3.2.0 ∫ A+B tan(e+fx)+C tan2(e+fx) (c+d tan(e+fx))3∕2 dx [119]

Integrand size = 35, antiderivative size = 157 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(c-i d)^{3/2} f}-\frac {(B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(c+i d)^{3/2} f}-\frac {2 \left (c^2 C-B c d+A d^2\right )}{d \left (c^2+d^2\right ) f \sqrt {c+d \tan (e+f x)}} \]

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

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

Rubi [A] (veriﬁed)

Time = 0.33 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3709, 3620, 3618, 65, 214} \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {(i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (c-i d)^{3/2}}-\frac {(B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (c+i d)^{3/2}}-\frac {2 \left (A d^2-B c d+c^2 C\right )}{d f \left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}} \]

Int[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(3/2),x]

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

cTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c + I*d]])/((c + I*d)^(3/2)*f) - (2*(c^2*C - B*c*d + A*d^2))/(d*(c^2 + d^2

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,

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

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]

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]

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f

_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2)

)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e +

f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2

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

Mathematica [C] (veriﬁed)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 1.11 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.39 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\frac {-i B \left (\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}-\frac {\text {arctanh}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}\right )-\frac {2 C}{\sqrt {c+d \tan (e+f x)}}+\frac {(B c+(-A+C) d) \left ((-i c+d) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c-i d}\right )+(i c+d) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {c+d \tan (e+f x)}{c+i d}\right )\right )}{\left (c^2+d^2\right ) \sqrt {c+d \tan (e+f x)}}}{d f} \]

Integrate[(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2)/(c + d*Tan[e + f*x])^(3/2),x]

((-I)*B*(ArcTanh[Sqrt[c + d*Tan[e + f*x]]/Sqrt[c - I*d]]/Sqrt[c - I*d] - 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]] + ((B*c + (-A + C)*d)*(((-I)*c + d)*Hypergeometric2

F1[-1/2, 1, 1/2, (c + d*Tan[e + f*x])/(c - I*d)] + (I*c + d)*Hypergeometric2F1[-1/2, 1, 1/2, (c + d*Tan[e + f*

x])/(c + I*d)]))/((c^2 + d^2)*Sqrt[c + d*Tan[e + f*x]]))/(d*f)

Maple [B] (veriﬁed)

Leaf count of result is larger than twice the leaf count of optimal. \(5612\) vs. \(2(136)=272\).

Time = 0.13 (sec) , antiderivative size = 5613, normalized size of antiderivative = 35.75


method	result	size

parts	\(\text {Expression too large to display}\)	\(5613\)
derivativedivides	\(\text {Expression too large to display}\)	\(11427\)
default	\(\text {Expression too large to display}\)	\(11427\)

int((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7982 vs. \(2 (129) = 258\).

Time = 1.83 (sec) , antiderivative size = 7982, normalized size of antiderivative = 50.84 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")

Sympy [F]

\[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*tan(f*x+e))**(3/2),x)

Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/(c + d*tan(e + f*x))**(3/2), x)

Maxima [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

integrate((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")

Giac [F(-1)]

Timed out. \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Timed out} \]

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

Mupad [B] (veriﬁcation not implemented)

Time = 18.19 (sec) , antiderivative size = 8588, normalized size of antiderivative = 54.70 \[ \int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {Too large to display} \]

int((A + B*tan(e + f*x) + C*tan(e + f*x)^2)/(c + d*tan(e + f*x))^(3/2),x)

(log(((((((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2

)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(64*C*c*d^11*f^4 - ((c + d*tan(e + f*x))^(1/2)*((

(96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(c^6*f^4

+ d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c

^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^7*f^4 + 256*C*c^7*d^5*f^4

+ 64*C*c^9*d^3*f^4))/4 + (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f^3 - 32*C^2*c^6*d^4*f^

3 - 16*C^2*c^8*d^2*f^3))*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 +

12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 8*C^3*d^9*f^2 - 24*C^3*c^2*

d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4

)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 + (l

og(((((-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)

/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(64*C*c*d^11*f^4 - ((c + d*tan(e + f*x))^(1/2)*(-(

(96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(c^6*f^4

+ d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c

^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^7*f^4 + 256*C*c^7*d^5*f^4

+ 64*C*c^9*d^3*f^4))/4 + (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f^3 - 32*C^2*c^6*d^4*f^

3 - 16*C^2*c^8*d^2*f^3))*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2

- 12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 8*C^3*d^9*f^2 - 24*C^3*c^2

