3.319 \(\int \tan (c+d x) \sqrt {a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx\)

Optimal. Leaf size=146 \[ -\frac {\sqrt {a-i b} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (A+i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d} \]

[Out]

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

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Rubi [A]  time = 0.28, antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3592, 3528, 3539, 3537, 63, 208} \[ -\frac {\sqrt {a-i b} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}-\frac {\sqrt {a+i b} (A+i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d}+\frac {2 A \sqrt {a+b \tan (c+d x)}}{d}+\frac {2 B (a+b \tan (c+d x))^{3/2}}{3 b d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]

[Out]

-((Sqrt[a - I*b]*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/d) - (Sqrt[a + I*b]*(A + I*B)*ArcT
anh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]])/d + (2*A*Sqrt[a + b*Tan[c + d*x]])/d + (2*B*(a + b*Tan[c + d*x])^
(3/2))/(3*b*d)

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 3528

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(d
*(a + b*Tan[e + f*x])^m)/(f*m), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

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 3592

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

Rubi steps

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

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Mathematica [A]  time = 0.50, size = 140, normalized size = 0.96 \[ \frac {2 \sqrt {a+b \tan (c+d x)} (a B+3 A b+b B \tan (c+d x))-3 b \sqrt {a-i b} (A-i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-3 b \sqrt {a+i b} (A+i B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{3 b d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]]*(A + B*Tan[c + d*x]),x]

[Out]

(-3*Sqrt[a - I*b]*b*(A - I*B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - 3*Sqrt[a + I*b]*b*(A + I*B)*Ar
cTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a + I*b]] + 2*Sqrt[a + b*Tan[c + d*x]]*(3*A*b + a*B + b*B*Tan[c + d*x]))/(
3*b*d)

