3.1250 \(\int \frac {a+b \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\)

Optimal. Leaf size=102 \[ \frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3539, 3537, 63, 208} \[ \frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f \sqrt {c+i d}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f \sqrt {c-i d}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Tan[e + f*x])/Sqrt[c + d*Tan[e + f*x]],x]

[Out]

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.13, size = 101, normalized size = 0.99 \[ \frac {i \left (\frac {(a+i b) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{\sqrt {c+i d}}-\frac {(a-i b) \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{\sqrt {c-i d}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Tan[e + f*x])/Sqrt[c + d*Tan[e + f*x]],x]

[Out]

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

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fricas [B]  time = 9.39, size = 8282, normalized size = 81.20 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*sqrt(2)*(c^2 + d^2)*f^4*sqrt(((2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt(
(a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2
*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2))*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d +
(a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))^(3/4)
*arctan(((2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^5 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^4*d + 4*(a^7*b
+ 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^3*d^2 - 2*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^2*d^3 + 2*(a^7*b + 3*a^5*b
^3 + 3*a^3*b^5 + a*b^7)*c*d^4 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*d^5)*f^4*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b -
a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4))*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 +
d^2)*f^4)) + (2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^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)*c^3*d + 2*(a^9*b + 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^2*d^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)*c*d^3)*f^2*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^
4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)) - sqrt(2)*((b*c^5 - a*c^4*d + 2*b*c^3*d^2 - 2*a*c^2*d
^3 + b*c*d^4 - a*d^5)*f^7*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2
*c^2*d^2 + d^4)*f^4))*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + ((a^2*b + b^3)*c^4 + 2*(a^2*b + b^3)*c
^2*d^2 + (a^2*b + b^3)*d^4)*f^5*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c
^4 + 2*c^2*d^2 + d^4)*f^4)))*sqrt(((2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a
^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b
^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2))*sqrt(((4*(a^4*b^2 + a^2*b^4)*c^4 - 4*(a^5*b - a
*b^5)*c^3*d + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d^2 - 4*(a^5*b - a*b^5)*c*d^3 + (a^6 - a^4*b^2 - a^2*b^4
+ b^6)*d^4)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))*cos(f*x + e) + sqrt(2)*((4*a^2*b^3*c^4 - 4*(a
^3*b^2 - a*b^4)*c^3*d + (a^4*b + 2*a^2*b^3 + b^5)*c^2*d^2 - 4*(a^3*b^2 - a*b^4)*c*d^3 + (a^4*b - 2*a^2*b^3 + b
