∫ A+B tan(e+fx)+C tan2(e+fx)_____ (a+b tan(e+fx))∘ __________

3.2.14 \(\int \frac {A+B \tan (e+f x)+C \tan ^2(e+f x)}{(a+b \tan (e+f x)) \sqrt {c+d \tan (e+f x)}} \, dx\) [114]

Optimal. Leaf size=210 \[ -\frac {(i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(a-i b) \sqrt {c-i d} f}-\frac {(A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) \sqrt {c+i d} f}-\frac {2 \left (A b^2-a (b B-a C)\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \left (a^2+b^2\right ) \sqrt {b c-a d} f} \]

[Out]

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

f*x+e))^(1/2)/(c+I*d)^(1/2))/(I*a-b)/f/(c+I*d)^(1/2)-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^2+b^2)/f/b^(1/2)/(-a*d+b*c)^(1/2)

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Rubi [A]
time = 0.41, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 47, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {3734, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {2 \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 )}{\sqrt {b} f \left (a^2+b^2\right ) \sqrt {b c-a d}}-\frac {(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) \sqrt {c-i d}}-\frac {(A+i B-C) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a) \sqrt {c+i d}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

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

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

Rubi steps

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

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Mathematica [A]
time = 0.29, size = 194, normalized size = 0.92 \begin {gather*} \frac {\frac {(-i a+b) (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+b) (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 \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 )}{\sqrt {b} \sqrt {b c-a d}}}{\left (a^2+b^2\right ) f} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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

h[(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])/Sqrt[b*c - a*d]])/(Sqrt[b]*Sqrt[b*c - a*d]))/((a^2 + b^2)*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3760\) vs. \(2(179)=358\).
time = 0.53, size = 3761, normalized size = 17.91


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(3761\)
default	\(\text {Expression too large to display}\)	\(3761\)

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)/(a+b*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/f*(2/(a^2+b^2)*(1/4/d^2/(c^2+d^2)^(3/2)*(-1/2*(A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^2*d+A*(2*

(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c*d^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^3+B*(2*

(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*d^3-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^2-B*(2*(c^2+d^2)^(1/2

)+2*c)^(1/2)*b*c^3*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*b

*c+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c^3

-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/

2)*b*c^2*d^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/

2)*b*c-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*d^3-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b

*c^3+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3-C*(2*(c^2+d^2)^(1/2)+2*c)

^(1/2)*b*c^2*d^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^4-C*(2*(c^2+d^2)^(1

/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^2*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c*d^2-B*(2*(c^2+d^2)^

(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c*d^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c^2*d-C*(2*(c^2+d^2

)^(1/2)+2*c)^(1/2)*b*d^4)*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*a*c^2*d^3-2*A*a*d^5-2*A*b*c^3*d^2-2*A*b*c*d^4+2*B*a*c^3*d^2+2*B*a*c*d^4-2*B*b*c^2*d^3-2*B*b*d^5+2*

C*a*c^2*d^3+2*C*a*d^5+2*C*b*c^3*d^2+2*C*b*c*d^4+1/2*(A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^2*d+A

*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c*d^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^3+B

*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*d^3-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^2-B*(2*(c^2+d^2)^

(1/2)+2*c)^(1/2)*b*c^3*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/

2)*b*c+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b

*c^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)

^(1/2)*b*c^2*d^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*a*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)

^(3/2)*b*c-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*d^3-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/

2)*b*c^3+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3-C*(2*(c^2+d^2)^(1/2)+

2*c)^(1/2)*b*c^2*d^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^4-C*(2*(c^2+d^2

)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^2*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c*d^2-B*(2*(c^2+d

^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c*d^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c^2*d-C*(2*(c^2

+d^2)^(1/2)+2*c)^(1/2)*b*d^4)*(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^2/(c^2+d^2)^(3/2)*(1/2*(A*(

2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^2*d+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c*d^2-B

*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^3+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*d^3-B*(

2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)

*b*c*d^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*b*c+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a