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fricas [B]  time = 13.61, size = 8737, normalized size = 59.84 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(12*sqrt(2)*b*d^5*sqrt(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B
^2 + B^4)*b^2)/d^4) + (A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A
*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4
)*b^2)/d^4)*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(3/4)*arctan(((2*(A^7*B + 3*A^5*
B^3 + 3*A^3*B^5 + A*B^7)*a^3 + (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b + 2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 +
A*B^7)*a*b^2 + (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^3)*d^4*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4
- 2*A^2*B^2 + B^4)*b^2)/d^4)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (2*(A^9*
B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^4 + (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*A^2*B^8 - B
^10)*a^3*b + 2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^2*b^2 + (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2
*A^4*B^6 - 3*A^2*B^8 - B^10)*a*b^3)*d^2*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*
b^2)/d^4) + sqrt(2)*((2*(A^4*B + A^2*B^3)*a + (A^5 - A*B^4)*b)*d^7*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b
+ (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (
2*(A^6*B + 2*A^4*B^3 + A^2*B^5)*a^2 + (A^7 - A^5*B^2 - 5*A^3*B^4 - 3*A*B^6)*a*b - (A^6*B + A^4*B^3 - A^2*B^5 -
B^7)*b^2)*d^5*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4))*sqrt(((2*A*B*b
- (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (A^4 + 2*A^2*B^2
+ B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^
2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^
4)*b^2)/d^4)^(3/4) + sqrt(2)*(A*d^7*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)
/d^4)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + ((A^3 + A*B^2)*a - (A^2*B + B^3)
*b)*d^5*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4))*sqrt(((2*A*B*b - (A^2
- B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (A^4 + 2*A^2*B^2 + B^4)
*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqr
t(((4*(A^4*B^2 + A^2*B^4)*a^4 + 4*(A^5*B - A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6)*a^2*b^2 + 4*(A^5
*B - A*B^5)*a*b^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^4)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*
B^2 + B^4)*b^2)/d^4)*cos(d*x + c) + sqrt(2)*((4*A^3*B^2*a^3 + 4*(A^4*B - 2*A^2*B^3)*a^2*b + (A^5 - 6*A^3*B^2 +
5*A*B^4)*a*b^2 - (A^4*B - 2*A^2*B^3 + B^5)*b^3)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^
4)*b^2)/d^4)*cos(d*x + c) + (4*(A^5*B^2 + A^3*B^4)*a^4 + 4*(A^6*B - A^2*B^5)*a^3*b + (A^7 + 3*A^5*B^2 + 3*A^3*
B^4 + A*B^6)*a^2*b^2 + 4*(A^6*B - A^2*B^5)*a*b^3 + (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^4)*d*cos(d*x + c))*sqrt
(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (A^4 +
2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2
+ B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^
2*B^2 + B^4)*b^2)/d^4)^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^5 + 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*
a^4*b + (A^8 + 4*A^6*B^2 + 6*A^4*B^4 + 4*A^2*B^6 + B^8)*a^3*b^2 + 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b^
3 + (A^8 - 2*A^4*B^4 + B^8)*a*b^4)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^4*b + 4*(A^7*B + A^5*B^
3 - A^3*B^5 - A*B^7)*a^3*b^2 + (A^8 + 4*A^6*B^2 + 6*A^4*B^4 + 4*A^2*B^6 + B^8)*a^2*b^3 + 4*(A^7*B + A^5*B^3 -
A^3*B^5 - A*B^7)*a*b^4 + (A^8 - 2*A^4*B^4 + B^8)*b^5)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c)))*(((A^4 + 2*A^2
*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(3/4))/(4*(A^10*B^2 + 4*A^8*B^4 + 6*A^6*B^6 + 4*A^4*B^8 +
A^2*B^10)*a^4*b + 4*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^3*b^2 + (A^12 + 6*A^10
*B^2 + 15*A^8*B^4 + 20*A^6*B^6 + 15*A^4*B^8 + 6*A^2*B^10 + B^12)*a^2*b^3 + 4*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 -
2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^4 + (A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8 + 2*A^2*B^10 + B^1