*d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f

^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 -

log(((((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(

16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(((96*C^4*c^2*d^

4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f

^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6

*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) + 64*C*c*d^11*f^4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^7*f^4 + 256*C*c^

7*d^5*f^4 + 64*C*c^9*d^3*f^4) - (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f^3 - 32*C^2*c^6*

d^4*f^3 - 16*C^2*c^8*d^2*f^3))*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) - 4*C^2*c^3

*f^2 + 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - 8*C^3*d^9*f^2 -

24*C^3*c^2*d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*

c^4*d^2*f^4)^(1/2) - 4*C^2*c^3*f^2 + 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*

f^4))^(1/2) - log(((-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C

^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((c + d*tan(e + f*x))^(1/2)*(

-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(16*c^

6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f

^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) + 64*C*c*d^11*f^4 + 256*C*c^3*d^9*f^4 + 384*C*c^5*d^

7*f^4 + 256*C*c^7*d^5*f^4 + 64*C*c^9*d^3*f^4) - (c + d*tan(e + f*x))^(1/2)*(16*C^2*d^10*f^3 + 32*C^2*c^2*d^8*f

^3 - 32*C^2*c^6*d^4*f^3 - 16*C^2*c^8*d^2*f^3))*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*d^6*f^4 - 144*C^4*c^4*d^2*f^4)^

(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) -

8*C^3*d^9*f^2 - 24*C^3*c^2*d^7*f^2 - 24*C^3*c^4*d^5*f^2 - 8*C^3*c^6*d^3*f^2)*(-((96*C^4*c^2*d^4*f^4 - 16*C^4*

d^6*f^4 - 144*C^4*c^4*d^2*f^4)^(1/2) + 4*C^2*c^3*f^2 - 12*C^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4

*f^4 + 48*c^4*d^2*f^4))^(1/2) + (log(8*A^3*d^9*f^2 - ((((((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d

^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(

((((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*

f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10

*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 64*A*c*d^11*f^4 + 256*A*c^3

*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(16*A^2*d

^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^

4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*

d^2*f^4))^(1/2))/4 + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*(((96*A^4*c^2*d^4*f^4 - 16*A

^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4

+ 3*c^4*d^2*f^4))^(1/2))/4 + (log(8*A^3*d^9*f^2 - ((((-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^

2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*((

(-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*

f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10

*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 64*A*c*d^11*f^4 + 256*A*c^3

*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4))/4 - (c + d*tan(e + f*x))^(1/2)*(16*A^2*d

^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f

^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4

*d^2*f^4))^(1/2))/4 + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*(-((96*A^4*c^2*d^4*f^4 - 16

*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f

^4 + 3*c^4*d^2*f^4))^(1/2))/4 - log(8*A^3*d^9*f^2 - ((((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*

f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1

/2)*(64*A*c*d^11*f^4 - (((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 1

2*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)

*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) +

256*A*c^3*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4) + (c + d*tan(e + f*x))^(1/2)*(16

*A^2*d^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(((96*A^4*c^2*d^4*f^4 - 16*A^4*

d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4

*f^4 + 48*c^4*d^2*f^4))^(1/2) + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*(((96*A^4*c^2*d^4

*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) - 4*A^2*c^3*f^2 + 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^

4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - log(8*A^3*d^9*f^2 - ((-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 1

44*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c

^4*d^2*f^4))^(1/2)*(64*A*c*d^11*f^4 - (-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4

*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan

(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*

c^11*d^2*f^5) + 256*A*c^3*d^9*f^4 + 384*A*c^5*d^7*f^4 + 256*A*c^7*d^5*f^4 + 64*A*c^9*d^3*f^4) + (c + d*tan(e +

f*x))^(1/2)*(16*A^2*d^10*f^3 + 32*A^2*c^2*d^8*f^3 - 32*A^2*c^6*d^4*f^3 - 16*A^2*c^8*d^2*f^3))*(-((96*A^4*c^2*

d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6

*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) + 24*A^3*c^2*d^7*f^2 + 24*A^3*c^4*d^5*f^2 + 8*A^3*c^6*d^3*f^2)*

(-((96*A^4*c^2*d^4*f^4 - 16*A^4*d^6*f^4 - 144*A^4*c^4*d^2*f^4)^(1/2) + 4*A^2*c^3*f^2 - 12*A^2*c*d^2*f^2)/(16*c