^5)*d^4)*f^3*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))*cos(f*x + e) + (4*(a^4*b^3 + a^2*b^5)*c^3 - 4*(2*
a^5*b^2 + a^3*b^4 - a*b^6)*c^2*d + (5*a^6*b - a^4*b^3 - 5*a^2*b^5 + b^7)*c*d^2 - (a^7 - a^5*b^2 - a^3*b^4 + a*
b^6)*d^3)*f*cos(f*x + e))*sqrt(((2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a^4
+ 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*
c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2))*sqrt((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e
))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))^(1/4) + (4*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^3 - 4*(a^7*b + a^5
*b^3 - a^3*b^5 - a*b^7)*c^2*d + (a^8 - 2*a^4*b^4 + b^8)*c*d^2)*cos(f*x + e) + (4*(a^6*b^2 + 2*a^4*b^4 + a^2*b^
6)*c^2*d - 4*(a^7*b + a^5*b^3 - a^3*b^5 - a*b^7)*c*d^2 + (a^8 - 2*a^4*b^4 + b^8)*d^3)*sin(f*x + e))/cos(f*x +
e))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))^(3/4) + sqrt(2)*((2*(a^3*b^2 + a*b^4)*c^6 - (3*a^4*b + 2*a^2*b
^3 - b^5)*c^5*d + (a^5 + 4*a^3*b^2 + 3*a*b^4)*c^4*d^2 - 2*(3*a^4*b + 2*a^2*b^3 - b^5)*c^3*d^3 + 2*(a^5 + a^3*b
^2)*c^2*d^4 - (3*a^4*b + 2*a^2*b^3 - b^5)*c*d^5 + (a^5 - a*b^4)*d^6)*f^7*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^
3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4))*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)
*f^4)) + (2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*c^5 - (a^6*b + a^4*b^3 - a^2*b^5 - b^7)*c^4*d + 4*(a^5*b^2 + 2*a^3*b
^4 + a*b^6)*c^3*d^2 - 2*(a^6*b + a^4*b^3 - a^2*b^5 - b^7)*c^2*d^3 + 2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*c*d^4 - (a
^6*b + a^4*b^3 - a^2*b^5 - b^7)*d^5)*f^5*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)
*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)))*sqrt(((2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^
2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)
/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2))*sqrt((c*cos(f*x + e) + d*sin(f*x + e))
/cos(f*x + e))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^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)*c^2*d - 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)*c*d^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)*d^3)) + 4*sqrt(2)*(c^2 + d^2)*f^4*sqrt(((2*a*b*
c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) +
(a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*
a^2*b^2 + b^4)*d^2))*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*
d^2 + d^4)*f^4))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))^(3/4)*arctan(-((2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5
+ a*b^7)*c^5 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^4*d + 4*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^3*d^2 -
2*(a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^2*d^3 + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c*d^4 - (a^8 + 2*a^
6*b^2 - 2*a^2*b^6 - b^8)*d^5)*f^4*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/(
(c^4 + 2*c^2*d^2 + d^4)*f^4))*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (2*(a^9*b + 4*a^7*b^3 + 6*a^5*
b^5 + 4*a^3*b^7 + a*b^9)*c^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)*c^3*d + 2*(a^9*b