2)*b^5))*cos(d*x + c) + 12*sqrt(2)*b*d^5*sqrt(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2
+ (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a
^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4
- 2*A^2*B^2 + B^4)*b^2)/d^4)*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(3/4)*arctan(-(
(2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^3 + (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b + 2*(A^7*B + 3*A^5*
B^3 + 3*A^3*B^5 + A*B^7)*a*b^2 + (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^3)*d^4*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B -
A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^
2)/d^4) + (2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^4 + (A^10 + 3*A^8*B^2 + 2*A^6*B^4 - 2*A^4*B
^6 - 3*A^2*B^8 - B^10)*a^3*b + 2*(A^9*B + 4*A^7*B^3 + 6*A^5*B^5 + 4*A^3*B^7 + A*B^9)*a^2*b^2 + (A^10 + 3*A^8*B
^2 + 2*A^6*B^4 - 2*A^4*B^6 - 3*A^2*B^8 - B^10)*a*b^3)*d^2*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 -
2*A^2*B^2 + B^4)*b^2)/d^4) - sqrt(2)*((2*(A^4*B + A^2*B^3)*a + (A^5 - A*B^4)*b)*d^7*sqrt((4*A^2*B^2*a^2 + 4*(
A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 +
B^4)*b^2)/d^4) + (2*(A^6*B + 2*A^4*B^3 + A^2*B^5)*a^2 + (A^7 - A^5*B^2 - 5*A^3*B^4 - 3*A*B^6)*a*b - (A^6*B + A
^4*B^3 - A^2*B^5 - B^7)*b^2)*d^5*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^
4))*sqrt(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)
+ (A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2
*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^
4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(3/4) - sqrt(2)*(A*d^7*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A
^2*B^2 + B^4)*b^2)/d^4)*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + ((A^3 + A*B^2)
*a - (A^2*B + B^3)*b)*d^5*sqrt((4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2)/d^4))*sqr
t(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (A^4
+ 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^
2 + B^4)*b^2))*sqrt(((4*(A^4*B^2 + A^2*B^4)*a^4 + 4*(A^5*B - A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6
)*a^2*b^2 + 4*(A^5*B - A*B^5)*a*b^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^4)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a
^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)*cos(d*x + c) - sqrt(2)*((4*A^3*B^2*a^3 + 4*(A^4*B - 2*A^2*B^3)*a^2*b +
(A^5 - 6*A^3*B^2 + 5*A*B^4)*a*b^2 - (A^4*B - 2*A^2*B^3 + B^5)*b^3)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^
4 + 2*A^2*B^2 + B^4)*b^2)/d^4)*cos(d*x + c) + (4*(A^5*B^2 + A^3*B^4)*a^4 + 4*(A^6*B - A^2*B^5)*a^3*b + (A^7 +
3*A^5*B^2 + 3*A^3*B^4 + A*B^6)*a^2*b^2 + 4*(A^6*B - A^2*B^5)*a*b^3 + (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^4)*d*
cos(d*x + c))*sqrt(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*
b^2)/d^4) + (A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b
+ (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)
*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^5 + 4*(A^7*B + A^5*B^3 -
A^3*B^5 - A*B^7)*a^4*b + (A^8 + 4*A^6*B^2 + 6*A^4*B^4 + 4*A^2*B^6 + B^8)*a^3*b^2 + 4*(A^7*B + A^5*B^3 - A^3*B
^5 - A*B^7)*a^2*b^3 + (A^8 - 2*A^4*B^4 + B^8)*a*b^4)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^4*b +
4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^2 + (A^8 + 4*A^6*B^2 + 6*A^4*B^4 + 4*A^2*B^6 + B^8)*a^2*b^3 + 4*(
A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^4 + (A^8 - 2*A^4*B^4 + B^8)*b^5)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c
)))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(3/4))/(4*(A^10*B^2 + 4*A^8*B^4 + 6*A^6*
B^6 + 4*A^4*B^8 + A^2*B^10)*a^4*b + 4*(A^11*B + 3*A^9*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a^3*b^
2 + (A^12 + 6*A^10*B^2 + 15*A^8*B^4 + 20*A^6*B^6 + 15*A^4*B^8 + 6*A^2*B^10 + B^12)*a^2*b^3 + 4*(A^11*B + 3*A^9
*B^3 + 2*A^7*B^5 - 2*A^5*B^7 - 3*A^3*B^9 - A*B^11)*a*b^4 + (A^12 + 2*A^10*B^2 - A^8*B^4 - 4*A^6*B^6 - A^4*B^8
+ 2*A^2*B^10 + B^12)*b^5))*cos(d*x + c) - 3*sqrt(2)*((2*A*B*b^2 - (A^2 - B^2)*a*b)*d^3*sqrt(((A^4 + 2*A^2*B^2
+ B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)*cos(d*x + c) - ((A^4 + 2*A^2*B^2 + B^4)*a^2*b + (A^4 + 2*A^2*B^
