3.344 \(\int \frac {\tan ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx\)
Optimal. Leaf size=166 \[ \frac {2 (3 A b-2 a B) \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {(-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}+\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d} \]
[Out]
(I*A+B)*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/d/(a-I*b)^(1/2)-(I*A-B)*arctanh((a+b*tan(d*x+c))^(1/2)/(
a+I*b)^(1/2))/d/(a+I*b)^(1/2)+2/3*(3*A*b-2*B*a)*(a+b*tan(d*x+c))^(1/2)/b^2/d+2/3*B*(a+b*tan(d*x+c))^(1/2)*tan(
d*x+c)/b/d
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Rubi [A] time = 0.36, antiderivative size = 166, normalized size of antiderivative = 1.00,
number of steps used = 9, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used =
{3607, 3630, 3539, 3537, 63, 208} \[ \frac {2 (3 A b-2 a B) \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {(B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {(-B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}+\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d} \]
Antiderivative was successfully verified.
[In]
Int[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]
[Out]
((I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) - ((I*A - B)*ArcTanh[Sqrt[a + b*
Tan[c + d*x]]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d) + (2*(3*A*b - 2*a*B)*Sqrt[a + b*Tan[c + d*x]])/(3*b^2*d) + (2*
B*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]])/(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 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 3607
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*B*(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^(n + 1))/(d*
f*(m + n)), x] + Dist[1/(d*(m + n)), Int[(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^n*Simp[a^2*A*d*(m +
n) - b*B*(b*c*(m - 1) + a*d*(n + 1)) + d*(m + n)*(2*a*A*b + B*(a^2 - b^2))*Tan[e + f*x] - (b*B*(b*c - a*d)*(m
- 1) - b*(A*b + a*B)*d*(m + n))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*
c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 1] && (IntegerQ[m] || IntegersQ[2*m, 2*n]) &&
!(IGtQ[n, 1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
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 {\tan ^2(c+d x) (A+B \tan (c+d x))}{\sqrt {a+b \tan (c+d x)}} \, dx &=\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}+\frac {2 \int \frac {-a B-\frac {3}{2} b B \tan (c+d x)+\frac {1}{2} (3 A b-2 a B) \tan ^2(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 b}\\ &=\frac {2 (3 A b-2 a B) \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}+\frac {2 \int \frac {-\frac {3 A b}{2}-\frac {3}{2} b B \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx}{3 b}\\ &=\frac {2 (3 A b-2 a B) \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}+\frac {1}{2} (-A-i B) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx+\frac {1}{2} (-A+i B) \int \frac {1+i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=\frac {2 (3 A b-2 a B) \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}+\frac {(i A-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 {(i A+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 (3 A b-2 a B) \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}+\frac {(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 {(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) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b} d}-\frac {(i A-B) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b} d}+\frac {2 (3 A b-2 a B) \sqrt {a+b \tan (c+d x)}}{3 b^2 d}+\frac {2 B \tan (c+d x) \sqrt {a+b \tan (c+d x)}}{3 b d}\\ \end {align*}
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Mathematica [A] time = 1.59, size = 139, normalized size = 0.84 \[ \frac {\frac {2 \sqrt {a+b \tan (c+d x)} (-2 a B+3 A b+b B \tan (c+d x))}{b^2}+\frac {3 (B+i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{\sqrt {a-i b}}+\frac {3 (B-i A) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{\sqrt {a+i b}}}{3 d} \]
Antiderivative was successfully verified.
