3.1239 \(\int \frac{(c+d \tan (e+f x))^{3/2}}{(a+b \tan (e+f x))^2} \, dx\)

Optimal. Leaf size=239 \[ -\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac{\sqrt{b c-a d} \left (a^2 (-d)+4 a b c+3 b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\sqrt{b} f \left (a^2+b^2\right )^2}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (a-i b)^2}+\frac{i (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (a+i b)^2} \]

[Out]

((-I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*(c + I*d)^(3/2)*Ar
cTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - (Sqrt[b*c - a*d]*(4*a*b*c - a^2*d + 3*b^2*d)*
ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)^2*f) - ((b*c - a*d)*Sqrt[c +
d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))

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Rubi [A]  time = 0.922341, antiderivative size = 239, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {3567, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac{(b c-a d) \sqrt{c+d \tan (e+f x)}}{f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac{\sqrt{b c-a d} \left (a^2 (-d)+4 a b c+3 b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )}{\sqrt{b} f \left (a^2+b^2\right )^2}-\frac{i (c-i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )}{f (a-i b)^2}+\frac{i (c+i d)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )}{f (a+i b)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I)*(c - I*d)^(3/2)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/((a - I*b)^2*f) + (I*(c + I*d)^(3/2)*Ar
cTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)^2*f) - (Sqrt[b*c - a*d]*(4*a*b*c - a^2*d + 3*b^2*d)*
ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*(a^2 + b^2)^2*f) - ((b*c - a*d)*Sqrt[c +
d*Tan[e + f*x]])/((a^2 + b^2)*f*(a + b*Tan[e + f*x]))

Rule 3567

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

Rule 3653

Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (
f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/(a^2 + b^2), Int[(c + d*Tan[e + f*
x])^n*Simp[b*B + a*(A - C) + (a*B - b*(A - C))*Tan[e + f*x], x], x], x] + Dist[(A*b^2 - a*b*B + a^2*C)/(a^2 +
b^2), Int[((c + d*Tan[e + f*x])^n*(1 + Tan[e + f*x]^2))/(a + b*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e,
f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] &&  !GtQ[n, 0] &&  !LeQ[n, -
1]

Rule 3539

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

Rule 3537

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 3634

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*
tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Dist[A/f, Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x]
/; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]

Rubi steps

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

Mathematica [A]  time = 3.05619, size = 316, normalized size = 1.32 \[ \frac{-\frac{4 \left (\frac{3}{4} b^{3/2} (b c-a d)^{3/2} \left (a^2 (-d)+4 a b c+3 b^2 d\right ) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d \tan (e+f x)}}{\sqrt{b c-a d}}\right )+\frac{3}{4} i b^2 (a+i b)^2 (c-i d)^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c-i d}}\right )-\frac{3}{4} i b^2 (a-i b)^2 (c+i d)^{3/2} (b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c+d \tan (e+f x)}}{\sqrt{c+i d}}\right )\right )}{b^2 \left (a^2+b^2\right )}-\frac{3 b^2 (c+d \tan (e+f x))^{5/2}}{a+b \tan (e+f x)}+3 d (b c-a d) \sqrt{c+d \tan (e+f x)}+3 b d (c+d \tan (e+f x))^{3/2}}{3 f \left (a^2+b^2\right ) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*(((3*I)/4)*(a + I*b)^2*b^2*(c - I*d)^(3/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]] -
((3*I)/4)*(a - I*b)^2*b^2*(c + I*d)^(3/2)*(b*c - a*d)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]] + (3*b^(
3/2)*(b*c - a*d)^(3/2)*(4*a*b*c - a^2*d + 3*b^2*d)*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]]
)/4))/(b^2*(a^2 + b^2)) + 3*d*(b*c - a*d)*Sqrt[c + d*Tan[e + f*x]] + 3*b*d*(c + d*Tan[e + f*x])^(3/2) - (3*b^2
*(c + d*Tan[e + f*x])^(5/2))/(a + b*Tan[e + f*x]))/(3*(a^2 + b^2)*(b*c - a*d)*f)