2 + B^4)*b^3)*d*cos(d*x + c))*sqrt(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2
*A^2*B^2 + B^4)*b^2)/d^4) + (A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3
*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d
^4)^(1/4)*log(((4*(A^4*B^2 + A^2*B^4)*a^4 + 4*(A^5*B - A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6)*a^2*
b^2 + 4*(A^5*B - A*B^5)*a*b^3 + (A^6 - A^4*B^2 - A^2*B^4 + B^6)*b^4)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (
A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)*cos(d*x + c) + sqrt(2)*((4*A^3*B^2*a^3 + 4*(A^4*B - 2*A^2*B^3)*a^2*b + (A^5 -
6*A^3*B^2 + 5*A*B^4)*a*b^2 - (A^4*B - 2*A^2*B^3 + B^5)*b^3)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*
A^2*B^2 + B^4)*b^2)/d^4)*cos(d*x + c) + (4*(A^5*B^2 + A^3*B^4)*a^4 + 4*(A^6*B - A^2*B^5)*a^3*b + (A^7 + 3*A^5*
B^2 + 3*A^3*B^4 + A*B^6)*a^2*b^2 + 4*(A^6*B - A^2*B^5)*a*b^3 + (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^4)*d*cos(d*
x + c))*sqrt(((2*A*B*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d
^4) + (A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4
- 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 +
(A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^5 + 4*(A^7*B + A^5*B^3 - A^3*B
^5 - A*B^7)*a^4*b + (A^8 + 4*A^6*B^2 + 6*A^4*B^4 + 4*A^2*B^6 + B^8)*a^3*b^2 + 4*(A^7*B + A^5*B^3 - A^3*B^5 - A
*B^7)*a^2*b^3 + (A^8 - 2*A^4*B^4 + B^8)*a*b^4)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^4*b + 4*(A^
7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^2 + (A^8 + 4*A^6*B^2 + 6*A^4*B^4 + 4*A^2*B^6 + B^8)*a^2*b^3 + 4*(A^7*B
+ A^5*B^3 - A^3*B^5 - A*B^7)*a*b^4 + (A^8 - 2*A^4*B^4 + B^8)*b^5)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) +
3*sqrt(2)*((2*A*B*b^2 - (A^2 - B^2)*a*b)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/
d^4)*cos(d*x + c) - ((A^4 + 2*A^2*B^2 + B^4)*a^2*b + (A^4 + 2*A^2*B^2 + B^4)*b^3)*d*cos(d*x + c))*sqrt(((2*A*B
*b - (A^2 - B^2)*a)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (A^4 + 2*A^2*B
^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*
b^2))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(1/4)*log(((4*(A^4*B^2 + A^2*B^4)*a^4
+ 4*(A^5*B - A*B^5)*a^3*b + (A^6 + 3*A^4*B^2 + 3*A^2*B^4 + B^6)*a^2*b^2 + 4*(A^5*B - A*B^5)*a*b^3 + (A^6 - A^4
*B^2 - A^2*B^4 + B^6)*b^4)*d^2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)*cos(d*x +
c) - sqrt(2)*((4*A^3*B^2*a^3 + 4*(A^4*B - 2*A^2*B^3)*a^2*b + (A^5 - 6*A^3*B^2 + 5*A*B^4)*a*b^2 - (A^4*B - 2*A
^2*B^3 + B^5)*b^3)*d^3*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)*cos(d*x + c) + (4
*(A^5*B^2 + A^3*B^4)*a^4 + 4*(A^6*B - A^2*B^5)*a^3*b + (A^7 + 3*A^5*B^2 + 3*A^3*B^4 + A*B^6)*a^2*b^2 + 4*(A^6*
B - A^2*B^5)*a*b^3 + (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^4)*d*cos(d*x + c))*sqrt(((2*A*B*b - (A^2 - B^2)*a)*d^
2*sqrt(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4) + (A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 +
2*A^2*B^2 + B^4)*b^2)/(4*A^2*B^2*a^2 + 4*(A^3*B - A*B^3)*a*b + (A^4 - 2*A^2*B^2 + B^4)*b^2))*sqrt((a*cos(d*x
+ c) + b*sin(d*x + c))/cos(d*x + c))*(((A^4 + 2*A^2*B^2 + B^4)*a^2 + (A^4 + 2*A^2*B^2 + B^4)*b^2)/d^4)^(1/4) +
(4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^5 + 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^4*b + (A^8 + 4*A^6*B^2 + 6*A
^4*B^4 + 4*A^2*B^6 + B^8)*a^3*b^2 + 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b^3 + (A^8 - 2*A^4*B^4 + B^8)*a*
b^4)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^4*b + 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^3*b^2 +
(A^8 + 4*A^6*B^2 + 6*A^4*B^4 + 4*A^2*B^6 + B^8)*a^2*b^3 + 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^4 + (A^8
- 2*A^4*B^4 + B^8)*b^5)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) - 8*(((A^4*B + 2*A^2*B^3 + B^5)*a^3 + 3*(A^5
+ 2*A^3*B^2 + A*B^4)*a^2*b + (A^4*B + 2*A^2*B^3 + B^5)*a*b^2 + 3*(A^5 + 2*A^3*B^2 + A*B^4)*b^3)*cos(d*x + c)
+ ((A^4*B + 2*A^2*B^3 + B^5)*a^2*b + (A^4*B + 2*A^2*B^3 + B^5)*b^3)*sin(d*x + c))*sqrt((a*cos(d*x + c) + b*sin
(d*x + c))/cos(d*x + c)))/(((A^4 + 2*A^2*B^2 + B^4)*a^2*b + (A^4 + 2*A^2*B^2 + B^4)*b^3)*d*cos(d*x + c))

