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

Optimal. Leaf size=115 \[ \frac {i \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3575, 910, 93, 205, 208} \[ \frac {i \tan ^{-1}\left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}-\frac {i \tanh ^{-1}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Tan[c + d*x]]/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 205

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

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 910

Int[((d_.) + (e_.)*(x_))^(m_)/(Sqrt[(f_.) + (g_.)*(x_)]*((a_.) + (c_.)*(x_)^2)), x_Symbol] :> Int[ExpandIntegr
and[1/(Sqrt[d + e*x]*Sqrt[f + g*x]), (d + e*x)^(m + 1/2)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] &
& NeQ[c*d^2 + a*e^2, 0] && IGtQ[m + 1/2, 0]

Rule 3575

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Wit
h[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[((a + b*ff*x)^m*(c + d*ff*x)^n)/(1 + ff^2*x^2), x]
, x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] &&
NeQ[c^2 + d^2, 0]

Rubi steps

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

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Mathematica [A]  time = 0.12, size = 123, normalized size = 1.07 \[ \frac {\sqrt [4]{-1} \left (\frac {\tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Tan[c + d*x]]/Sqrt[a + b*Tan[c + d*x]],x]

[Out]

((-1)^(1/4)*(-(ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[-a + I*b])
+ ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]/Sqrt[a + I*b]))/d

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

[Out]

Timed out

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

[Out]