+ 4*a^7*b^3 + 6*a^5*b^5 + 4*a^3*b^7 + a*b^9)*c^2*d^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)*c*d^3)*f^2*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2
+ d^4)*f^4)) + sqrt(2)*((b*c^5 - a*c^4*d + 2*b*c^3*d^2 - 2*a*c^2*d^3 + b*c*d^4 - a*d^5)*f^7*sqrt((4*a^2*b^2*c
^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4))*sqrt((a^4 + 2*a^2*b^2
+ b^4)/((c^2 + d^2)*f^4)) + ((a^2*b + b^3)*c^4 + 2*(a^2*b + b^3)*c^2*d^2 + (a^2*b + b^3)*d^4)*f^5*sqrt((4*a^2
*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)))*sqrt(((2*a*b*c
^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) +
(a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a
^2*b^2 + b^4)*d^2))*sqrt(((4*(a^4*b^2 + a^2*b^4)*c^4 - 4*(a^5*b - a*b^5)*c^3*d + (a^6 + 3*a^4*b^2 + 3*a^2*b^4
+ b^6)*c^2*d^2 - 4*(a^5*b - a*b^5)*c*d^3 + (a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^4)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^
4)/((c^2 + d^2)*f^4))*cos(f*x + e) - sqrt(2)*((4*a^2*b^3*c^4 - 4*(a^3*b^2 - a*b^4)*c^3*d + (a^4*b + 2*a^2*b^3
+ b^5)*c^2*d^2 - 4*(a^3*b^2 - a*b^4)*c*d^3 + (a^4*b - 2*a^2*b^3 + b^5)*d^4)*f^3*sqrt((a^4 + 2*a^2*b^2 + b^4)/(
(c^2 + d^2)*f^4))*cos(f*x + e) + (4*(a^4*b^3 + a^2*b^5)*c^3 - 4*(2*a^5*b^2 + a^3*b^4 - a*b^6)*c^2*d + (5*a^6*b
- a^4*b^3 - 5*a^2*b^5 + b^7)*c*d^2 - (a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3)*f*cos(f*x + e))*sqrt(((2*a*b*c^2*
d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^
4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*
b^2 + b^4)*d^2))*sqrt((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^
4))^(1/4) + (4*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^3 - 4*(a^7*b + a^5*b^3 - a^3*b^5 - a*b^7)*c^2*d + (a^8 - 2*a^
4*b^4 + b^8)*c*d^2)*cos(f*x + e) + (4*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^2*d - 4*(a^7*b + a^5*b^3 - a^3*b^5 - a
*b^7)*c*d^2 + (a^8 - 2*a^4*b^4 + b^8)*d^3)*sin(f*x + e))/cos(f*x + e))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f
^4))^(3/4) - sqrt(2)*((2*(a^3*b^2 + a*b^4)*c^6 - (3*a^4*b + 2*a^2*b^3 - b^5)*c^5*d + (a^5 + 4*a^3*b^2 + 3*a*b^
4)*c^4*d^2 - 2*(3*a^4*b + 2*a^2*b^3 - b^5)*c^3*d^3 + 2*(a^5 + a^3*b^2)*c^2*d^4 - (3*a^4*b + 2*a^2*b^3 - b^5)*c
*d^5 + (a^5 - a*b^4)*d^6)*f^7*sqrt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4
+ 2*c^2*d^2 + d^4)*f^4))*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*c
^5 - (a^6*b + a^4*b^3 - a^2*b^5 - b^7)*c^4*d + 4*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*c^3*d^2 - 2*(a^6*b + a^4*b^3 -
a^2*b^5 - b^7)*c^2*d^3 + 2*(a^5*b^2 + 2*a^3*b^4 + a*b^6)*c*d^4 - (a^6*b + a^4*b^3 - a^2*b^5 - b^7)*d^5)*f^5*sq
rt((4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2)/((c^4 + 2*c^2*d^2 + d^4)*f^4)))*sqrt(
((2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)
*f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (
a^4 - 2*a^2*b^2 + b^4)*d^2))*sqrt((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))*((a^4 + 2*a^2*b^2 + b^4)/((c
^2 + d^2)*f^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)*c^2*d - 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)*c*d^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)*d^3)) - sqrt(2)*(a^4 + 2*a^2*b^2 + b^4 - (2*a*b*d + (a^2 - b^2)*c)*f^2*sqrt((a^4 + 2*a^2*b