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

[Out]

Timed out

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maple [B]  time = 0.28, size = 989, normalized size = 6.77 \[ \frac {2 B \left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{3 b d}+\frac {2 A \sqrt {a +b \tan \left (d x +c \right )}}{d}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}+\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}-\frac {\ln \left (b \tan \left (d x +c \right )+a +\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}+\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}-\frac {\arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {b \arctan \left (\frac {2 \sqrt {a +b \tan \left (d x +c \right )}+\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) A \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}}{4 d}-\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, \sqrt {a^{2}+b^{2}}}{4 d b}+\frac {\ln \left (\sqrt {a +b \tan \left (d x +c \right )}\, \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-b \tan \left (d x +c \right )-a -\sqrt {a^{2}+b^{2}}\right ) B \sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}\, a}{4 d b}+\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A \sqrt {a^{2}+b^{2}}}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}-\frac {\arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) A a}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}+\frac {b \arctan \left (\frac {\sqrt {2 \sqrt {a^{2}+b^{2}}+2 a}-2 \sqrt {a +b \tan \left (d x +c \right )}}{\sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}}\right ) B}{d \sqrt {2 \sqrt {a^{2}+b^{2}}-2 a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*B*(a+b*tan(d*x+c))^(3/2)/b/d+2*A*(a+b*tan(d*x+c))^(1/2)/d-1/4/d*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(
2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*t
an(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2
)-1/4/d/b*ln(b*tan(d*x+c)+a+(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*B*(2*(a^2+b^
2)^(1/2)+2*a)^(1/2)*a-1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*
a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*(a^2+b^2)^(1/2)+1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan((2*(a+b*ta
n(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*a-1/d*b/(2*(a^2+b^2)^(1/2)-2*a
)^(1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B+1/4/d
*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*A*(2*(a^2+b^2)^(1/2)+
2*a)^(1/2)-1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(
2*(a^2+b^2)^(1/2)+2*a)^(1/2)*(a^2+b^2)^(1/2)+1/4/d/b*ln((a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-b
*tan(d*x+c)-a-(a^2+b^2)^(1/2))*B*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+1/d/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*
(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*A*(a^2+b^2)^(1/2)-1/d/(2*(
a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2
*a)^(1/2))*A*a+1/d*b/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1
/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*tan(d*x + c) + A)*sqrt(b*tan(d*x + c) + a)*tan(d*x + c), x)

