3.13.32 \(\int \frac {\sqrt {c+d \tan (e+f x)}}{a+b \tan (e+f x)} \, dx\) [1232]

Optimal. Leaf size=170 \[ \frac {\sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{(i a+b) f}-\frac {\sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{(i a-b) f}-\frac {2 \sqrt {b} \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) f} \]

[Out]

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

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Rubi [A]
time = 0.33, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3653, 3620, 3618, 65, 214, 3715} \begin {gather*} -\frac {2 \sqrt {b} \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{f \left (a^2+b^2\right )}+\frac {\sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )}{f (b+i a)}-\frac {\sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )}{f (-b+i a)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 65

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

Rule 214

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

Rule 3618

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

Rule 3620

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

Rule 3653

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

Rule 3715

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

Rubi steps

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

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Mathematica [A]
time = 0.34, size = 158, normalized size = 0.93 \begin {gather*} \frac {(-i a+b) \sqrt {c-i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d}}\right )+(i a+b) \sqrt {c+i d} \tanh ^{-1}\left (\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {c+i d}}\right )-2 \sqrt {b} \sqrt {b c-a d} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d \tan (e+f x)}}{\sqrt {b c-a d}}\right )}{\left (a^2+b^2\right ) f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(692\) vs. \(2(142)=284\).
time = 0.55, size = 693, normalized size = 4.08

method	result	size
derivativedivides	\(\frac {2 d^{2} \left (-\frac {\left (a d -b c \right ) b \arctan \left (\frac {b \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {\left (a d -b c \right ) b}}\right )}{d^{2} \left (a^{2}+b^{2}\right ) \sqrt {\left (a d -b c \right ) b}}+\frac {\frac {-\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-2 \sqrt {c^{2}+d^{2}}\, b d +\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}+\frac {\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (2 \sqrt {c^{2}+d^{2}}\, b d -\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}}{\left (a^{2}+b^{2}\right ) d^{2}}\right )}{f}\)	\(693\)
default	\(\frac {2 d^{2} \left (-\frac {\left (a d -b c \right ) b \arctan \left (\frac {b \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {\left (a d -b c \right ) b}}\right )}{d^{2} \left (a^{2}+b^{2}\right ) \sqrt {\left (a d -b c \right ) b}}+\frac {\frac {-\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \ln \left (\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-d \tan \left (f x +e \right )-c -\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (-2 \sqrt {c^{2}+d^{2}}\, b d +\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}-2 \sqrt {c +d \tan \left (f x +e \right )}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}+\frac {\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \ln \left (d \tan \left (f x +e \right )+c +\sqrt {c +d \tan \left (f x +e \right )}\, \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}+\sqrt {c^{2}+d^{2}}\right )}{2}+\frac {2 \left (2 \sqrt {c^{2}+d^{2}}\, b d -\frac {\left (-\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, \sqrt {c^{2}+d^{2}}\, a +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, a c +\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}\, b d \right ) \sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{2}\right ) \arctan \left (\frac {2 \sqrt {c +d \tan \left (f x +e \right )}+\sqrt {2 \sqrt {c^{2}+d^{2}}+2 c}}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}\right )}{\sqrt {2 \sqrt {c^{2}+d^{2}}-2 c}}}{4 d}}{\left (a^{2}+b^{2}\right ) d^{2}}\right )}{f}\)	\(693\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/f*d^2*(-(a*d-b*c)*b/d^2/(a^2+b^2)/((a*d-b*c)*b)^(1/2)*arctan(b*(c+d*tan(f*x+e))^(1/2)/((a*d-b*c)*b)^(1/2))+1
/(a^2+b^2)/d^2*(1/4/d*(-1/2*(-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*
c+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d)*ln((c+d*tan(f*x+e))^(1/2)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2)-d*tan(f*x+e)-c-(c
^2+d^2)^(1/2))+2*(-2*(c^2+d^2)^(1/2)*b*d+1/2*(-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+(2*(c^2+d^2)^(1
/2)+2*c)^(1/2)*a*c+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*b*d)*(2*(c^2+d^2)^(1/2)+2*c)^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(
1/2)*arctan(((2*(c^2+d^2)^(1/2)+2*c)^(1/2)-2*(c+d*tan(f*x+e))^(1/2))/(2*(c^2+d^2)^(1/2)-2*c)^(1/2)))+1/4/d*(1/
2*(-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+(2*(c^2+d^2)^(1/2)+2*c)^
(1/2)*b*d)*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*(c^2+d
^2)^(1/2)*b*d-1/2*(-(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*(c^2+d^2)^(1/2)*a+(2*(c^2+d^2)^(1/2)+2*c)^(1/2)*a*c+(2*(c^2+
d^2)^(1/2)+2*c)^(1/2)*b*d)*(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((c+d*tan(f*x+e))^(1/2)/(a+b*tan(f*x+e)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more detail