[In]
Integrate[(Tan[c + d*x]^2*(A + B*Tan[c + d*x]))/Sqrt[a + b*Tan[c + d*x]],x]
[Out]
((3*(I*A + B)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]])/Sqrt[a - I*b] + (3*((-I)*A + B)*ArcTanh[Sqrt[a
+ b*Tan[c + d*x]]/Sqrt[a + I*b]])/Sqrt[a + I*b] + (2*Sqrt[a + b*Tan[c + d*x]]*(3*A*b - 2*a*B + b*B*Tan[c + d*x
]))/b^2)/(3*d)
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fricas [B] time = 5.47, size = 8468, normalized size = 51.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")
[Out]
-1/12*(12*sqrt(2)*(a^2*b^2 + b^4)*d^5*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^
2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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^5 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^4*b + 4
*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a^3*b^2 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^3 + 2*(A^7*B +
3*A^5*B^3 + 3*A^3*B^5 + A*B^7)*a*b^4 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^5)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/(
(a^2 + 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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*((B*a^5 - A*a^4*b + 2*B*a^3*b^2 - 2*
A*a^2*b^3 + B*a*b^4 - A*b^5)*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)/((
a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + ((A^2*B + B^3)*a^4 + 2*(A^2*B +
B^3)*a^2*b^2 + (A^2*B + B^3)*b^4)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*
sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((4*A^2*B^3*a^4
- 4*(A^3*B^2 - A*B^4)*a^3*b + (A^4*B + 2*A^2*B^3 + B^5)*a^2*b^2 - 4*(A^3*B^2 - A*B^4)*a*b^3 + (A^4*B - 2*A^2*
B^3 + B^5)*b^4)*d^3*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^4*B^3 + A^2*B^5)*a^3
- 4*(2*A^5*B^2 + A^3*B^4 - A*B^6)*a^2*b + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)*a*b^2 - (A^7 - A^5*B^2 - A^3*B
^4 + A*B^6)*b^3)*d*cos(d*x + c))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqr
t((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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 + b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^3 - 4*(A^7*
B + A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b + (A^8 - 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 +
A^2*B^6)*a^2*b - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(d*x + c))/cos
(d*x + c))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))^(3/4) + sqrt(2)*((2*(A^3*B^2 + A*B^4)*a^6 - (3*A^4*B +
2*A^2*B^3 - B^5)*a^5*b + (A^5 + 4*A^3*B^2 + 3*A*B^4)*a^4*b^2 - 2*(3*A^4*B + 2*A^2*B^3 - B^5)*a^3*b^3 + 2*(A^5
+ A^3*B^2)*a^2*b^4 - (3*A^4*B + 2*A^2*B^3 - B^5)*a*b^5 + (A^5 - A*B^4)*b^6)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2
+ b^2)*d^4)) + (2*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*a^5 - (A^6*B + A^4*B^3 - A^2*B^5 - B^7)*a^4*b + 4*(A^5*B^2 +
2*A^3*B^4 + A*B^6)*a^3*b^2 - 2*(A^6*B + A^4*B^3 - A^2*B^5 - B^7)*a^2*b^3 + 2*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*a*b
^4 - (A^6*B + A^4*B^3 - A^2*B^5 - B^7)*b^5)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*
b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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 + 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^2*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*b^2 + (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^3))*cos(d*x + c) + 12*sqrt(2)*(a^2*b^2
+ b^4)*d^5*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 +
B^4)/((a^2 + 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)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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^5 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^4*b + 4*(A^7*B + 3*A^5*B^3 + 3*A
^3*B^5 + A*B^7)*a^3*b^2 - 2*(A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*a^2*b^3 + 2*(A^7*B + 3*A^5*B^3 + 3*A^3*B^5 + A
*B^7)*a*b^4 - (A^8 + 2*A^6*B^2 - 2*A^2*B^6 - B^8)*b^5)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*((B*a^5 - A*a^4*b + 2*B*a^3*b^2 - 2*A*a^2*b^3 + B*a*b^4 - A*b
^5)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^
4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + ((A^2*B + B^3)*a^4 + 2*(A^2*B + B^3)*a^2*b^2 + (A^2*B +
B^3)*b^4)*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)/((a^4 + 2*a^2*b^2 + b
