3.1.94 \(\int \frac {\sqrt {c+d \tan (e+f x)} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{a+b \tan (e+f x)} \, dx\) [94]

Optimal. Leaf size=234 \[ -\frac {(i A+B-i C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b) f}+\frac {(i A-B-i C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(a+i b) f}-\frac {2 \left (A b^2-a (b B-a C)\right ) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{b^{3/2} \left (a^2+b^2\right ) f}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{b f} \]

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

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Rubi [A]
time = 0.72, antiderivative size = 234, 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 = {3728, 3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {2 \sqrt {b c-a d} \left (A b^2-a (b B-a 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 )}-\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)}+\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)}+\frac {2 C \sqrt {c+d \tan (e+f x)}}{b f} \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]),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)*f)) + ((I*A - B -
I*C)*Sqrt[c + I*d]*ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((a + I*b)*f) - (2*(A*b^2 - a*(b*B - a*C)
)*Sqrt[b*c - a*d]*ArcTanh[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(b^(3/2)*(a^2 + b^2)*f) + (2*C*
Sqrt[c + d*Tan[e + f*x]])/(b*f)

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 3728

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*
tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d
*Tan[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Tan[e + f*x])^(m - 1)*(c +
d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C*(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f
*x] - (C*m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, 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[m, 0] &&  !(IGtQ[n, 0] && ( !Intege
rQ[m] || (EqQ[c, 0] && NeQ[a, 0])))

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

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Mathematica [A]
time = 0.46, size = 233, normalized size = 1.00 \begin {gather*} \frac {b^{3/2} (-i a+b) (A-i B-C) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+b^{3/2} (i a+b) (A+i B-C) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )-2 \left (A b^2+a (-b B+a C)\right ) \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )+2 \sqrt {b} \left (a^2+b^2\right ) C \sqrt {c+d \tan (e+f x)}}{b^{3/2} \left (a^2+b^2\right ) f} \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]),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1393\) vs. \(2(200)=400\).
time = 0.62, size = 1394, normalized size = 5.96 Too large to display

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)),x,method=_RETURNVERBOSE)

[Out]

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

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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)),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

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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)),x, algorithm="fricas")

[Out]

Timed out

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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 )}{a + b \tan {\left (e + f x \right )}}\, 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)),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)), x)

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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)),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

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Mupad [B]
time = 36.22, size = 2500, normalized size = 10.68 \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)),x)

[Out]

