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

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

[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)

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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 = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3539, 3537, 63, 208} \[ \frac {(-b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}-\frac {(b+i a) \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}} \]

Antiderivative was successfully verified.

[In]

Int[(-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)

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 (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 {1}{2} (-a+i b) \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\\ &=-\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 {(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}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 109, normalized size = 1.07 \[ \frac {i \left ((a+i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )-(a-i b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )\right )}{d \sqrt {a-i b} \sqrt {a+i b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(I*((a + I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c + d*x]]/Sqrt[a - I*b]] - (a - I*b)^(3/2)*ArcTanh[Sqrt[a + b*Tan[c
+ d*x]]/Sqrt[a + I*b]]))/(Sqrt[a - I*b]*Sqrt[a + I*b]*d)

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fricas [B]  time = 0.78, size = 3540, normalized size = 34.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(4*sqrt(2)*(a^2 + b^2)*d^4*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt
((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^2 + b^2)/d^4)^(3/4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/(
(a^4 + 2*a^2*b^2 + b^4)*d^4))*arctan(((3*a^8 + 8*a^6*b^2 + 6*a^4*b^4 - b^8)*d^4*sqrt((a^2 + b^2)/d^4)*sqrt((9*
a^4*b^2 - 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (3*a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((
9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + sqrt(2)*(2*a*d^7*sqrt((a^2 + b^2)/d^4)*sqrt((9*a
^4*b^2 - 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^2 + b^2)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/
((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt
((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(((9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10
)*d^2*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + sqrt(2)*((9*a^6*b^3 + 3*a^4*b^5 - 5*a^2*b^7 + b^9)*d^3*sqrt((a^2 +
b^2)/d^4)*cos(d*x + c) + 2*(9*a^7*b^3 + 3*a^5*b^5 - 5*a^3*b^7 + a*b^9)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(
(a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^2 + b^2)/d^4)^(1/4) + (9*a^9*b^2 + 12*a^7*b^4 - 2*a^5*b^6
- 4*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^8*b^3 + 12*a^6*b^5 - 2*a^4*b^7 - 4*a^2*b^9 + b^11)*sin(d*x + c))/cos
(d*x + c))*((a^2 + b^2)/d^4)^(3/4) + sqrt(2)*(2*(3*a^5*b + 2*a^3*b^3 - a*b^5)*d^7*sqrt((a^2 + b^2)/d^4)*sqrt((
9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (3*a^6*b + 5*a^4*b^3 + a^2*b^5 - b^7)*d^5*sqrt((
9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 -
2*a^3*b^2 - 3*a*b^4)*d^2*sqrt((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*
x + c))/cos(d*x + c))*((a^2 + b^2)/d^4)^(3/4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10)) + 4*sq
rt(2)*(a^2 + b^2)*d^4*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt((a^2 + b^
2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^2 + b^2)/d^4)^(3/4)*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^4 + 2*a
^2*b^2 + b^4)*d^4))*arctan(-((3*a^8 + 8*a^6*b^2 + 6*a^4*b^4 - b^8)*d^4*sqrt((a^2 + b^2)/d^4)*sqrt((9*a^4*b^2 -
6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (3*a^9 + 8*a^7*b^2 + 6*a^5*b^4 - a*b^8)*d^2*sqrt((9*a^4*b^2
- 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) - sqrt(2)*(2*a*d^7*sqrt((a^2 + b^2)/d^4)*sqrt((9*a^4*b^2 -
6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (a^2 + b^2)*d^5*sqrt((9*a^4*b^2 - 6*a^2*b^4 + b^6)/((a^4 + 2