^4)*d^4)))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B
^4)/((a^2 + 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*s
qrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((4*A^2*B^3*a^4 - 4*(A^3*B^2 - A*B^4)*a^
3*b + (A^4*B + 2*A^2*B^3 + B^5)*a^2*b^2 - 4*(A^3*B^2 - A*B^4)*a*b^3 + (A^4*B - 2*A^2*B^3 + B^5)*b^4)*d^3*sqrt(
(A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^4*B^3 + A^2*B^5)*a^3 - 4*(2*A^5*B^2 + A^3*B^4
- A*B^6)*a^2*b + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)*a*b^2 - (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^3)*d*cos(d*
x + c))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)
/((a^2 + 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 + b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A
*B^7)*a^2*b + (A^8 - 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b - 4*(A^7*
B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(d*x + c))/cos(d*x + c))*((A^4 + 2*A^2*
B^2 + B^4)/((a^2 + b^2)*d^4))^(3/4) - sqrt(2)*((2*(A^3*B^2 + A*B^4)*a^6 - (3*A^4*B + 2*A^2*B^3 - B^5)*a^5*b +
(A^5 + 4*A^3*B^2 + 3*A*B^4)*a^4*b^2 - 2*(3*A^4*B + 2*A^2*B^3 - B^5)*a^3*b^3 + 2*(A^5 + A^3*B^2)*a^2*b^4 - (3*A
^4*B + 2*A^2*B^3 - B^5)*a*b^5 + (A^5 - A*B^4)*b^6)*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)/((a^4 + 2*a^2*b^2 + b^4)*d^4))*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4)) + (2*(A^5*B
^2 + 2*A^3*B^4 + A*B^6)*a^5 - (A^6*B + A^4*B^3 - A^2*B^5 - B^7)*a^4*b + 4*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*a^3*b^
2 - 2*(A^6*B + A^4*B^3 - A^2*B^5 - B^7)*a^2*b^3 + 2*(A^5*B^2 + 2*A^3*B^4 + A*B^6)*a*b^4 - (A^6*B + A^4*B^3 - A
^2*B^5 - B^7)*b^5)*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)/((a^4 + 2*a^
2*b^2 + b^4)*d^4)))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^
2*B^2 + B^4)/((a^2 + 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 + 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
^2*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*b^2 + (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^3))*cos(d*x + c) + 3*sqrt(2)*((2*A*B*b^3 + (A^2 - B^2)*a*b^2)*d
^3*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) - (A^4 + 2*A^2*B^2 + B^4)*b^2*d*cos(d*x + c))*
sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 +
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 + 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 + b^2)*d^4))*cos(d*x + c) + sqrt(2)*((
4*A^2*B^3*a^4 - 4*(A^3*B^2 - A*B^4)*a^3*b + (A^4*B + 2*A^2*B^3 + B^5)*a^2*b^2 - 4*(A^3*B^2 - A*B^4)*a*b^3 + (A
^4*B - 2*A^2*B^3 + B^5)*b^4)*d^3*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^4*B^3 +
A^2*B^5)*a^3 - 4*(2*A^5*B^2 + A^3*B^4 - A*B^6)*a^2*b + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)*a*b^2 - (A^7 - A^
5*B^2 - A^3*B^4 + A*B^6)*b^3)*d*cos(d*x + c))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a
*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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 + b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*
a^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a^2*b + (A^8 - 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2
+ 2*A^4*B^4 + A^2*B^6)*a^2*b - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(
d*x + c))/cos(d*x + c)) - 3*sqrt(2)*((2*A*B*b^3 + (A^2 - B^2)*a*b^2)*d^3*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 +
b^2)*d^4))*cos(d*x + c) - (A^4 + 2*A^2*B^2 + B^4)*b^2*d*cos(d*x + c))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 -
B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/((a^2 + 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 + 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*sqr
t((A^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) - sqrt(2)*((4*A^2*B^3*a^4 - 4*(A^3*B^2 - A*B^4)*a^3*
b + (A^4*B + 2*A^2*B^3 + B^5)*a^2*b^2 - 4*(A^3*B^2 - A*B^4)*a*b^3 + (A^4*B - 2*A^2*B^3 + B^5)*b^4)*d^3*sqrt((A
^4 + 2*A^2*B^2 + B^4)/((a^2 + b^2)*d^4))*cos(d*x + c) + (4*(A^4*B^3 + A^2*B^5)*a^3 - 4*(2*A^5*B^2 + A^3*B^4 -
A*B^6)*a^2*b + (5*A^6*B - A^4*B^3 - 5*A^2*B^5 + B^7)*a*b^2 - (A^7 - A^5*B^2 - A^3*B^4 + A*B^6)*b^3)*d*cos(d*x
+ c))*sqrt(((2*A*B*a^2*b + 2*A*B*b^3 + (A^2 - B^2)*a^3 + (A^2 - B^2)*a*b^2)*d^2*sqrt((A^4 + 2*A^2*B^2 + B^4)/(