Timed out

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maple [B]  time = 0.77, size = 940034, normalized size = 8174.21 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 43.19, size = 4102, normalized size = 35.67 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(((-1/(d^2*(a*1i + b)))^(1/2)*(((-1/(d^2*(a*1i + b)))^(1/2)*(((-1/(d^2*(a*1i + b)))^(1/2)*(((-1/(d^2*(a*1i
+ b)))^(1/2)*(((-1/(d^2*(a*1i + b)))^(1/2)*(536870912*a^8*b^16*(-1/(d^2*(a*1i + b)))^(1/2)*(8*a^2 + 7*b^2)*(8
*a^2 + 7*b^2 - (17*b^3*tan(c + d*x))/((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2 - (16*a^2*b*tan(c + d*x))/((a +
b*tan(c + d*x))^(1/2) - a^(1/2))^2) + (1073741824*a^7*b^17*tan(c + d*x)^(1/2)*(48*a^4 + 5*b^4 + 52*a^2*b^2))/(
d*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))))/2 + (268435456*a^6*b^17*(144*a^4 - b^4 + 112*a^2*b^2))/d^2 + (2684
35456*a^6*b^16*tan(c + d*x)*(256*a^6 + b^6 - 270*a^2*b^4 - 32*a^4*b^2))/(d^2*((a + b*tan(c + d*x))^(1/2) - a^(
1/2))^2)))/2 + (536870912*a^7*b^18*tan(c + d*x)^(1/2)*(48*a^2 + 5*b^2))/(d^3*((a + b*tan(c + d*x))^(1/2) - a^(
1/2)))))/2 + (134217728*a^6*b^16*(32*a^4 - b^4 + 40*a^2*b^2))/d^4 + (134217728*a^6*b^17*tan(c + d*x)*(192*a^4
+ b^4 - 92*a^2*b^2))/(d^4*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)))/2 + (67108864*a^7*b^17*tan(c + d*x)^(1/2
)*(48*a^2 + 5*b^2))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))))/2 + (16777216*a^6*b^17*(16*a^2 - b^2))/d^6
+ (16777216*a^6*b^16*tan(c + d*x)*(16*a^2 - b^2)^2)/(d^6*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2))*(-1/(a*d^2
*1i + b*d^2))^(1/2))/2 - log(((-1/(d^2*(a*1i + b)))^(1/2)*(((-1/(d^2*(a*1i + b)))^(1/2)*(((-1/(d^2*(a*1i + b))
)^(1/2)*(((-1/(d^2*(a*1i + b)))^(1/2)*(((-1/(d^2*(a*1i + b)))^(1/2)*(536870912*a^8*b^16*(-1/(d^2*(a*1i + b)))^
(1/2)*(8*a^2 + 7*b^2)*(8*a^2 + 7*b^2 - (17*b^3*tan(c + d*x))/((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2 - (16*a^
2*b*tan(c + d*x))/((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2) - (1073741824*a^7*b^17*tan(c + d*x)^(1/2)*(48*a^4
+ 5*b^4 + 52*a^2*b^2))/(d*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))))/2 + (268435456*a^6*b^17*(144*a^4 - b^4 + 1
12*a^2*b^2))/d^2 + (268435456*a^6*b^16*tan(c + d*x)*(256*a^6 + b^6 - 270*a^2*b^4 - 32*a^4*b^2))/(d^2*((a + b*t
an(c + d*x))^(1/2) - a^(1/2))^2)))/2 - (536870912*a^7*b^18*tan(c + d*x)^(1/2)*(48*a^2 + 5*b^2))/(d^3*((a + b*t
an(c + d*x))^(1/2) - a^(1/2)))))/2 + (134217728*a^6*b^16*(32*a^4 - b^4 + 40*a^2*b^2))/d^4 + (134217728*a^6*b^1
7*tan(c + d*x)*(192*a^4 + b^4 - 92*a^2*b^2))/(d^4*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)))/2 - (67108864*a^
7*b^17*tan(c + d*x)^(1/2)*(48*a^2 + 5*b^2))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))))/2 + (16777216*a^6*b
^17*(16*a^2 - b^2))/d^6 + (16777216*a^6*b^16*tan(c + d*x)*(16*a^2 - b^2)^2)/(d^6*((a + b*tan(c + d*x))^(1/2) -
a^(1/2))^2))*(-1/(4*(a*d^2*1i + b*d^2)))^(1/2) + atan(((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i +
b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b
^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((167
77216*(3136*a^8*b^20*d^6 + 7168*a^10*b^18*d^6 + 4096*a^12*b^16*d^6))/d^6 - (16777216*tan(c + d*x)*(7616*a^8*b^
21*d^6 + 15872*a^10*b^19*d^6 + 8192*a^12*b^17*d^6))/(d^6*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)) - (6710886
4*tan(c + d*x)^(1/2)*(80*a^7*b^21*d^4 + 832*a^9*b^19*d^4 + 768*a^11*b^17*d^4))/(d^5*((a + b*tan(c + d*x))^(1/2
) - a^(1/2)))) + (16777216*(1792*a^8*b^19*d^4 - 16*a^6*b^21*d^4 + 2304*a^10*b^17*d^4))/d^6 + (16777216*tan(c +
d*x)*(16*a^6*b^22*d^4 - 4320*a^8*b^20*d^4 - 512*a^10*b^18*d^4 + 4096*a^12*b^16*d^4))/(d^6*((a + b*tan(c + d*x
))^(1/2) - a^(1/2))^2)) - (67108864*tan(c + d*x)^(1/2)*(40*a^7*b^20*d^2 + 384*a^9*b^18*d^2))/(d^5*((a + b*tan(
c + d*x))^(1/2) - a^(1/2)))) + (16777216*(320*a^8*b^18*d^2 - 8*a^6*b^20*d^2 + 256*a^10*b^16*d^2))/d^6 + (16777
216*tan(c + d*x)*(8*a^6*b^21*d^2 - 736*a^8*b^19*d^2 + 1536*a^10*b^17*d^2))/(d^6*((a + b*tan(c + d*x))^(1/2) -
a^(1/2))^2)) - (67108864*tan(c + d*x)^(1/2)*(5*a^7*b^19 + 48*a^9*b^17))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(