atan(((((((32*(4*C*a*b^8*d^11*f^4 - 4*C*b^9*c*d^10*f^4 + 8*C*a^3*b^6*d^11*f^4 + 4*C*a^5*b^4*d^11*f^4 - 4*C*b^9
*c^3*d^8*f^4 + 4*C*a*b^8*c^2*d^9*f^4 - 8*C*a^2*b^7*c*d^10*f^4 - 4*C*a^4*b^5*c*d^10*f^4 - 8*C*a^2*b^7*c^3*d^8*f
^4 + 8*C*a^3*b^6*c^2*d^9*f^4 - 4*C*a^4*b^5*c^3*d^8*f^4 + 4*C*a^5*b^4*c^2*d^9*f^4))/(b*f^5) - (32*(c + d*tan(e
+ f*x))^(1/2)*((((8*C^2*a^2*c*f^2 - 8*C^2*b^2*c*f^2 + 16*C^2*a*b*d*f^2)^2/4 - (C^4*c^2 + C^4*d^2)*(16*a^4*f^4
+ 16*b^4*f^4 + 32*a^2*b^2*f^4))^(1/2) - 4*C^2*a^2*c*f^2 + 4*C^2*b^2*c*f^2 - 8*C^2*a*b*d*f^2)/(16*(a^4*f^4 + b^
4*f^4 + 2*a^2*b^2*f^4)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10
*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*
c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4 + 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*((((8*C^2*a^2*c*f^2 -
8*C^2*b^2*c*f^2 + 16*C^2*a*b*d*f^2)^2/4 - (C^4*c^2 + C^4*d^2)*(16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4))^(1/
2) - 4*C^2*a^2*c*f^2 + 4*C^2*b^2*c*f^2 - 8*C^2*a*b*d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2) - (3
2*(c + d*tan(e + f*x))^(1/2)*(14*C^2*a*b^7*d^11*f^2 - 2*C^2*a^5*b^3*d^11*f^2 - 10*C^2*b^8*c^3*d^8*f^2 - 4*C^2*
a^3*b^5*d^11*f^2 - 16*C^2*a^7*b*d^11*f^2 + 8*C^2*a^8*c*d^10*f^2 - 6*C^2*b^8*c*d^10*f^2 + 18*C^2*a*b^7*c^2*d^9*
f^2 + 12*C^2*a^2*b^6*c*d^10*f^2 + 2*C^2*a^4*b^4*c*d^10*f^2 + 24*C^2*a^6*b^2*c*d^10*f^2 - 16*C^2*a^7*b*c^2*d^9*
f^2 + 4*C^2*a^2*b^6*c^3*d^8*f^2 + 4*C^2*a^3*b^5*c^2*d^9*f^2 - 10*C^2*a^4*b^4*c^3*d^8*f^2 + 2*C^2*a^5*b^3*c^2*d
^9*f^2 + 8*C^2*a^6*b^2*c^3*d^8*f^2))/(b*f^4))*((((8*C^2*a^2*c*f^2 - 8*C^2*b^2*c*f^2 + 16*C^2*a*b*d*f^2)^2/4 -
(C^4*c^2 + C^4*d^2)*(16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4))^(1/2) - 4*C^2*a^2*c*f^2 + 4*C^2*b^2*c*f^2 - 8*
C^2*a*b*d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2) + (32*(15*C^3*a^4*b^3*d^12*f^2 - C^3*a^2*b^5*d^
12*f^2 + C^3*b^7*c^2*d^10*f^2 + C^3*b^7*c^4*d^8*f^2 - 12*C^3*a^6*b*d^12*f^2 - 24*C^3*a^3*b^4*c*d^11*f^2 + 24*C
^3*a^5*b^2*c*d^11*f^2 - 12*C^3*a^6*b*c^2*d^10*f^2 + 8*C^3*a^2*b^5*c^2*d^10*f^2 + 9*C^3*a^2*b^5*c^4*d^8*f^2 - 2
4*C^3*a^3*b^4*c^3*d^9*f^2 + 3*C^3*a^4*b^3*c^2*d^10*f^2 - 12*C^3*a^4*b^3*c^4*d^8*f^2 + 24*C^3*a^5*b^2*c^3*d^9*f
^2))/(b*f^5))*((((8*C^2*a^2*c*f^2 - 8*C^2*b^2*c*f^2 + 16*C^2*a*b*d*f^2)^2/4 - (C^4*c^2 + C^4*d^2)*(16*a^4*f^4
+ 16*b^4*f^4 + 32*a^2*b^2*f^4))^(1/2) - 4*C^2*a^2*c*f^2 + 4*C^2*b^2*c*f^2 - 8*C^2*a*b*d*f^2)/(16*(a^4*f^4 + b^
4*f^4 + 2*a^2*b^2*f^4)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(C^4*b^6*d^12 - 2*C^4*a^6*d^12 + 2*C^4*a^6*c^2