^2 + b^4)/((c^2 + d^2)*f^4)))*sqrt(((2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((
a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*
b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))^(1
/4)*log(((4*(a^4*b^2 + a^2*b^4)*c^4 - 4*(a^5*b - a*b^5)*c^3*d + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*c^2*d^2 -
4*(a^5*b - a*b^5)*c*d^3 + (a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^4)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f
^4))*cos(f*x + e) + sqrt(2)*((4*a^2*b^3*c^4 - 4*(a^3*b^2 - a*b^4)*c^3*d + (a^4*b + 2*a^2*b^3 + b^5)*c^2*d^2 -
4*(a^3*b^2 - a*b^4)*c*d^3 + (a^4*b - 2*a^2*b^3 + b^5)*d^4)*f^3*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))
*cos(f*x + e) + (4*(a^4*b^3 + a^2*b^5)*c^3 - 4*(2*a^5*b^2 + a^3*b^4 - a*b^6)*c^2*d + (5*a^6*b - a^4*b^3 - 5*a^
2*b^5 + b^7)*c*d^2 - (a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3)*f*cos(f*x + e))*sqrt(((2*a*b*c^2*d + 2*a*b*d^3 + (
a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^4 + 2*a^2*b^2 + b
^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2))*
sqrt((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))*((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4))^(1/4) + (4*(a
^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^3 - 4*(a^7*b + a^5*b^3 - a^3*b^5 - a*b^7)*c^2*d + (a^8 - 2*a^4*b^4 + b^8)*c*d^
2)*cos(f*x + e) + (4*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^2*d - 4*(a^7*b + a^5*b^3 - a^3*b^5 - a*b^7)*c*d^2 + (a^
8 - 2*a^4*b^4 + b^8)*d^3)*sin(f*x + e))/cos(f*x + e)) + sqrt(2)*(a^4 + 2*a^2*b^2 + b^4 - (2*a*b*d + (a^2 - b^2
)*c)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)))*sqrt(((2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (
a^2 - b^2)*c*d^2)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2
*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a^4 - 2*a^2*b^2 + b^4)*d^2))*((a^4 + 2*a^2*b^2
+ b^4)/((c^2 + d^2)*f^4))^(1/4)*log(((4*(a^4*b^2 + a^2*b^4)*c^4 - 4*(a^5*b - a*b^5)*c^3*d + (a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*c^2*d^2 - 4*(a^5*b - a*b^5)*c*d^3 + (a^6 - a^4*b^2 - a^2*b^4 + b^6)*d^4)*f^2*sqrt((a^4 + 2*a
^2*b^2 + b^4)/((c^2 + d^2)*f^4))*cos(f*x + e) - sqrt(2)*((4*a^2*b^3*c^4 - 4*(a^3*b^2 - a*b^4)*c^3*d + (a^4*b +
2*a^2*b^3 + b^5)*c^2*d^2 - 4*(a^3*b^2 - a*b^4)*c*d^3 + (a^4*b - 2*a^2*b^3 + b^5)*d^4)*f^3*sqrt((a^4 + 2*a^2*b
^2 + b^4)/((c^2 + d^2)*f^4))*cos(f*x + e) + (4*(a^4*b^3 + a^2*b^5)*c^3 - 4*(2*a^5*b^2 + a^3*b^4 - a*b^6)*c^2*d
+ (5*a^6*b - a^4*b^3 - 5*a^2*b^5 + b^7)*c*d^2 - (a^7 - a^5*b^2 - a^3*b^4 + a*b^6)*d^3)*f*cos(f*x + e))*sqrt((
(2*a*b*c^2*d + 2*a*b*d^3 + (a^2 - b^2)*c^3 + (a^2 - b^2)*c*d^2)*f^2*sqrt((a^4 + 2*a^2*b^2 + b^4)/((c^2 + d^2)*
f^4)) + (a^4 + 2*a^2*b^2 + b^4)*c^2 + (a^4 + 2*a^2*b^2 + b^4)*d^2)/(4*a^2*b^2*c^2 - 4*(a^3*b - a*b^3)*c*d + (a
^4 - 2*a^2*b^2 + b^4)*d^2))*sqrt((c*cos(f*x + e) + d*sin(f*x + e))/cos(f*x + e))*((a^4 + 2*a^2*b^2 + b^4)/((c^
2 + d^2)*f^4))^(1/4) + (4*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^3 - 4*(a^7*b + a^5*b^3 - a^3*b^5 - a*b^7)*c^2*d +
(a^8 - 2*a^4*b^4 + b^8)*c*d^2)*cos(f*x + e) + (4*(a^6*b^2 + 2*a^4*b^4 + a^2*b^6)*c^2*d - 4*(a^7*b + a^5*b^3 -
a^3*b^5 - a*b^7)*c*d^2 + (a^8 - 2*a^4*b^4 + b^8)*d^3)*sin(f*x + e))/cos(f*x + e)))/(a^4 + 2*a^2*b^2 + b^4)

