3.113 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\)

Optimal. Leaf size=133 \[ -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f} \]

[Out]

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

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Rubi [A]  time = 0.22, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3630, 3539, 3537, 63, 208} \[ -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}}-\frac {(B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((I*A + B - I*C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/(Sqrt[c - I*d]*f)) - ((B - I*(A - C))*ArcT
anh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/(Sqrt[c + I*d]*f) + (2*C*Sqrt[c + d*Tan[e + f*x]])/(d*f)

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 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 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 3630

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

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 129, normalized size = 0.97 \[ \frac {-\frac {i (A-i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}+\frac {i (A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{d}}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(((-I)*(A - I*B - C)*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]])/Sqrt[c - I*d] + (I*(A + I*B - C)*ArcTanh
[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/Sqrt[c + I*d] + (2*C*Sqrt[c + d*Tan[e + f*x]])/d)/f

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.37, size = 5570, normalized size = 41.88 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A}{\sqrt {d \tan \left (f x + e\right ) + c}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)/sqrt(d*tan(f*x + e) + c), x)

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mupad [B]  time = 14.21, size = 4326, normalized size = 32.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atanh((32*C^2*d^2*((-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(
1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^4) - (4*C*d^3*f^2*(-16*C^4*d^2*f^4)^(1/2
))/(c^2*f^5 + d^2*f^5)) + (8*c*d^2*((-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4
+ d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*C^4*d^2*f^4)^(1/2))/((16*C^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4
) - (4*C*d^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*C^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*
C*c^2*d^3*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*C^2*c^2*d^2*f^2*((-16*C^4*d^2*f^4)^(1/2)/(16
*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^5*
f^5)/(c^2*f^4 + d^2*f^4) - (4*C*d^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*C^3*c^3*d^3*f^5)/(c
^2*f^4 + d^2*f^4) - (4*C*c^2*d^3*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*((-16*C^4*d^2*f^4)^(1/2)/(
16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((8*c*d^2*(- (-16*C^4*d^2*f^4)^(
1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*C^4
*d^2*f^4)^(1/2))/((16*C^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*d^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^
2*f^5) + (16*C^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*c^2*d^3*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f
^5)) - (32*C^2*d^2*(- (-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^
(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^4) + (4*C*d^3*f^2*(-16*C^4*d^2*f^4)^(1/
2))/(c^2*f^5 + d^2*f^5)) + (32*C^2*c^2*d^2*f^2*(- (-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*c*f^
2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*C^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*d
^5*f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*C^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) + (4*C*c^2*d^3*
f^4*(-16*C^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*(- (-16*C^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (C^2*
c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((32*A^2*d^2*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4))
- (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*A^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^
4) - (4*A*d^3*f^2*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (8*c*d^2*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f
^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*A^4*d^2*f^4)^(1/2)
)/((16*A^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*d^5*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*A^
3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*c^2*d^3*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*A^2*
c^2*d^2*f^2*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c
+ d*tan(e + f*x))^(1/2))/((16*A^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*d^5*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*
f^5 + d^2*f^5) + (16*A^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*A*c^2*d^3*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5
+ d^2*f^5)))*((-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) +
2*atanh((8*c*d^2*(- (-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(
1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*A^4*d^2*f^4)^(1/2))/((16*A^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*A*d^5*f^
4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*A^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) + (4*A*c^2*d^3*f^4*(
-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*A^2*d^2*(- (-16*A^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4))
- (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*A^3*c*d^3*f^3)/(c^2*f^4 + d^2*f
^4) + (4*A*d^3*f^2*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (32*A^2*c^2*d^2*f^2*(- (-16*A^4*d^2*f^4)^(1
/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*A^3
*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*A*d^5*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*A^3*c^3*d^3*
f^5)/(c^2*f^4 + d^2*f^4) + (4*A*c^2*d^3*f^4*(-16*A^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*(- (-16*A^4*d^2*f^4
)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (A^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((32*B^2*d^2*((B^2*c*f^
2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2
))/((16*B^3*d^2)/f - (16*B^3*c^2*d^2*f^3)/(c^2*f^4 + d^2*f^4) + (4*B*c*d^2*f^2*(-16*B^4*d^2*f^4)^(1/2))/(c^2*f
^5 + d^2*f^5)) + (8*c*d^2*((B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^
4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*B^4*d^2*f^4)^(1/2))/(16*B^3*d^4*f + 16*B^3*c^2*d^2*f - (16*B^3*c^2*
d^4*f^5)/(c^2*f^4 + d^2*f^4) - (16*B^3*c^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*B*c*d^4*f^4*(-16*B^4*d^2*f^4)^(1/
2))/(c^2*f^5 + d^2*f^5) + (4*B*c^3*d^2*f^4*(-16*B^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*B^2*c^2*d^2*f^2
*((B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e +
f*x))^(1/2))/(16*B^3*d^4*f + 16*B^3*c^2*d^2*f - (16*B^3*c^2*d^4*f^5)/(c^2*f^4 + d^2*f^4) - (16*B^3*c^4*d^2*f^
5)/(c^2*f^4 + d^2*f^4) + (4*B*c*d^4*f^4*(-16*B^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (4*B*c^3*d^2*f^4*(-16*B
^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*((B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*B^4*d^2*f^4)^(1/2)/(16*(c
^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((8*c*d^2*((-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) + (B^2*c*f^2)/(
4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-16*B^4*d^2*f^4)^(1/2))/((16*B^3*c^2*d^4*f^5)/(c^2*f
^4 + d^2*f^4) - 16*B^3*c^2*d^2*f - 16*B^3*d^4*f + (16*B^3*c^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*B*c*d^4*f^4*(-
16*B^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (4*B*c^3*d^2*f^4*(-16*B^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) -
(32*B^2*d^2*((-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) + (B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c
+ d*tan(e + f*x))^(1/2))/((16*B^3*c^2*d^2*f^3)/(c^2*f^4 + d^2*f^4) - (16*B^3*d^2)/f + (4*B*c*d^2*f^2*(-16*B^4*
d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (32*B^2*c^2*d^2*f^2*((-16*B^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) +
(B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*B^3*c^2*d^4*f^5)/(c^2*f^4 + d^2*f
^4) - 16*B^3*c^2*d^2*f - 16*B^3*d^4*f + (16*B^3*c^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*B*c*d^4*f^4*(-16*B^4*d^2
*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (4*B*c^3*d^2*f^4*(-16*B^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*((-16*B^4*d
^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) + (B^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) + (2*C*(c + d*tan(e + f*x)
)^(1/2))/(d*f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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