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 8361 vs. \(2 (139) = 278\).
time = 102.60, size = 16728, normalized size = 98.40 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/4*(4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*f^5*sqrt((((a^6 + a^4*b^2 - a^2*b^4 - b^6)*c + 2*(a^5*b +
2*a^3*b^3 + a*b^5)*d)*f^2*sqrt((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*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)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)
*f^4))*((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4))^(3/4)*arctan(-((2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c
^3 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^2*d + 2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c*d^2 - (a^8 + 2*a^
6*b^2 - 2*a^2*b^6 - b^8)*d^3)*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)/(
(a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4))*sqrt((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4)) + (2*(a^
5*b + 2*a^3*b^3 + a*b^5)*c^4 - (a^6 + a^4*b^2 - a^2*b^4 - b^6)*c^3*d + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*c^2*d^2 -
(a^6 + a^4*b^2 - a^2*b^4 - b^6)*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)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4)) + sqrt(2)*((2*(a^9*b^2 + 4*a^7*b^4 + 6*a^5*b^6
+ 4*a^3*b^8 + a*b^10)*c - (a^10*b + 3*a^8*b^3 + 2*a^6*b^5 - 2*a^4*b^7 - 3*a^2*b^9 - b^11)*d)*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)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8
)*f^4))*sqrt((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4)) + (2*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*c^2 - (
3*a^8*b + 8*a^6*b^3 + 6*a^4*b^5 - b^9)*c*d + (a^9 + 2*a^7*b^2 - 2*a^3*b^6 - a*b^8)*d^2)*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)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4)
))*sqrt((((a^6 + a^4*b^2 - a^2*b^4 - b^6)*c + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d)*f^2*sqrt((c^2 + d^2)/((a^4 + 2*
a^2*b^2 + b^4)*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))*((c^2 + d^2)/(
(a^4 + 2*a^2*b^2 + b^4)*f^4))^(3/4) - sqrt(2)*((a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*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)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 +
b^8)*f^4))*sqrt((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4)) + ((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*c - (a^7 +
3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*d)*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)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4)))*sqrt((((a^6 + a^4*b^2 - a^2*b^4 - b^6)*c + 2*(a^5*
b + 2*a^3*b^3 + a*b^5)*d)*f^2*sqrt((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*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((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4))*cos(f*x
+ e) + sqrt(2)*((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^3*sqrt((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4)
