3.1.95 \(\int \frac {\sqrt {c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{(a+b \tan (e+f x))^2} \, dx\) [95]

Optimal. Leaf size=317 \[ -\frac {(i A+B-i C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}-\frac {(B-i (A-C)) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\left (a^3 b B d+a^4 C d+b^4 (2 B c+A d)+a b^3 (4 A c-4 c C-3 B d)-a^2 b^2 (2 B c+3 A d-5 C d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))} \]

[Out]

-(I*A+B-I*C)*arctanh((c+d*tan(f*x+e))^(1/2)/(c-I*d)^(1/2))*(c-I*d)^(1/2)/(a-I*b)^2/f-(B-I*(A-C))*arctanh((c+d*
tan(f*x+e))^(1/2)/(c+I*d)^(1/2))*(c+I*d)^(1/2)/(a+I*b)^2/f-(a^3*b*B*d+a^4*C*d+b^4*(A*d+2*B*c)+a*b^3*(4*A*c-3*B
*d-4*C*c)-a^2*b^2*(3*A*d+2*B*c-5*C*d))*arctanh(b^(1/2)*(c+d*tan(f*x+e))^(1/2)/(-a*d+b*c)^(1/2))/b^(3/2)/(a^2+b
^2)^2/f/(-a*d+b*c)^(1/2)-(A*b^2-a*(B*b-C*a))*(c+d*tan(f*x+e))^(1/2)/b/(a^2+b^2)/f/(a+b*tan(f*x+e))

________________________________________________________________________________________

Rubi [A]
time = 0.98, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.149, Rules used = {3726, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b f \left (a^2+b^2\right ) (a+b \tan (e+f x))}-\frac {\left (a^4 C d+a^3 b B d-a^2 b^2 (3 A d+2 B c-5 C d)+a b^3 (4 A c-3 B d-4 c C)+b^4 (A d+2 B c)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} f \left (a^2+b^2\right )^2 \sqrt {b c-a d}}-\frac {\sqrt {c-i d} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (a-i b)^2}-\frac {\sqrt {c+i d} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (a+i b)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]

[Out]

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

Rule 65

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 214

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

Rule 3618

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c*(
d/f), Subst[Int[(a + (b/d)*x)^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 3620

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 3715

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]

