3.13.50 \(\int \frac {a+b \tan (e+f x)}{\sqrt {c+d \tan (e+f x)}} \, dx\) [1250]

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

[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 = 0.11, 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 = {3620, 3618, 65, 214} \begin {gather*} \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}} \end {gather*}

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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 214

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

Rule 3618

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

Rule 3620

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

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) \text {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) \text {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) \text {Subst}\left (\int \frac {1}{-1-\frac {i c}{d}+\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{d f}-\frac {(a+i b) \text {Subst}\left (\int \frac {1}{-1+\frac {i c}{d}-\frac {i x^2}{d}} \, dx,x,\sqrt {c+d \tan (e+f x)}\right )}{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 = 0.15, size = 101, normalized size = 0.99 \begin {gather*} \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} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1398\) vs. \(2(84)=168\).
time = 0.46, size = 1399, normalized size = 13.72 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,method=_RETURNVERBOSE)

[Out]

1/f*(1/2/(c^2+d^2)^(3/2)/d^2*(1/2*(-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c-(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*a*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)
^(1/2)+2*c)^(1/2)*b*c^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1
/2)*b*d^4)*ln(d*tan(f*x+e)+c-(c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(2*a*c^2*
d^3+2*a*d^5-2*b*c^3*d^2-2*b*c*d^4+1/2*(-(c^2+d^2)^(3/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c-(c^2+d^2)^(1/2)*(2*(
c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*d^3+(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b*c^3+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c*d^2+(2*(c^2+d^2)^(1/2)+2*c)^(1/2
)*a*c^3*d+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^2+(2*(c^2+d^2)^(1/2)+2*c
)^(1/2)*b*d^4)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+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)))+1/2/(c^2+d^2)^(3/2)/d^2*(1/2*((c^2+d^2)^(3/2)*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d+(c^2+d^2)^(1/2)*(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*a*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(
1/2)+2*c)^(1/2)*b*c*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3-(2*(c^2+d^
2)^(1/2)+2*c)^(1/2)*b*c^2*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4)*ln(d*tan(f*x+e)+c+(c+d*tan(f*x+e))^(1/2)*(2
*(c^2+d^2)^(1/2)+2*c)^(1/2)+(c^2+d^2)^(1/2))+2*(2*a*c^2*d^3+2*a*d^5-2*b*c^3*d^2-2*b*c*d^4-1/2*((c^2+d^2)^(3/2)
*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c+(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^2*d+(c^2+d^2)^(1/2)*(2*(c
^2+d^2)^(1/2)+2*c)^(1/2)*a*d^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*c^3-(c^2+d^2)^(1/2)*(2*(c^2+d^2
)^(1/2)+2*c)^(1/2)*b*c*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c^3*d-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c*d^3-(2*(c^2
+d^2)^(1/2)+2*c)^(1/2)*b*c^2*d^2-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d^4)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d
^2)^(1/2)-2*c)^(1/2)*arctan((2*(c+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))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))/(c+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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8324 vs. \(2 (81) = 162\).
time = 4.68, size = 8324, normalized size = 81.61 \begin {gather*} \text {Too large to display} \end {gather*}

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...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*tan(f*x+e))/(c+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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Warning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The choi
ce was done

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Mupad [B]
time = 7.35, size = 2909, normalized size = 28.52 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*tan(e + f*x))/(c + 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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