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Maple [B]  time = 0.081, size = 3224, normalized size = 13.5 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4/f*d/(a^2+b^2)^2*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(
2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2+1/4/f*d/(a^2+b^2)^2*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2+1/4/f*d/(a^2+b^2)^2*ln((c+d*tan(f*x+e))^(1/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2+1/4/f/d/(a^2+
b^2)^2*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a^2*c-1/4/f/d/(a^2+b^2)^2*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b^2*c-1/4/f/d/(a^2+b^2)^2*ln(
d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(
1/2)*(c^2+d^2)^(1/2)*a^2*c+1/f*d^2/(a^2+b^2)^2*(c+d*tan(f*x+e))^(1/2)/(tan(f*x+e)*b*d+a*d)*a^3+1/f*d^2/(a^2+b^
2)^2/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^3+1/4/f/d/(a^2+b^2)^2*ln(d*tan
(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
(c^2+d^2)^(1/2)*b^2*c-2/f/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c
+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a*b*c+2/f/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-
2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2
+d^2)^(1/2)*a*b*c-1/f*d/(a^2+b^2)^2*(c+d*tan(f*x+e))^(1/2)/(tan(f*x+e)*b*d+a*d)*a^2*b*c-5/f*d/(a^2+b^2)^2/((a*
d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*a^2*b*c+1/f/(a^2+b^2)^2*ln(d*tan(f*x+e)+c
+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c-1/f
*d/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(
2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*b^2-2/f*d/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(
c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a^2*c+2/f*d/(a^2+b^2)^2/(2*
(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-
2*c)^(1/2))*b^2*c+2/f*d^2/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d
^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b-2/f*d^2/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arc
tan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b+1/4/f/d/(a^2+b
^2)^2*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*a^2*c^2-1/4/f/d/(a^2+b^2)^2*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/
2)+(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2-1/4/f*d/(a^2+b^2)^2*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^
2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2-2/f*d/(a^2+b^2)^2/(2
*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)
-2*c)^(1/2))*b^2*c-1/f*d/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^
2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+d^2)^(1/2)*a^2+1/f*d/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2
*c)^(1/2)*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*(c^2+
d^2)^(1/2)*b^2-1/4/f/d/(a^2+b^2)^2*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2
+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c^2+1/4/f/d/(a^2+b^2)^2*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^
(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c^2+1/f*d/(a^2+b^2)^2/(2*(c
^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*
c)^(1/2))*(c^2+d^2)^(1/2)*a^2+4/f/(a^2+b^2)^2/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b
)^(1/2))*a*b^2*c^2+1/2/f/(a^2+b^2)^2*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c
^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b-2/f/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)
*arctan((2*(c+d*tan(f*x+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b*c^2-1/2/f/
(a^2+b^2)^2*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))*(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*b+2/f*d/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+d*tan(f*x
+e))^(1/2)+(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a^2*c+1/f*d^2/(a^2+b^2)^2*(c+d*tan(f*
x+e))^(1/2)/(tan(f*x+e)*b*d+a*d)*b^2*a-1/f*d/(a^2+b^2)^2*(c+d*tan(f*x+e))^(1/2)/(tan(f*x+e)*b*d+a*d)*b^3*c-3/f
*d^2/(a^2+b^2)^2/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*b^2*a+3/f*d/(a^2+b^2
)^2/((a*d-b*c)*b)^(1/2)*arctan((c+d*tan(f*x+e))^(1/2)*b/((a*d-b*c)*b)^(1/2))*b^3*c-1/f/(a^2+b^2)^2*ln((c+d*tan
(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c^2+d^2)^(1/2))*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b
*c+2/f/(a^2+b^2)^2/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2
))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2))*a*b*c^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((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^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((c+d*tan(f*x+e))^(3/2)/(a+b*tan(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [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((c+d*tan(f*x+e))**(3/2)/(a+b*tan(f*x+e))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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