)*cos(f*x + e) + (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*cos(f*x + e))*sqrt((((a^6 + a^4*b^2 - a^2*b^4 - b^6)*c + 2*(a
^5*b + 2*a^3*b^3 + a*b^5)*d)*f^2*sqrt((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*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))*((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4))^(1/4) + (4*a^2*b^2*c
^5 - 4*(a^3*b - a*b^3)*c^4*d + (a^4 + 2*a^2*b^2 + b^4)*c^3*d^2 - 4*(a^3*b - a*b^3)*c^2*d^3 + (a^4 - 2*a^2*b^2
+ b^4)*c*d^4)*cos(f*x + e) + (4*a^2*b^2*c^4*d - 4*(a^3*b - a*b^3)*c^3*d^2 + (a^4 + 2*a^2*b^2 + b^4)*c^2*d^3 -
4*(a^3*b - a*b^3)*c*d^4 + (a^4 - 2*a^2*b^2 + b^4)*d^5)*sin(f*x + e))/((c^2 + d^2)*cos(f*x + e)))*((c^2 + d^2)/
((a^4 + 2*a^2*b^2 + b^4)*f^4))^(3/4))/(4*a^2*b^2*c^4*d - 4*(a^3*b - a*b^3)*c^3*d^2 + (a^4 + 2*a^2*b^2 + b^4)*c
^2*d^3 - 4*(a^3*b - a*b^3)*c*d^4 + (a^4 - 2*a^2*b^2 + b^4)*d^5)) + 4*sqrt(2)*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^
6)*f^5*sqrt((((a^6 + a^4*b^2 - a^2*b^4 - b^6)*c + 2*(a^5*b + 2*a^3*b^3 + a*b^5)*d)*f^2*sqrt((c^2 + d^2)/((a^4
+ 2*a^2*b^2 + b^4)*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)/((a^8 + 4*a^6*b^2 + 6*a^4*b^4 + 4*a^2*b^6 + b^8)*f^4))*((c^2 + d^2)/((a^4 + 2*a^2*b^2 + b^4)*f^4))
^(3/4)*arctan(((2*(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c^3 - (a^8 + 2*a^6*b^2 - 2*a^2*b^6 - b^8)*c^2*d + 2*
(a^7*b + 3*a^5*b^3 + 3*a^3*b^5 + a*b^7)*c*d^2 -...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 9.76, size = 2500, normalized size = 14.71 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan(((((((32*(12*a*b^7*d^11*f^4 - 12*b^8*c*d^10*f^4 + 24*a^3*b^5*d^11*f^4 + 12*a^5*b^3*d^11*f^4 - 12*b^8*c^3*
d^8*f^4 - 24*a^2*b^6*c^3*d^8*f^4 + 24*a^3*b^5*c^2*d^9*f^4 - 12*a^4*b^4*c^3*d^8*f^4 + 12*a^5*b^3*c^2*d^9*f^4 +
12*a*b^7*c^2*d^9*f^4 - 24*a^2*b^6*c*d^10*f^4 - 12*a^4*b^4*c*d^10*f^4))/f^5 - (32*(c + d*tan(e + f*x))^(1/2)*(-
(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5*d^1
0*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*b^3*
c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)*(-(
c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(14*a*b^6*d^11*f^2 - 6*
b^7*c*d^10*f^2 - 20*a^3*b^4*d^11*f^2 - 2*a^5*b^2*d^11*f^2 - 18*b^7*c^3*d^8*f^2 + 12*a^2*b^5*c^3*d^8*f^2 - 12*a
^3*b^4*c^2*d^9*f^2 - 2*a^4*b^3*c^3*d^8*f^2 + 2*a^5*b^2*c^2*d^9*f^2 + 18*a*b^6*c^2*d^9*f^2 + 36*a^2*b^5*c*d^10*
f^2 + 10*a^4*b^3*c*d^10*f^2))/f^4)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(13*a^2*b^4*
d^12*f^2 + a^4*b^2*d^12*f^2 + 3*b^6*c^2*d^10*f^2 + 3*b^6*c^4*d^8*f^2 + 12*a^2*b^4*c^2*d^10*f^2 - a^2*b^4*c^4*d
^8*f^2 + a^4*b^2*c^2*d^10*f^2 - 16*a*b^5*c*d^11*f^2 - 16*a*b^5*c^3*d^9*f^2))/f^5)*(-(c + d*1i)/(4*(a^2*f^2 - b
^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^12 - 2*a^2*b^3*d^12 + 3*b^5*c^4*d^8 - 4*a
*b^4*c^3*d^9 + 2*a^2*b^3*c^2*d^10 + 4*a*b^4*c*d^11))/f^4)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(
1/2)*1i - (((((32*(12*a*b^7*d^11*f^4 - 12*b^8*c*d^10*f^4 + 24*a^3*b^5*d^11*f^4 + 12*a^5*b^3*d^11*f^4 - 12*b^8*
c^3*d^8*f^4 - 24*a^2*b^6*c^3*d^8*f^4 + 24*a^3*b^5*c^2*d^9*f^4 - 12*a^4*b^4*c^3*d^8*f^4 + 12*a^5*b^3*c^2*d^9*f^