Rule 3726

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

Rule 3734

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {c+d \tan (e+f x)} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{(a+b \tan (e+f x))^2} \, dx &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\int \frac {\frac {1}{2} \left (2 (b B-a C) \left (b c-\frac {a d}{2}\right )+2 A b \left (a c+\frac {b d}{2}\right )\right )-b ((A-C) (b c-a d)-B (a c+b d)) \tan (e+f x)-\frac {1}{2} \left (A b^2-a b B-a^2 C-2 b^2 C\right ) d \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {\int \frac {b \left (a^2 (A c-c C-B d)-b^2 (A c-c C-B d)+2 a b (B c+(A-C) d)\right )-b \left (2 a b (A c-c C-B d)-a^2 (B c+(A-C) d)+b^2 (B c+(A-C) d)\right ) \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{b \left (a^2+b^2\right )^2}+\frac {\left (a^3 b B d+a^4 C d+b^4 (2 B c+A d)+a b^3 (4 A c-4 c C-3 B d)-a^2 b^2 (2 B c+3 A d-5 C d)\right ) \int \frac {1+\tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx}{2 b \left (a^2+b^2\right )^2}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}+\frac {((A-i B-C) (c-i d)) \int \frac {1+i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a-i b)^2}+\frac {((A+i B-C) (c+i d)) \int \frac {1-i \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx}{2 (a+i b)^2}+\frac {\left (a^3 b B d+a^4 C d+b^4 (2 B c+A d)+a b^3 (4 A c-4 c C-3 B d)-a^2 b^2 (2 B c+3 A d-5 C d)\right ) \text {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 b \left (a^2+b^2\right )^2 f}\\ &=-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {(i (A+i B-C) (c+i d)) \text {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 {((A-i B-C) (i c+d)) \text {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 (a^3 b B d+a^4 C d+b^4 (2 B c+A d)+a b^3 (4 A c-4 c C-3 B d)-a^2 b^2 (2 B c+3 A d-5 C d)\right ) \text {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 )}{b \left (a^2+b^2\right )^2 d f}\\ &=-\frac {\left (a^3 b B d+a^4 C d+b^4 (2 B c+A d)+a b^3 (4 A c-4 c C-3 B d)-a^2 b^2 (2 B c+3 A d-5 C d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}-\frac {((A+i B-C) (c+i d)) \text {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 A+B-i C) (i c+d)) \text {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 {(B+i (A-C)) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b)^2 f}-\frac {(B-i (A-C)) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b)^2 f}-\frac {\left (a^3 b B d+a^4 C d+b^4 (2 B c+A d)+a b^3 (4 A c-4 c C-3 B d)-a^2 b^2 (2 B c+3 A d-5 C d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right )^2 \sqrt {b c-a d} f}-\frac {\left (A b^2-a (b B-a C)\right ) \sqrt {c+d \tan (e+f x)}}{b \left (a^2+b^2\right ) f (a+b \tan (e+f x))}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 4.11, size = 362, normalized size = 1.14 \begin {gather*} \frac {2 \left (\frac {\frac {i b \left (-(a+i b)^2 (A-i B-C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+(a-i b)^2 (A+i B-C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )\right )}{2 f}-\frac {\left (a^3 b B d+a^4 C d+b^4 (2 B c+A d)+a b^3 (4 A c-4 c C-3 B d)+a^2 b^2 (-2 B c-3 A d+5 C d)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{2 \sqrt {b} \sqrt {b c-a d} f}}{\left (a^2+b^2\right )^2}-\frac {C \sqrt {c+d \tan (e+f x)}}{f (a+b \tan (e+f x))}+\frac {\left (-A b^2+a b B+a^2 C+2 b^2 C\right ) \sqrt {c+d \tan (e+f x)}}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))}\right )}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + d*Tan[e + f*x]]*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/(a + b*Tan[e + f*x])^2,x]

[Out]

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

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2142\) vs. \(2(284)=568\).
time = 0.58, size = 2143, normalized size = 6.76

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2143\)
default	\(\text {Expression too large to display}\)	\(2143\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c+d*tan(f*x+e))^(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

2/f*d*(1/d/(a^2+b^2)^2*(1/4/d*(-1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2+A*(c^2+d^2)^(1/2)*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d-
A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c-2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b-B*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*a^2*d+2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d+C*(c^2+d^2)^
(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2-C*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*a^2*c-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c)*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*(-4*A*(c^2+d^2)^(1/2)*a*b*d+2*B
*(c^2+d^2)^(1/2)*a^2*d-2*B*(c^2+d^2)^(1/2)*b^2*d+4*C*(c^2+d^2)^(1/2)*a*b*d+1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*a^2+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2
*c+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c-2*B*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*a*b-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d+2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c+B*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d
^2)^(1/2)+2*c)^(1/2)*b^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d+C*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(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)))+1/4/d*(1/2*(-A*(c^2+d^2)^(1/2)*
(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2+A*(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*a^2*c+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c-2*B*(c^2+d^2)^(1/2
)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*d+2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
b*c+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d+C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2-C*(c^2+d^2)^(1/2
)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
b*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c)*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*(4*A*(c^2+d^2)^(1/2)*a*b*d-2*B*(c^2+d^2)^(1/2)*a^2*d+2*B*(c^2+d^2)^(1/2)*b^2*d-4*C*(c^2
+d^2)^(1/2)*a*b*d-1/2*(-A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2+A*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/
2)+2*c)^(1/2)*b^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a^2*c+2*A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d-A*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b^2*c-2*B*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a^2*d+2*B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*c+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*d+C*(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*a^2-C*(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*
a^2*c-2*C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*b*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b^2*c)*(2*(c^2+d^2)^(1/2)+2*c)^(
1/2))/(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))))-1/d/(a^2+b^2)^2*(1/2*d*(A*a^2*b^2+A*b^4-B*a^3*b-B*a*b^3+C*a^4+C*a^2*b^2)/b*(c+d*tan(f*x
+e))^(1/2)/(b*(c+d*tan(f*x+e))+a*d-b*c)+1/2*(3*A*a^2*b^2*d-4*A*a*b^3*c-A*b^4*d-B*a^3*b*d+2*B*a^2*b^2*c+3*B*a*b
^3*d-2*B*b^4*c-C*a^4*d-5*C*a^2*b^2*d+4*C*a*b^3*c)/b/((a*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e))^(1/2)/((a*d-
b*c)*b)^(1/2))))