^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) + (log(- (((c + d*tan(e + f*x))^(1/2)*(16*B^2*d^

10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d^4*f^3 - 16*B^2*c^8*d^2*f^3) + ((((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f

^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4

*d^2*f^4))^(1/2)*(((((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^

2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*

f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5))/4 + 32*B*d^12

*f^4 + 96*B*c^2*d^10*f^4 + 64*B*c^4*d^8*f^4 - 64*B*c^6*d^6*f^4 - 96*B*c^8*d^4*f^4 - 32*B*c^10*d^2*f^4))/4)*(((

96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(c^6*f^4

+ d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 24*B^3*c^3*d^6*f^2 - 24*B^3*c^5*d^4*f^2 - 8*B^3*c^7*d^2

*f^2 - 8*B^3*c*d^8*f^2)*(((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 -

12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 + (log(- (((c + d*tan(e + f*x)

)^(1/2)*(16*B^2*d^10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d^4*f^3 - 16*B^2*c^8*d^2*f^3) + ((-((96*B^4*c^2*d^4

*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*

c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(((-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4

*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x

))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2

*f^5))/4 + 32*B*d^12*f^4 + 96*B*c^2*d^10*f^4 + 64*B*c^4*d^8*f^4 - 64*B*c^6*d^6*f^4 - 96*B*c^8*d^4*f^4 - 32*B*c

^10*d^2*f^4))/4)*(-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2

*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - 24*B^3*c^3*d^6*f^2 - 24*B^3*c^5*d^

4*f^2 - 8*B^3*c^7*d^2*f^2 - 8*B^3*c*d^8*f^2)*(-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1

/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(c^6*f^4 + d^6*f^4 + 3*c^2*d^4*f^4 + 3*c^4*d^2*f^4))^(1/2))/4 - log(((

c + d*tan(e + f*x))^(1/2)*(16*B^2*d^10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d^4*f^3 - 16*B^2*c^8*d^2*f^3) + (

((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6

*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*

c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*

f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 640*c^5*d^8*f^5 + 640*c^7*d^6*f^5 +

320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 32*B*d^12*f^4 - 96*B*c^2*d^10*f^4 - 64*B*c^4*d^8*f^4 + 64*B*c^6*d^6*f^4

+ 96*B*c^8*d^4*f^4 + 32*B*c^10*d^2*f^4))*(((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) +

4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - 24*B^3

*c^3*d^6*f^2 - 24*B^3*c^5*d^4*f^2 - 8*B^3*c^7*d^2*f^2 - 8*B^3*c*d^8*f^2)*(((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^

4 - 144*B^4*c^4*d^2*f^4)^(1/2) + 4*B^2*c^3*f^2 - 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 +

48*c^4*d^2*f^4))^(1/2) - log(((c + d*tan(e + f*x))^(1/2)*(16*B^2*d^10*f^3 + 32*B^2*c^2*d^8*f^3 - 32*B^2*c^6*d

^4*f^3 - 16*B^2*c^8*d^2*f^3) + (-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^

3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*((-((96*B^4*c^2*d

^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*

f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(64*c*d^12*f^5 + 320*c^3*d^10*f^5 + 6

40*c^5*d^8*f^5 + 640*c^7*d^6*f^5 + 320*c^9*d^4*f^5 + 64*c^11*d^2*f^5) - 32*B*d^12*f^4 - 96*B*c^2*d^10*f^4 - 64

*B*c^4*d^8*f^4 + 64*B*c^6*d^6*f^4 + 96*B*c^8*d^4*f^4 + 32*B*c^10*d^2*f^4))*(-((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6

*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^

4 + 48*c^4*d^2*f^4))^(1/2) - 24*B^3*c^3*d^6*f^2 - 24*B^3*c^5*d^4*f^2 - 8*B^3*c^7*d^2*f^2 - 8*B^3*c*d^8*f^2)*(-

((96*B^4*c^2*d^4*f^4 - 16*B^4*d^6*f^4 - 144*B^4*c^4*d^2*f^4)^(1/2) - 4*B^2*c^3*f^2 + 12*B^2*c*d^2*f^2)/(16*c^6

*f^4 + 16*d^6*f^4 + 48*c^2*d^4*f^4 + 48*c^4*d^2*f^4))^(1/2) - (2*A*d)/(f*(c^2 + d^2)*(c + d*tan(e + f*x))^(1/2

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

\(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(c+d \tan (e+f x))^{3/2}} \, dx\) [119]

Optimal result