(a^2 + 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 + b^2)*d^4))^(1/4) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^3 - 4*(A^7*B + A^5*B^3 - A^3*B^5 - A*B
^7)*a^2*b + (A^8 - 2*A^4*B^4 + B^8)*a*b^2)*cos(d*x + c) + (4*(A^6*B^2 + 2*A^4*B^4 + A^2*B^6)*a^2*b - 4*(A^7*B
+ A^5*B^3 - A^3*B^5 - A*B^7)*a*b^2 + (A^8 - 2*A^4*B^4 + B^8)*b^3)*sin(d*x + c))/cos(d*x + c)) - 8*((A^4*B + 2*
A^2*B^3 + B^5)*b*sin(d*x + c) - (2*(A^4*B + 2*A^2*B^3 + B^5)*a - 3*(A^5 + 2*A^3*B^2 + A*B^4)*b)*cos(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)*b^2*d*cos(d*x + c))
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")
[Out]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*p
i/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Unab
le to check sign: (2*pi/x/2)>(-2*pi/x/2)Unable to check sign: (2*pi/x/2)>(-2*pi/x/2)Warning, need to choose a
branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [d]=[-49
,-25]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need
to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done assum
ing [d]=[46,-92]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym
2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const g
en & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_
m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur
& l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Ar
gument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2pol
y/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen &
e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m &
i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l
) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argume
nt Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueUnable to c
heck sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unabl
e to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2
)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_no
step/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*p
i/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)
>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nos
tep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi
/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign:
(2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check
sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to
check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Una
ble to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep
/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_
nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2
*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/
2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_n
ostep/2)>(-2*pi/t_nostep/2)Unable to check sign: (2*pi/t_nostep/2)>(-2*pi/t_nostep/2)Warning, integration of a
bs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep^2-1)]Disco
ntinuities at zeroes of t_nostep^2-1 were not checkedEvaluation time: 81.9Done
________________________________________________________________________________________
maple [B] time = 0.33, size = 4040, normalized size = 24.34 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x)
[Out]
-1/d*b/(a^2+b^2)^(1/2)/(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+1/d/b^2/(a^2+b^2)/(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*a^4+1/4/d/b/(a^2+b^2)^(3/2)*ln(b*ta
n(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)*a^3+1/4/d*b/(a^2+b^2)^(3/2)*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)*a+1/4/d*b/(a^2+b^2)*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)-2/d*b^3/(a^2+b^2)^(3/2)/(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+1/d*b/(a^2+b^2)^(1/2)/(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-1/d/(a^2+b^2)^(1/2)/(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*a-1/4/d/(a^2+b^2)*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/b/(a^2+b^2)^(3/2)/(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^4+1/d/b^2*(a^2+b^2)^(1/2)/(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*a-1/d/b^2/(a^2+b^2)^(1/2)/(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*a^3+2/d/(a^2+b^2)/(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*ta
n(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a^2+1/d/(a^2+b^2)^(1/2)/(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*a-1/4/d*b^2/(a^2+b^