*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d-A*(2*(c^2+d

^2)^(1/2)+2*c)^(1/2)*a*c*d^3+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^2+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^

2)^(3/2)*a*c+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(3/2)*b*c-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/

2)*a*d^3-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c^3+C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d+C*(2*(c

^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^2+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d

^4-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^4-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^2*d-C*(2*(c^2+d^2

)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c*d^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c*d^2+B*(2*(c^2+d

^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c^2*d-C*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4)*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*a*c^2*d^3+2*A*a*d^5+2*A*b*c^3*d^2+2*A*b*c

*d^4-2*B*a*c^3*d^2-2*B*a*c*d^4+2*B*b*c^2*d^3+2*B*b*d^5-2*C*a*c^2*d^3-2*C*a*d^5-2*C*b*c^3*d^2-2*C*b*c*d^4-1/2*(

A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^2*d+A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*c*d^

2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a*c^3+B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*b*d^3-

B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d^2-B*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3*d-B*(2*(c^2+d^2)^(1/2)+2*c)^(1

/2)*b*c*d^3-A*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^...

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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((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2)/(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((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2)/(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 {A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}}{\left (a + b \tan {\left (e + f x \right )}\right ) \sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \end {gather*}

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)/(a+b*tan(f*x+e)),x)

[Out]

Integral((A + B*tan(e + f*x) + C*tan(e + f*x)**2)/((a + b*tan(e + f*x))*sqrt(c + d*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((A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e))^(1/2)/(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 = 69.14, size = 2500, normalized size = 11.90 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log((((((((((128*C*b^2*d^8*(a*d + b*c)^2*(a^2 + b^2)^2)/f - 64*b^2*d^8*(a^2 + b^2)^2*(c + d*tan(e + f*x))^(1/

2)*((4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f^

4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2)*(3*b^3*c^2 + 2*b^3*d^2 - a^2*b*c^2 - 2*a^2*b*d^2 + a^3*c*d + a*b^2*c*d))*(

(4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f^4*(a

^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 + (64*C^2*b*d^8*(c + d*tan(e + f*x))^(1/2)*(5*b^6*c - 4*a^6*c - 2*a^2*b^4*c

+ 5*a^4*b^2*c - 2*a^3*b^3*d + 7*a*b^5*d + 7*a^5*b*d))/f^2)*((4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 + (32*C^3*b*d^

8*(4*a^5*d - b^5*c - 9*a^2*b^3*c - 15*a^3*b^2*d + 12*a^4*b*c + a*b^4*d))/f^3)*((4*(-C^4*f^4*(a^2*d - b^2*d + 2

*a*b*c)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2)

)/4 - (32*C^4*b*d^8*(2*a^4 + b^4)*(c + d*tan(e + f*x))^(1/2))/f^4)*((4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^

(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 + (32*C

^5*a^2*b^2*d^8)/f^5)*(((32*C^4*a^2*b^2*d^2*f^4 - 16*C^4*b^4*d^2*f^4 - 64*C^4*a^2*b^2*c^2*f^4 - 16*C^4*a^4*d^2*

f^4 + 64*C^4*a*b^3*c*d*f^4 - 64*C^4*a^3*b*c*d*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

)/(a^4*c^2*f^4 + a^4*d^2*f^4 + b^4*c^2*f^4 + b^4*d^2*f^4 + 2*a^2*b^2*c^2*f^4 + 2*a^2*b^2*d^2*f^4))^(1/2))/4 +

(log((((((((((128*C*b^2*d^8*(a*d + b*c)^2*(a^2 + b^2)^2)/f - 64*b^2*d^8*(a^2 + b^2)^2*(c + d*tan(e + f*x))^(1/

2)*(-(4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f

^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2)*(3*b^3*c^2 + 2*b^3*d^2 - a^2*b*c^2 - 2*a^2*b*d^2 + a^3*c*d + a*b^2*c*d))*

(-(4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f^4*

(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 + (64*C^2*b*d^8*(c + d*tan(e + f*x))^(1/2)*(5*b^6*c - 4*a^6*c - 2*a^2*b^4