________________________________________________________________________________________

Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c + d \tan {\left (e + f x \right )}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\left (a + b \tan {\left (e + f x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c+d*tan(f*x+e))**(1/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*tan(f*x+e))**2,x)

[Out]

Integral(sqrt(c + d*tan(e + f*x))*(A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(a + b*tan(e + f*x))**2, x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

________________________________________________________________________________________

Mupad [B]
time = 45.42, size = 2500, normalized size = 7.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c + d*tan(e + f*x))^(1/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/(a + b*tan(e + f*x))^2,x)

[Out]

atan(((((8*(156*B^3*a^2*b^9*d^12*f^2 - 16*B^3*a^4*b^7*d^12*f^2 - 120*B^3*a^6*b^5*d^12*f^2 + 48*B^3*a^8*b^3*d^1
2*f^2 + 12*B^3*b^11*c^2*d^10*f^2 + 12*B^3*b^11*c^4*d^8*f^2 - 4*B^3*a^10*b*d^12*f^2 - 124*B^3*a*b^10*c*d^11*f^2
- 124*B^3*a*b^10*c^3*d^9*f^2 + 224*B^3*a^3*b^8*c*d^11*f^2 + 200*B^3*a^5*b^6*c*d^11*f^2 - 128*B^3*a^7*b^4*c*d^
11*f^2 + 20*B^3*a^9*b^2*c*d^11*f^2 - 4*B^3*a^10*b*c^2*d^10*f^2 + 44*B^3*a^2*b^9*c^2*d^10*f^2 - 112*B^3*a^2*b^9
*c^4*d^8*f^2 + 224*B^3*a^3*b^8*c^3*d^9*f^2 - 40*B^3*a^4*b^7*c^2*d^10*f^2 - 24*B^3*a^4*b^7*c^4*d^8*f^2 + 200*B^
3*a^5*b^6*c^3*d^9*f^2 - 40*B^3*a^6*b^5*c^2*d^10*f^2 + 80*B^3*a^6*b^5*c^4*d^8*f^2 - 128*B^3*a^7*b^4*c^3*d^9*f^2
+ 28*B^3*a^8*b^3*c^2*d^10*f^2 - 20*B^3*a^8*b^3*c^4*d^8*f^2 + 20*B^3*a^9*b^2*c^3*d^9*f^2))/(a^8*f^5 + b^8*f^5
+ 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) + (((8*(80*B*a*b^14*d^11*f^4 - 48*B*b^15*c*d^10*f^4 + 384*B*a
^3*b^12*d^11*f^4 + 720*B*a^5*b^10*d^11*f^4 + 640*B*a^7*b^8*d^11*f^4 + 240*B*a^9*b^6*d^11*f^4 - 16*B*a^13*b^2*d
^11*f^4 - 48*B*b^15*c^3*d^8*f^4 + 80*B*a*b^14*c^2*d^9*f^4 - 224*B*a^2*b^13*c*d^10*f^4 - 400*B*a^4*b^11*c*d^10*