2)^(3/2)*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)+2/d*b^3/(a^2+b^2)^(3/2)/(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+1/4/d/b^2*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/b^2/(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*a^2
-1/4/d/b^2*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/b^2/(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*a^2-1/d*b^2/(a^2+b^2)/(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*b/(a^2+b^2)*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/d*b^2/(a^2+b^2)/(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+1/4/d*b^2/(a^2+b^2)^(3/2)*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)+1/d/b/(a^2+b^2)^(3/2)/(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^4-1/d/b^2*(a^2+b^2)^(1/2)/(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*ta
n(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2)-2*a)^(1/2))*B*a+1/d/b^2/(a^2+b^2)^(1/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arct
an(((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*a^3+1/d/b/(a^2+b^
2)^(1/2)/(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^2-1/4/d/(a^2+b^2)^(3/2)*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^2-3/d*b/(a^2+b^2)^(3/2)/(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^2*A+1
/d*b^2/(a^2+b^2)^(3/2)/(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*B-1/4/d/b/(a^2+b^2)*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)*a^2-1/d/b^2/(a^2+b^2)/(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*a
^4+1/4/d/b^2/(a^2+b^2)*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^3-1/4/d/b/(a^2+b^2)^(3/2)*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)*a^3-1/4/d*b/(a^2+b^2)^(3/2)*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)*a
-1/4/d/b^2/(a^2+b^2)*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^3+1/4/d/b/(a^2+b^2)*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)*a^2-1/d/b/(a^2+b^2)^(1/2)/(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+
3/d*b/(a^2+b^2)^(3/2)/(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^2*A-1/d*b^2/(a^2+b^2)^(3/2)/(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*B+2/d/b*A*(a+b*tan(d*x+c))
^(1/2)-1/d/(a^2+b^2)^(3/2)/(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^3*B-2/d/(a^2+b^2)/(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*a^2+1/4/d/(a^2+b^2)*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
+1/4/d/(a^2+b^2)^(3/2)*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^2+1/d/(a^2+b^2)^(3/2)/(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^3*B+2/3/d/b^2*B*(a+b*tan(d*x+c))^(3/
2)-2/d/b^2*B*(a+b*tan(d*x+c))^(1/2)*a
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{2}}{\sqrt {b \tan \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)^2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")
[Out]
integrate((B*tan(d*x + c) + A)*tan(d*x + c)^2/sqrt(b*tan(d*x + c) + a), x)
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mupad [B] time = 10.55, size = 2981, normalized size = 17.96 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((tan(c + d*x)^2*(A + B*tan(c + d*x)))/(a + b*tan(c + d*x))^(1/2),x)
[Out]
2*atanh((32*A^2*b^2*((-16*A^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) - (A^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(
1/2)*(a + b*tan(c + d*x))^(1/2))/((16*A^3*a*b^3*d^3)/(a^2*d^4 + b^2*d^4) - (4*A*b^3*d^2*(-16*A^4*b^2*d^4)^(1/2
))/(a^2*d^5 + b^2*d^5)) + (8*a*b^2*((-16*A^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) - (A^2*a*d^2)/(4*(a^2*d^4
+ b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(-16*A^4*b^2*d^4)^(1/2))/((16*A^3*a*b^5*d^5)/(a^2*d^4 + b^2*d^4
) - (4*A*b^5*d^4*(-16*A^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (16*A^3*a^3*b^3*d^5)/(a^2*d^4 + b^2*d^4) - (4*
A*a^2*b^3*d^4*(-16*A^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) - (32*A^2*a^2*b^2*d^2*((-16*A^4*b^2*d^4)^(1/2)/(16
*(a^2*d^4 + b^2*d^4)) - (A^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*A^3*a*b^5*
d^5)/(a^2*d^4 + b^2*d^4) - (4*A*b^5*d^4*(-16*A^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (16*A^3*a^3*b^3*d^5)/(a
^2*d^4 + b^2*d^4) - (4*A*a^2*b^3*d^4*(-16*A^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)))*((-16*A^4*b^2*d^4)^(1/2)/(