4 + 12*a*b^7*c^2*d^9*f^4 - 24*a^2*b^6*c*d^10*f^4 - 12*a^4*b^4*c*d^10*f^4))/f^5 + (32*(c + d*tan(e + f*x))^(1/2
)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4*b^5
*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*a^6*
b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/f^4)
*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(14*a*b^6*d^11*f^2
- 6*b^7*c*d^10*f^2 - 20*a^3*b^4*d^11*f^2 - 2*a^5*b^2*d^11*f^2 - 18*b^7*c^3*d^8*f^2 + 12*a^2*b^5*c^3*d^8*f^2 -
12*a^3*b^4*c^2*d^9*f^2 - 2*a^4*b^3*c^3*d^8*f^2 + 2*a^5*b^2*c^2*d^9*f^2 + 18*a*b^6*c^2*d^9*f^2 + 36*a^2*b^5*c*d
^10*f^2 + 10*a^4*b^3*c*d^10*f^2))/f^4)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(13*a^2*
b^4*d^12*f^2 + a^4*b^2*d^12*f^2 + 3*b^6*c^2*d^10*f^2 + 3*b^6*c^4*d^8*f^2 + 12*a^2*b^4*c^2*d^10*f^2 - a^2*b^4*c
^4*d^8*f^2 + a^4*b^2*c^2*d^10*f^2 - 16*a*b^5*c*d^11*f^2 - 16*a*b^5*c^3*d^9*f^2))/f^5)*(-(c + d*1i)/(4*(a^2*f^2
- b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^12 - 2*a^2*b^3*d^12 + 3*b^5*c^4*d^8 -
4*a*b^4*c^3*d^9 + 2*a^2*b^3*c^2*d^10 + 4*a*b^4*c*d^11))/f^4)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)
))^(1/2)*1i)/((((((32*(12*a*b^7*d^11*f^4 - 12*b^8*c*d^10*f^4 + 24*a^3*b^5*d^11*f^4 + 12*a^5*b^3*d^11*f^4 - 12*
b^8*c^3*d^8*f^4 - 24*a^2*b^6*c^3*d^8*f^4 + 24*a^3*b^5*c^2*d^9*f^4 - 12*a^4*b^4*c^3*d^8*f^4 + 12*a^5*b^3*c^2*d^
9*f^4 + 12*a*b^7*c^2*d^9*f^4 - 24*a^2*b^6*c*d^10*f^4 - 12*a^4*b^4*c*d^10*f^4))/f^5 - (32*(c + d*tan(e + f*x))^
(1/2)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2)*(16*b^9*d^10*f^4 + 16*a^2*b^7*d^10*f^4 - 16*a^4
*b^5*d^10*f^4 - 16*a^6*b^3*d^10*f^4 + 24*b^9*c^2*d^8*f^4 + 40*a^2*b^7*c^2*d^8*f^4 + 8*a^4*b^5*c^2*d^8*f^4 - 8*
a^6*b^3*c^2*d^8*f^4 + 8*a*b^8*c*d^9*f^4 + 24*a^3*b^6*c*d^9*f^4 + 24*a^5*b^4*c*d^9*f^4 + 8*a^7*b^2*c*d^9*f^4))/
f^4)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(14*a*b^6*d^11*
f^2 - 6*b^7*c*d^10*f^2 - 20*a^3*b^4*d^11*f^2 - 2*a^5*b^2*d^11*f^2 - 18*b^7*c^3*d^8*f^2 + 12*a^2*b^5*c^3*d^8*f^
2 - 12*a^3*b^4*c^2*d^9*f^2 - 2*a^4*b^3*c^3*d^8*f^2 + 2*a^5*b^2*c^2*d^9*f^2 + 18*a*b^6*c^2*d^9*f^2 + 36*a^2*b^5
*c*d^10*f^2 + 10*a^4*b^3*c*d^10*f^2))/f^4)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) + (32*(13*
a^2*b^4*d^12*f^2 + a^4*b^2*d^12*f^2 + 3*b^6*c^2*d^10*f^2 + 3*b^6*c^4*d^8*f^2 + 12*a^2*b^4*c^2*d^10*f^2 - a^2*b
^4*c^4*d^8*f^2 + a^4*b^2*c^2*d^10*f^2 - 16*a*b^5*c*d^11*f^2 - 16*a*b^5*c^3*d^9*f^2))/f^5)*(-(c + d*1i)/(4*(a^2
*f^2 - b^2*f^2 + a*b*f^2*2i)))^(1/2) - (32*(c + d*tan(e + f*x))^(1/2)*(b^5*d^12 - 2*a^2*b^3*d^12 + 3*b^5*c^4*d
^8 - 4*a*b^4*c^3*d^9 + 2*a^2*b^3*c^2*d^10 + 4*a*b^4*c*d^11))/f^4)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b*f^2
*2i)))^(1/2) + (((((32*(12*a*b^7*d^11*f^4 - 12*b^8*c*d^10*f^4 + 24*a^3*b^5*d^11*f^4 + 12*a^5*b^3*d^11*f^4 - 12
*b^8*c^3*d^8*f^4 - 24*a^2*b^6*c^3*d^8*f^4 + 24*a^3*b^5*c^2*d^9*f^4 - 12*a^4*b^4*c^3*d^8*f^4 + 12*a^5*b^3*c^2*d
^9*f^4 + 12*a*b^7*c^2*d^9*f^4 - 24*a^2*b^6*c*d^10*f^4 - 12*a^4*b^4*c*d^10*f^4))/f^5 + (32*(c + d*tan(e + f*x))
^(1/2)*(-(c + d*1i)/(4*(a^2*f^2 - b^2*f^2 + a*b...

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