1/2))))*1i - (-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i
+ b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4
*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((16777216*(3136*a^8*b^20*d^6 + 7168*a^10*b^18*d
^6 + 4096*a^12*b^16*d^6))/d^6 - (16777216*tan(c + d*x)*(7616*a^8*b^21*d^6 + 15872*a^10*b^19*d^6 + 8192*a^12*b^
17*d^6))/(d^6*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)) + (67108864*tan(c + d*x)^(1/2)*(80*a^7*b^21*d^4 + 832
*a^9*b^19*d^4 + 768*a^11*b^17*d^4))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))) + (16777216*(1792*a^8*b^19*d
^4 - 16*a^6*b^21*d^4 + 2304*a^10*b^17*d^4))/d^6 + (16777216*tan(c + d*x)*(16*a^6*b^22*d^4 - 4320*a^8*b^20*d^4
- 512*a^10*b^18*d^4 + 4096*a^12*b^16*d^4))/(d^6*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)) + (67108864*tan(c +
d*x)^(1/2)*(40*a^7*b^20*d^2 + 384*a^9*b^18*d^2))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))) + (16777216*(3
20*a^8*b^18*d^2 - 8*a^6*b^20*d^2 + 256*a^10*b^16*d^2))/d^6 + (16777216*tan(c + d*x)*(8*a^6*b^21*d^2 - 736*a^8*
b^19*d^2 + 1536*a^10*b^17*d^2))/(d^6*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)) + (67108864*tan(c + d*x)^(1/2)
*(5*a^7*b^19 + 48*a^9*b^17))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(1/2))))*1i)/((-(a*1i + b)/(4*a^2*d^2 + 4*b^
2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*
1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 +
4*b^2*d^2))^(1/2)*((16777216*(3136*a^8*b^20*d^6 + 7168*a^10*b^18*d^6 + 4096*a^12*b^16*d^6))/d^6 - (16777216*t
an(c + d*x)*(7616*a^8*b^21*d^6 + 15872*a^10*b^19*d^6 + 8192*a^12*b^17*d^6))/(d^6*((a + b*tan(c + d*x))^(1/2) -
a^(1/2))^2)) - (67108864*tan(c + d*x)^(1/2)*(80*a^7*b^21*d^4 + 832*a^9*b^19*d^4 + 768*a^11*b^17*d^4))/(d^5*((
a + b*tan(c + d*x))^(1/2) - a^(1/2)))) + (16777216*(1792*a^8*b^19*d^4 - 16*a^6*b^21*d^4 + 2304*a^10*b^17*d^4))
/d^6 + (16777216*tan(c + d*x)*(16*a^6*b^22*d^4 - 4320*a^8*b^20*d^4 - 512*a^10*b^18*d^4 + 4096*a^12*b^16*d^4))/
(d^6*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)) - (67108864*tan(c + d*x)^(1/2)*(40*a^7*b^20*d^2 + 384*a^9*b^18
*d^2))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))) + (16777216*(320*a^8*b^18*d^2 - 8*a^6*b^20*d^2 + 256*a^10
*b^16*d^2))/d^6 + (16777216*tan(c + d*x)*(8*a^6*b^21*d^2 - 736*a^8*b^19*d^2 + 1536*a^10*b^17*d^2))/(d^6*((a +
b*tan(c + d*x))^(1/2) - a^(1/2))^2)) - (67108864*tan(c + d*x)^(1/2)*(5*a^7*b^19 + 48*a^9*b^17))/(d^5*((a + b*t
an(c + d*x))^(1/2) - a^(1/2)))) - (33554432*(a^6*b^19 - 16*a^8*b^17))/d^6 + (-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^
2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i +
b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*((-(a*1i + b)/(4*a^2*d^2 + 4*b
^2*d^2))^(1/2)*((16777216*(3136*a^8*b^20*d^6 + 7168*a^10*b^18*d^6 + 4096*a^12*b^16*d^6))/d^6 - (16777216*tan(c
+ d*x)*(7616*a^8*b^21*d^6 + 15872*a^10*b^19*d^6 + 8192*a^12*b^17*d^6))/(d^6*((a + b*tan(c + d*x))^(1/2) - a^(
1/2))^2)) + (67108864*tan(c + d*x)^(1/2)*(80*a^7*b^21*d^4 + 832*a^9*b^19*d^4 + 768*a^11*b^17*d^4))/(d^5*((a +
b*tan(c + d*x))^(1/2) - a^(1/2)))) + (16777216*(1792*a^8*b^19*d^4 - 16*a^6*b^21*d^4 + 2304*a^10*b^17*d^4))/d^6
+ (16777216*tan(c + d*x)*(16*a^6*b^22*d^4 - 4320*a^8*b^20*d^4 - 512*a^10*b^18*d^4 + 4096*a^12*b^16*d^4))/(d^6
*((a + b*tan(c + d*x))^(1/2) - a^(1/2))^2)) + (67108864*tan(c + d*x)^(1/2)*(40*a^7*b^20*d^2 + 384*a^9*b^18*d^2
))/(d^5*((a + b*tan(c + d*x))^(1/2) - a^(1/2)))) + (16777216*(320*a^8*b^18*d^2 - 8*a^6*b^20*d^2 + 256*a^10*b^1
6*d^2))/d^6 + (16777216*tan(c + d*x)*(8*a^6*b^21*d^2 - 736*a^8*b^19*d^2 + 1536*a^10*b^17*d^2))/(d^6*((a + b*ta
n(c + d*x))^(1/2) - a^(1/2))^2)) + (67108864*tan(c + d*x)^(1/2)*(5*a^7*b^19 + 48*a^9*b^17))/(d^5*((a + b*tan(c
+ d*x))^(1/2) - a^(1/2)))) + (33554432*tan(c + d*x)*(a^6*b^20 - 32*a^8*b^18 + 256*a^10*b^16))/(d^6*((a + b*ta
n(c + d*x))^(1/2) - a^(1/2))^2)))*(-(a*1i + b)/(4*a^2*d^2 + 4*b^2*d^2))^(1/2)*2i

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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