f^4 - 320*B*a^6*b^9*c*d^10*f^4 - 80*B*a^8*b^7*c*d^10*f^4 + 32*B*a^10*b^5*c*d^10*f^4 + 16*B*a^12*b^3*c*d^10*f^4
- 224*B*a^2*b^13*c^3*d^8*f^4 + 384*B*a^3*b^12*c^2*d^9*f^4 - 400*B*a^4*b^11*c^3*d^8*f^4 + 720*B*a^5*b^10*c^2*d
^9*f^4 - 320*B*a^6*b^9*c^3*d^8*f^4 + 640*B*a^7*b^8*c^2*d^9*f^4 - 80*B*a^8*b^7*c^3*d^8*f^4 + 240*B*a^9*b^6*c^2*
d^9*f^4 + 32*B*a^10*b^5*c^3*d^8*f^4 + 16*B*a^12*b^3*c^3*d^8*f^4 - 16*B*a^13*b^2*c^2*d^9*f^4))/(a^8*f^5 + b^8*f
^5 + 4*a^2*b^6*f^5 + 6*a^4*b^4*f^5 + 4*a^6*b^2*f^5) - (16*(c + d*tan(e + f*x))^(1/2)*(-(((8*B^2*a^4*c*f^2 + 8*
B^2*b^4*c*f^2 - 32*B^2*a*b^3*d*f^2 + 32*B^2*a^3*b*d*f^2 - 48*B^2*a^2*b^2*c*f^2)^2/4 - (B^4*c^2 + B^4*d^2)*(16*
a^8*f^4 + 16*b^8*f^4 + 64*a^2*b^6*f^4 + 96*a^4*b^4*f^4 + 64*a^6*b^2*f^4))^(1/2) - 4*B^2*a^4*c*f^2 - 4*B^2*b^4*
c*f^2 + 16*B^2*a*b^3*d*f^2 - 16*B^2*a^3*b*d*f^2 + 24*B^2*a^2*b^2*c*f^2)/(16*(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4
+ 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)))^(1/2)*(32*b^17*d^10*f^4 + 160*a^2*b^15*d^10*f^4 + 288*a^4*b^13*d^10*f^4 +
160*a^6*b^11*d^10*f^4 - 160*a^8*b^9*d^10*f^4 - 288*a^10*b^7*d^10*f^4 - 160*a^12*b^5*d^10*f^4 - 32*a^14*b^3*d^1
0*f^4 + 48*b^17*c^2*d^8*f^4 + 272*a^2*b^15*c^2*d^8*f^4 + 624*a^4*b^13*c^2*d^8*f^4 + 720*a^6*b^11*c^2*d^8*f^4 +
400*a^8*b^9*c^2*d^8*f^4 + 48*a^10*b^7*c^2*d^8*f^4 - 48*a^12*b^5*c^2*d^8*f^4 - 16*a^14*b^3*c^2*d^8*f^4 + 16*a*
b^16*c*d^9*f^4 + 112*a^3*b^14*c*d^9*f^4 + 336*a^5*b^12*c*d^9*f^4 + 560*a^7*b^10*c*d^9*f^4 + 560*a^9*b^8*c*d^9*
f^4 + 336*a^11*b^6*c*d^9*f^4 + 112*a^13*b^4*c*d^9*f^4 + 16*a^15*b^2*c*d^9*f^4))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6
*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(((8*B^2*a^4*c*f^2 + 8*B^2*b^4*c*f^2 - 32*B^2*a*b^3*d*f^2 + 32*B^2*a^
3*b*d*f^2 - 48*B^2*a^2*b^2*c*f^2)^2/4 - (B^4*c^2 + B^4*d^2)*(16*a^8*f^4 + 16*b^8*f^4 + 64*a^2*b^6*f^4 + 96*a^4
*b^4*f^4 + 64*a^6*b^2*f^4))^(1/2) - 4*B^2*a^4*c*f^2 - 4*B^2*b^4*c*f^2 + 16*B^2*a*b^3*d*f^2 - 16*B^2*a^3*b*d*f^
2 + 24*B^2*a^2*b^2*c*f^2)/(16*(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)))^(1/2) - (1
