

[Out]

integrate(tan(e*x + d)^2/(c*tan(e*x + d)^2 + b*tan(e*x + d) + a)^(3/2), x)