*c + 5*a^4*b^2*c - 2*a^3*b^3*d + 7*a*b^5*d + 7*a^5*b*d))/f^2)*(-(4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 + (32*C^3*b

*d^8*(4*a^5*d - b^5*c - 9*a^2*b^3*c - 15*a^3*b^2*d + 12*a^4*b*c + a*b^4*d))/f^3)*(-(4*(-C^4*f^4*(a^2*d - b^2*d

+ 2*a*b*c)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(

1/2))/4 - (32*C^4*b*d^8*(2*a^4 + b^4)*(c + d*tan(e + f*x))^(1/2))/f^4)*(-(4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c

)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 +

(32*C^5*a^2*b^2*d^8)/f^5)*(-((32*C^4*a^2*b^2*d^2*f^4 - 16*C^4*b^4*d^2*f^4 - 64*C^4*a^2*b^2*c^2*f^4 - 16*C^4*a^

4*d^2*f^4 + 64*C^4*a*b^3*c*d*f^4 - 64*C^4*a^3*b*c*d*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)/(a^4*c^2*f^4 + a^4*d^2*f^4 + b^4*c^2*f^4 + b^4*d^2*f^4 + 2*a^2*b^2*c^2*f^4 + 2*a^2*b^2*d^2*f^4))^(1/2)

)/4 - log((((((((((128*C*b^2*d^8*(a*d + b*c)^2*(a^2 + b^2)^2)/f + 64*b^2*d^8*(a^2 + b^2)^2*(c + d*tan(e + f*x)

)^(1/2)*((4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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

)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2)*(3*b^3*c^2 + 2*b^3*d^2 - a^2*b*c^2 - 2*a^2*b*d^2 + a^3*c*d + a*b^2*c*

d))*((4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f

^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 - (64*C^2*b*d^8*(c + d*tan(e + f*x))^(1/2)*(5*b^6*c - 4*a^6*c - 2*a^2*

b^4*c + 5*a^4*b^2*c - 2*a^3*b^3*d + 7*a*b^5*d + 7*a^5*b*d))/f^2)*((4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 + (32*C^3

*b*d^8*(4*a^5*d - b^5*c - 9*a^2*b^3*c - 15*a^3*b^2*d + 12*a^4*b*c + a*b^4*d))/f^3)*((4*(-C^4*f^4*(a^2*d - b^2*

d + 2*a*b*c)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^

(1/2))/4 + (32*C^4*b*d^8*(2*a^4 + b^4)*(c + d*tan(e + f*x))^(1/2))/f^4)*((4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c

)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2))/4 +

(32*C^5*a^2*b^2*d^8)/f^5)*(((32*C^4*a^2*b^2*d^2*f^4 - 16*C^4*b^4*d^2*f^4 - 64*C^4*a^2*b^2*c^2*f^4 - 16*C^4*a^4

*d^2*f^4 + 64*C^4*a*b^3*c*d*f^4 - 64*C^4*a^3*b*c*d*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*c^2*f^4 + 16*a^4*d^2*f^4 + 16*b^4*c^2*f^4 + 16*b^4*d^2*f^4 + 32*a^2*b^2*c^2*f^4 + 32*a^2*b^2*d^

2*f^4))^(1/2) - log((((((((((128*C*b^2*d^8*(a*d + b*c)^2*(a^2 + b^2)^2)/f + 64*b^2*d^8*(a^2 + b^2)^2*(c + d*ta

n(e + f*x))^(1/2)*(-(4*(-C^4*f^4*(a^2*d - b^2*d + 2*a*b*c)^2)^(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)/(f^4*(a^2 + b^2)^2*(c^2 + d^2)))^(1/2)*(3*b^3*c^2 + 2*b^3*d^2 - a^2*b*c^2 - 2*a^2*b*d^2 + a^3*c*d

+ a*b^2*c*d))*(-(4*(-C^4*f^4*(a^2*d - b^2*d + ...

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