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

[Out]

Timed out

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maple [B]  time = 0.26, size = 3976, normalized size = 38.98 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))/(c+d*tan(f*x+e))^(1/2),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(d-c>0)', see `assume?` for mor
e details)Is d-c positive, negative or zero?

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mupad [B]  time = 7.35, size = 2909, normalized size = 28.52 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*atanh((8*c*d^2*(- (-16*a^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (a^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1
/2)*(c + d*tan(e + f*x))^(1/2)*(-16*a^4*d^2*f^4)^(1/2))/((16*a^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*a*d^5*f^4
*(-16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*a^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) + (4*a*c^2*d^3*f^4*(-
16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*a^2*d^2*(- (-16*a^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4))
- (a^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*a^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^
4) + (4*a*d^3*f^2*(-16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (32*a^2*c^2*d^2*f^2*(- (-16*a^4*d^2*f^4)^(1/
2)/(16*(c^2*f^4 + d^2*f^4)) - (a^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*a^3*
c*d^5*f^5)/(c^2*f^4 + d^2*f^4) + (4*a*d^5*f^4*(-16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*a^3*c^3*d^3*f
^5)/(c^2*f^4 + d^2*f^4) + (4*a*c^2*d^3*f^4*(-16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*(- (-16*a^4*d^2*f^4)
^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (a^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((32*a^2*d^2*((-16*a^4*d
^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (a^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)
)/((16*a^3*c*d^3*f^3)/(c^2*f^4 + d^2*f^4) - (4*a*d^3*f^2*(-16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (8*c*
d^2*((-16*a^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (a^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(
e + f*x))^(1/2)*(-16*a^4*d^2*f^4)^(1/2))/((16*a^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) - (4*a*d^5*f^4*(-16*a^4*d^2*f
^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*a^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*a*c^2*d^3*f^4*(-16*a^4*d^2*f^4)
^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*a^2*c^2*d^2*f^2*((-16*a^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (a^2*c*
f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*a^3*c*d^5*f^5)/(c^2*f^4 + d^2*f^4) - (4*a
*d^5*f^4*(-16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (16*a^3*c^3*d^3*f^5)/(c^2*f^4 + d^2*f^4) - (4*a*c^2*d^
3*f^4*(-16*a^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*((-16*a^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) - (a^2*
c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((32*b^2*d^2*((b^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*b^4*d^
2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*b^3*d^2)/f - (16*b^3*c^2*d^2*f^3
)/(c^2*f^4 + d^2*f^4) + (4*b*c*d^2*f^2*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (8*c*d^2*((b^2*c*f^2)/(
4*(c^2*f^4 + d^2*f^4)) - (-16*b^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2)*(-
16*b^4*d^2*f^4)^(1/2))/(16*b^3*d^4*f + 16*b^3*c^2*d^2*f - (16*b^3*c^2*d^4*f^5)/(c^2*f^4 + d^2*f^4) - (16*b^3*c
^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*b*c*d^4*f^4*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (4*b*c^3*d^2*f
^4*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*b^2*c^2*d^2*f^2*((b^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) -
(-16*b^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/(16*b^3*d^4*f + 16*b^3*c^2
*d^2*f - (16*b^3*c^2*d^4*f^5)/(c^2*f^4 + d^2*f^4) - (16*b^3*c^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*b*c*d^4*f^4*
(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (4*b*c^3*d^2*f^4*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))
*((b^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)) - (-16*b^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)))^(1/2) - 2*atanh((8*c
*d^2*((-16*b^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) + (b^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan
(e + f*x))^(1/2)*(-16*b^4*d^2*f^4)^(1/2))/((16*b^3*c^2*d^4*f^5)/(c^2*f^4 + d^2*f^4) - 16*b^3*c^2*d^2*f - 16*b^
3*d^4*f + (16*b^3*c^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*b*c*d^4*f^4*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^
5) + (4*b*c^3*d^2*f^4*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) - (32*b^2*d^2*((-16*b^4*d^2*f^4)^(1/2)/(16
*(c^2*f^4 + d^2*f^4)) + (b^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)*(c + d*tan(e + f*x))^(1/2))/((16*b^3*c^2*d^
2*f^3)/(c^2*f^4 + d^2*f^4) - (16*b^3*d^2)/f + (4*b*c*d^2*f^2*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)) + (
32*b^2*c^2*d^2*f^2*((-16*b^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) + (b^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1
/2)*(c + d*tan(e + f*x))^(1/2))/((16*b^3*c^2*d^4*f^5)/(c^2*f^4 + d^2*f^4) - 16*b^3*c^2*d^2*f - 16*b^3*d^4*f +
(16*b^3*c^4*d^2*f^5)/(c^2*f^4 + d^2*f^4) + (4*b*c*d^4*f^4*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5) + (4*b*
c^3*d^2*f^4*(-16*b^4*d^2*f^4)^(1/2))/(c^2*f^5 + d^2*f^5)))*((-16*b^4*d^2*f^4)^(1/2)/(16*(c^2*f^4 + d^2*f^4)) +
(b^2*c*f^2)/(4*(c^2*f^4 + d^2*f^4)))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))/(c+d*tan(f*x+e))**(1/2),x)

[Out]

Integral((a + b*tan(e + f*x))/sqrt(c + d*tan(e + f*x)), x)

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