16*(a^2*d^4 + b^2*d^4)) - (A^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2) - atan((a*b^2*((-16*B^4*b^2*d^4)^(1/2)/(1
6*(a^2*d^4 + b^2*d^4)) + (B^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(-16*B^4*b^2*d^
4)^(1/2)*8i)/((16*B^3*a^2*b^4*d^5)/(a^2*d^4 + b^2*d^4) - 16*B^3*a^2*b^2*d - 16*B^3*b^4*d + (16*B^3*a^4*b^2*d^5
)/(a^2*d^4 + b^2*d^4) + (4*B*a^3*b^2*d^4*(-16*B^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (4*B*a*b^4*d^4*(-16*B^
4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) - (B^2*b^2*((-16*B^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) + (B^2*a*d
^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((16*B^3*a^2*b^2*d^3)/(a^2*d^4 + b^2*d^4) -
(16*B^3*b^2)/d + (4*B*a*b^2*d^2*(-16*B^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) + (B^2*a^2*b^2*d^2*((-16*B^4*b^
2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) + (B^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*
32i)/((16*B^3*a^2*b^4*d^5)/(a^2*d^4 + b^2*d^4) - 16*B^3*a^2*b^2*d - 16*B^3*b^4*d + (16*B^3*a^4*b^2*d^5)/(a^2*d
^4 + b^2*d^4) + (4*B*a^3*b^2*d^4*(-16*B^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (4*B*a*b^4*d^4*(-16*B^4*b^2*d^
4)^(1/2))/(a^2*d^5 + b^2*d^5)))*((-16*B^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) + (B^2*a*d^2)/(4*(a^2*d^4 +
b^2*d^4)))^(1/2)*2i - atan((B^2*b^2*((B^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)) - (-16*B^4*b^2*d^4)^(1/2)/(16*(a^2*d^
4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/((16*B^3*b^2)/d - (16*B^3*a^2*b^2*d^3)/(a^2*d^4 + b^2*d^4
) + (4*B*a*b^2*d^2*(-16*B^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) + (a*b^2*((B^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4))
- (-16*B^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(-16*B^4*b^2*d^4)^(1/2)*
8i)/(16*B^3*b^4*d + 16*B^3*a^2*b^2*d - (16*B^3*a^2*b^4*d^5)/(a^2*d^4 + b^2*d^4) - (16*B^3*a^4*b^2*d^5)/(a^2*d^
4 + b^2*d^4) + (4*B*a^3*b^2*d^4*(-16*B^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (4*B*a*b^4*d^4*(-16*B^4*b^2*d^4
)^(1/2))/(a^2*d^5 + b^2*d^5)) - (B^2*a^2*b^2*d^2*((B^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)) - (-16*B^4*b^2*d^4)^(1/2
)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*32i)/(16*B^3*b^4*d + 16*B^3*a^2*b^2*d - (16*B^3*a
^2*b^4*d^5)/(a^2*d^4 + b^2*d^4) - (16*B^3*a^4*b^2*d^5)/(a^2*d^4 + b^2*d^4) + (4*B*a^3*b^2*d^4*(-16*B^4*b^2*d^4
)^(1/2))/(a^2*d^5 + b^2*d^5) + (4*B*a*b^4*d^4*(-16*B^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)))*((B^2*a*d^2)/(4*(
a^2*d^4 + b^2*d^4)) - (-16*B^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2)*2i - 2*atanh((8*a*b^2*(- (-16*A^
4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) - (A^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1
/2)*(-16*A^4*b^2*d^4)^(1/2))/((16*A^3*a*b^5*d^5)/(a^2*d^4 + b^2*d^4) + (4*A*b^5*d^4*(-16*A^4*b^2*d^4)^(1/2))/(
a^2*d^5 + b^2*d^5) + (16*A^3*a^3*b^3*d^5)/(a^2*d^4 + b^2*d^4) + (4*A*a^2*b^3*d^4*(-16*A^4*b^2*d^4)^(1/2))/(a^2
*d^5 + b^2*d^5)) - (32*A^2*b^2*(- (-16*A^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) - (A^2*a*d^2)/(4*(a^2*d^4 +
b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*A^3*a*b^3*d^3)/(a^2*d^4 + b^2*d^4) + (4*A*b^3*d^2*(-16*A^4*
b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) + (32*A^2*a^2*b^2*d^2*(- (-16*A^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4))
- (A^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*A^3*a*b^5*d^5)/(a^2*d^4 + b^2*d
^4) + (4*A*b^5*d^4*(-16*A^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (16*A^3*a^3*b^3*d^5)/(a^2*d^4 + b^2*d^4) + (
4*A*a^2*b^3*d^4*(-16*A^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)))*(- (-16*A^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d
^4)) - (A^2*a*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2) + (2*A*(a + b*tan(c + d*x))^(1/2))/(b*d) + (2*B*(a + b*tan(c
+ d*x))^(3/2))/(3*b^2*d) - (2*B*a*(a + b*tan(c + d*x))^(1/2))/(b^2*d)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{2}{\left (c + d x \right )}}{\sqrt {a + b \tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(tan(d*x+c)**2*(A+B*tan(d*x+c))/(a+b*tan(d*x+c))**(1/2),x)
[Out]
Integral((A + B*tan(c + d*x))*tan(c + d*x)**2/sqrt(a + b*tan(c + d*x)), x)
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