*d^10 + 2*C^4*b^6*c^2*d^10 + C^4*b^6*c^4*d^8 - 2*C^4*a^4*b^2*c^2*d^10 + 2*C^4*a^4*b^2*c^4*d^8 + 4*C^4*a^5*b*c*
d^11 - 4*C^4*a^5*b*c^3*d^9))/(b*f^4))*((((8*C^2*a^2*c*f^2 - 8*C^2*b^2*c*f^2 + 16*C^2*a*b*d*f^2)^2/4 - (C^4*c^2
+ C^4*d^2)*(16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4))^(1/2) - 4*C^2*a^2*c*f^2 + 4*C^2*b^2*c*f^2 - 8*C^2*a*b*
d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2)*1i - (((((32*(4*C*a*b^8*d^11*f^4 - 4*C*b^9*c*d^10*f^4 +
8*C*a^3*b^6*d^11*f^4 + 4*C*a^5*b^4*d^11*f^4 - 4*C*b^9*c^3*d^8*f^4 + 4*C*a*b^8*c^2*d^9*f^4 - 8*C*a^2*b^7*c*d^1
0*f^4 - 4*C*a^4*b^5*c*d^10*f^4 - 8*C*a^2*b^7*c^3*d^8*f^4 + 8*C*a^3*b^6*c^2*d^9*f^4 - 4*C*a^4*b^5*c^3*d^8*f^4 +
4*C*a^5*b^4*c^2*d^9*f^4))/(b*f^5) + (32*(c + d*tan(e + f*x))^(1/2)*((((8*C^2*a^2*c*f^2 - 8*C^2*b^2*c*f^2 + 16
*C^2*a*b*d*f^2)^2/4 - (C^4*c^2 + C^4*d^2)*(16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4))^(1/2) - 4*C^2*a^2*c*f^2
+ 4*C^2*b^2*c*f^2 - 8*C^2*a*b*d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2)*(16*b^10*d^10*f^4 + 16*a^
2*b^8*d^10*f^4 - 16*a^4*b^6*d^10*f^4 - 16*a^6*b^4*d^10*f^4 + 24*b^10*c^2*d^8*f^4 + 40*a^2*b^8*c^2*d^8*f^4 + 8*
a^4*b^6*c^2*d^8*f^4 - 8*a^6*b^4*c^2*d^8*f^4 + 8*a*b^9*c*d^9*f^4 + 24*a^3*b^7*c*d^9*f^4 + 24*a^5*b^5*c*d^9*f^4
+ 8*a^7*b^3*c*d^9*f^4))/(b*f^4))*((((8*C^2*a^2*c*f^2 - 8*C^2*b^2*c*f^2 + 16*C^2*a*b*d*f^2)^2/4 - (C^4*c^2 + C^
4*d^2)*(16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^2*f^4))^(1/2) - 4*C^2*a^2*c*f^2 + 4*C^2*b^2*c*f^2 - 8*C^2*a*b*d*f^2
)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(14*C^2*a*b^7*d^11*f^2 - 2*
C^2*a^5*b^3*d^11*f^2 - 10*C^2*b^8*c^3*d^8*f^2 - 4*C^2*a^3*b^5*d^11*f^2 - 16*C^2*a^7*b*d^11*f^2 + 8*C^2*a^8*c*d
^10*f^2 - 6*C^2*b^8*c*d^10*f^2 + 18*C^2*a*b^7*c^2*d^9*f^2 + 12*C^2*a^2*b^6*c*d^10*f^2 + 2*C^2*a^4*b^4*c*d^10*f
^2 + 24*C^2*a^6*b^2*c*d^10*f^2 - 16*C^2*a^7*b*c^2*d^9*f^2 + 4*C^2*a^2*b^6*c^3*d^8*f^2 + 4*C^2*a^3*b^5*c^2*d^9*
f^2 - 10*C^2*a^4*b^4*c^3*d^8*f^2 + 2*C^2*a^5*b^3*c^2*d^9*f^2 + 8*C^2*a^6*b^2*c^3*d^8*f^2))/(b*f^4))*((((8*C^2*
a^2*c*f^2 - 8*C^2*b^2*c*f^2 + 16*C^2*a*b*d*f^2)^2/4 - (C^4*c^2 + C^4*d^2)*(16*a^4*f^4 + 16*b^4*f^4 + 32*a^2*b^
2*f^4))^(1/2) - 4*C^2*a^2*c*f^2 + 4*C^2*b^2*c*f^2 - 8*C^2*a*b*d*f^2)/(16*(a^4*f^4 + b^4*f^4 + 2*a^2*b^2*f^4)))
^(1/2) + (32*(15*C^3*a^4*b^3*d^12*f^2 - C^3*a^2*b^5*d^12*f^2 + C^3*b^7*c^2*d^10*f^2 + C^3*b^7*c^4*d^8*f^2 - 12
*C^3*a^6*b*d^12*f^2 - 24*C^3*a^3*b^4*c*d^11*f^2 + 24*C^3*a^5*b^2*c*d^11*f^2 - 12*C^3*a^6*b*c^2*d^10*f^2 + 8*C^
3*a^2*b^5*c^2*d^10*f^2 + 9*C^3*a^2*b^5*c^4*d^8*...

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