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mupad [B]  time = 12.05, size = 864, normalized size = 5.92 \[ \mathrm {atanh}\left (\frac {d^3\,\left (\frac {16\,\left (B^2\,b^4-B^2\,a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}+\frac {16\,a\,b^2\,\left (\sqrt {-B^4\,b^2\,d^4}+B^2\,a\,d^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^4}\right )\,\sqrt {-\frac {\sqrt {-B^4\,b^2\,d^4}+B^2\,a\,d^2}{d^4}}}{16\,\left (B^3\,a^2\,b^3+B^3\,b^5\right )}\right )\,\sqrt {-\frac {\sqrt {-B^4\,b^2\,d^4}+B^2\,a\,d^2}{d^4}}+\mathrm {atanh}\left (\frac {d^3\,\left (\frac {16\,\left (B^2\,b^4-B^2\,a^2\,b^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^2}-\frac {16\,a\,b^2\,\left (\sqrt {-B^4\,b^2\,d^4}-B^2\,a\,d^2\right )\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d^4}\right )\,\sqrt {\frac {\sqrt {-B^4\,b^2\,d^4}-B^2\,a\,d^2}{d^4}}}{16\,\left (B^3\,a^2\,b^3+B^3\,b^5\right )}\right )\,\sqrt {\frac {\sqrt {-B^4\,b^2\,d^4}-B^2\,a\,d^2}{d^4}}-2\,\mathrm {atanh}\left (\frac {32\,A^2\,b^4\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}+\frac {A^2\,a}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d^3}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d^3}}+\frac {32\,a\,b^2\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}+\frac {A^2\,a}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-A^4\,b^2\,d^4}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d}}\right )\,\sqrt {\frac {\sqrt {-A^4\,b^2\,d^4}+A^2\,a\,d^2}{4\,d^4}}+2\,\mathrm {atanh}\left (\frac {32\,A^2\,b^4\,\sqrt {\frac {A^2\,a}{4\,d^2}-\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d^3}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d^3}}-\frac {32\,a\,b^2\,\sqrt {\frac {A^2\,a}{4\,d^2}-\frac {\sqrt {-A^4\,b^2\,d^4}}{4\,d^4}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-A^4\,b^2\,d^4}}{\frac {16\,A\,b^4\,\sqrt {-A^4\,b^2\,d^4}}{d}+\frac {16\,A\,a^2\,b^2\,\sqrt {-A^4\,b^2\,d^4}}{d}}\right )\,\sqrt {-\frac {\sqrt {-A^4\,b^2\,d^4}-A^2\,a\,d^2}{4\,d^4}}+\frac {2\,A\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {2\,B\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{3\,b\,d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atanh((d^3*((16*(B^2*b^4 - B^2*a^2*b^2)*(a + b*tan(c + d*x))^(1/2))/d^2 + (16*a*b^2*((-B^4*b^2*d^4)^(1/2) + B^
2*a*d^2)*(a + b*tan(c + d*x))^(1/2))/d^4)*(-((-B^4*b^2*d^4)^(1/2) + B^2*a*d^2)/d^4)^(1/2))/(16*(B^3*b^5 + B^3*
a^2*b^3)))*(-((-B^4*b^2*d^4)^(1/2) + B^2*a*d^2)/d^4)^(1/2) + atanh((d^3*((16*(B^2*b^4 - B^2*a^2*b^2)*(a + b*ta
n(c + d*x))^(1/2))/d^2 - (16*a*b^2*((-B^4*b^2*d^4)^(1/2) - B^2*a*d^2)*(a + b*tan(c + d*x))^(1/2))/d^4)*(((-B^4
*b^2*d^4)^(1/2) - B^2*a*d^2)/d^4)^(1/2))/(16*(B^3*b^5 + B^3*a^2*b^3)))*(((-B^4*b^2*d^4)^(1/2) - B^2*a*d^2)/d^4
)^(1/2) - 2*atanh((32*A^2*b^4*((-A^4*b^2*d^4)^(1/2)/(4*d^4) + (A^2*a)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2
))/((16*A*b^4*(-A^4*b^2*d^4)^(1/2))/d^3 + (16*A*a^2*b^2*(-A^4*b^2*d^4)^(1/2))/d^3) + (32*a*b^2*((-A^4*b^2*d^4)
^(1/2)/(4*d^4) + (A^2*a)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(-A^4*b^2*d^4)^(1/2))/((16*A*b^4*(-A^4*b^2*
d^4)^(1/2))/d + (16*A*a^2*b^2*(-A^4*b^2*d^4)^(1/2))/d))*(((-A^4*b^2*d^4)^(1/2) + A^2*a*d^2)/(4*d^4))^(1/2) + 2
*atanh((32*A^2*b^4*((A^2*a)/(4*d^2) - (-A^4*b^2*d^4)^(1/2)/(4*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*A*b
^4*(-A^4*b^2*d^4)^(1/2))/d^3 + (16*A*a^2*b^2*(-A^4*b^2*d^4)^(1/2))/d^3) - (32*a*b^2*((A^2*a)/(4*d^2) - (-A^4*b
^2*d^4)^(1/2)/(4*d^4))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(-A^4*b^2*d^4)^(1/2))/((16*A*b^4*(-A^4*b^2*d^4)^(1/2))
/d + (16*A*a^2*b^2*(-A^4*b^2*d^4)^(1/2))/d))*(-((-A^4*b^2*d^4)^(1/2) - A^2*a*d^2)/(4*d^4))^(1/2) + (2*A*(a + b
*tan(c + d*x))^(1/2))/d + (2*B*(a + b*tan(c + d*x))^(3/2))/(3*b*d)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \sqrt {a + b \tan {\left (c + d x \right )}} \tan {\left (c + d x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((A + B*tan(c + d*x))*sqrt(a + b*tan(c + d*x))*tan(c + d*x), x)

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