6*(c + d*tan(e + f*x))^(1/2)*(44*B^2*a^9*b^4*d^11*f^2 - 168*B^2*a^5*b^8*d^11*f^2 - 40*B^2*a^7*b^6*d^11*f^2 - 2
0*B^2*a^3*b^10*d^11*f^2 - 4*B^2*a^11*b^2*d^11*f^2 - 36*B^2*b^13*c^3*d^8*f^2 + 60*B^2*a*b^12*d^11*f^2 - 12*B^2*
b^13*c*d^10*f^2 + 4*B^2*a^12*b*c*d^10*f^2 + 100*B^2*a*b^12*c^2*d^9*f^2 + 120*B^2*a^2*b^11*c*d^10*f^2 + 156*B^2
*a^4*b^9*c*d^10*f^2 - 112*B^2*a^6*b^7*c*d^10*f^2 - 148*B^2*a^8*b^5*c*d^10*f^2 - 8*B^2*a^10*b^3*c*d^10*f^2 + 68
*B^2*a^2*b^11*c^3*d^8*f^2 + 124*B^2*a^3*b^10*c^2*d^9*f^2 + 184*B^2*a^4*b^9*c^3*d^8*f^2 + 8*B^2*a^5*b^8*c^2*d^9
*f^2 + 40*B^2*a^6*b^7*c^3*d^8*f^2 + 24*B^2*a^7*b^6*c^2*d^9*f^2 - 20*B^2*a^8*b^5*c^3*d^8*f^2 + 20*B^2*a^9*b^4*c
^2*d^9*f^2 + 20*B^2*a^10*b^3*c^3*d^8*f^2 - 20*B^2*a^11*b^2*c^2*d^9*f^2))/(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 +
6*a^4*b^4*f^4 + 4*a^6*b^2*f^4))*(-(((8*B^2*a^4*c*f^2 + 8*B^2*b^4*c*f^2 - 32*B^2*a*b^3*d*f^2 + 32*B^2*a^3*b*d*f
^2 - 48*B^2*a^2*b^2*c*f^2)^2/4 - (B^4*c^2 + B^4*d^2)*(16*a^8*f^4 + 16*b^8*f^4 + 64*a^2*b^6*f^4 + 96*a^4*b^4*f^
4 + 64*a^6*b^2*f^4))^(1/2) - 4*B^2*a^4*c*f^2 - 4*B^2*b^4*c*f^2 + 16*B^2*a*b^3*d*f^2 - 16*B^2*a^3*b*d*f^2 + 24*
B^2*a^2*b^2*c*f^2)/(16*(a^8*f^4 + b^8*f^4 + 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)))^(1/2))*(-(((8*B^2
*a^4*c*f^2 + 8*B^2*b^4*c*f^2 - 32*B^2*a*b^3*d*f^2 + 32*B^2*a^3*b*d*f^2 - 48*B^2*a^2*b^2*c*f^2)^2/4 - (B^4*c^2
+ B^4*d^2)*(16*a^8*f^4 + 16*b^8*f^4 + 64*a^2*b^6*f^4 + 96*a^4*b^4*f^4 + 64*a^6*b^2*f^4))^(1/2) - 4*B^2*a^4*c*f
^2 - 4*B^2*b^4*c*f^2 + 16*B^2*a*b^3*d*f^2 - 16*B^2*a^3*b*d*f^2 + 24*B^2*a^2*b^2*c*f^2)/(16*(a^8*f^4 + b^8*f^4
+ 4*a^2*b^6*f^4 + 6*a^4*b^4*f^4 + 4*a^6*b^2*f^4)))^(1/2) - (16*(c + d*tan(e + f*x))^(1/2)*(2*B^4*b^9*d^12 - 5*
B^4*a^2*b^7*d^12 + 17*B^4*a^4*b^5*d^12 - 7*B^4*a^6*b^3*d^12 + 6*B^4*b^9*c^4*d^8 + B^4*a^8*b*d^12 + 77*B^4*a^2*
b^7*c^2*d^10 - 8*B^4*a^2*b^7*c^4*d^8 + 60*B^4*a^3*b^6*c^3*d^9 - 87*B^4*a^4*b^5*c^2*d^10 + 14*B^4*a^4*b^5*c^4*d
^8 - 36*B^4*a^5*b^4*c^3*d^9 + 27*B^4*a^6*b^3*c^...

________________________________________________________________________________________