*a^2*b^2 + b^4)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt((a^2 + b
^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt(((9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10)*d^2*sqr
t((a^2 + b^2)/d^4)*cos(d*x + c) - sqrt(2)*((9*a^6*b^3 + 3*a^4*b^5 - 5*a^2*b^7 + b^9)*d^3*sqrt((a^2 + b^2)/d^4)
*cos(d*x + c) + 2*(9*a^7*b^3 + 3*a^5*b^5 - 5*a^3*b^7 + a*b^9)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^
4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*
x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^2 + b^2)/d^4)^(1/4) + (9*a^9*b^2 + 12*a^7*b^4 - 2*a^5*b^6 - 4*a^3*b
^8 + a*b^10)*cos(d*x + c) + (9*a^8*b^3 + 12*a^6*b^5 - 2*a^4*b^7 - 4*a^2*b^9 + b^11)*sin(d*x + c))/cos(d*x + c)
)*((a^2 + b^2)/d^4)^(3/4) - sqrt(2)*(2*(3*a^5*b + 2*a^3*b^3 - a*b^5)*d^7*sqrt((a^2 + b^2)/d^4)*sqrt((9*a^4*b^2
- 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)) + (3*a^6*b + 5*a^4*b^3 + a^2*b^5 - b^7)*d^5*sqrt((9*a^4*b^2
- 6*a^2*b^4 + b^6)/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2
- 3*a*b^4)*d^2*sqrt((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/c
os(d*x + c))*((a^2 + b^2)/d^4)^(3/4))/(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10)) + sqrt(2)*(a^4
+ 2*a^2*b^2 + b^4 - (a^3 - 3*a*b^2)*d^2*sqrt((a^2 + b^2)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5
- 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^2 + b^2)/d^4)^(1/4)*log((
(9*a^8*b^2 + 12*a^6*b^4 - 2*a^4*b^6 - 4*a^2*b^8 + b^10)*d^2*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + sqrt(2)*((9*a
^6*b^3 + 3*a^4*b^5 - 5*a^2*b^7 + b^9)*d^3*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + 2*(9*a^7*b^3 + 3*a^5*b^5 - 5*a^
3*b^7 + a*b^9)*d*cos(d*x + c))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt(
(a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^2 +
b^2)/d^4)^(1/4) + (9*a^9*b^2 + 12*a^7*b^4 - 2*a^5*b^6 - 4*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^8*b^3 + 12*a^6
*b^5 - 2*a^4*b^7 - 4*a^2*b^9 + b^11)*sin(d*x + c))/cos(d*x + c)) - sqrt(2)*(a^4 + 2*a^2*b^2 + b^4 - (a^3 - 3*a
*b^2)*d^2*sqrt((a^2 + b^2)/d^4))*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqr
t((a^2 + b^2)/d^4))/(9*a^4*b^2 - 6*a^2*b^4 + b^6))*((a^2 + b^2)/d^4)^(1/4)*log(((9*a^8*b^2 + 12*a^6*b^4 - 2*a^
4*b^6 - 4*a^2*b^8 + b^10)*d^2*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) - sqrt(2)*((9*a^6*b^3 + 3*a^4*b^5 - 5*a^2*b^7
+ b^9)*d^3*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + 2*(9*a^7*b^3 + 3*a^5*b^5 - 5*a^3*b^7 + a*b^9)*d*cos(d*x + c))
*sqrt((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6 + (a^5 - 2*a^3*b^2 - 3*a*b^4)*d^2*sqrt((a^2 + b^2)/d^4))/(9*a^4*b^2 -
6*a^2*b^4 + b^6))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*((a^2 + b^2)/d^4)^(1/4) + (9*a^9*b^2 +
12*a^7*b^4 - 2*a^5*b^6 - 4*a^3*b^8 + a*b^10)*cos(d*x + c) + (9*a^8*b^3 + 12*a^6*b^5 - 2*a^4*b^7 - 4*a^2*b^9 +
b^11)*sin(d*x + c))/cos(d*x + c)))/(a^4 + 2*a^2*b^2 + b^4)

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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((-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)Warning, need to choose a branch for the root of
a polynomial with parameters. This might be wrong.The choice was done assuming [d]=[-84,-49]sym2poly/r2sym(co
nst 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 assuming [d]=[82,-6]sym2pol
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 Valuesym2poly/r2
sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,c
onst index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,co
nst vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Er
ror: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument V
aluesym2poly/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 ValueWarning, integration of abs or sig
n assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep^2-1)]Discontinuitie
s at zeroes of t_nostep^2-1 were not checkedEvaluation time: 77.12Done

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maple [B]  time = 0.34, size = 1905, normalized size = 18.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))*(2*
(a^2+b^2)^(1/2)+2*a)^(1/2)*a^3-1/4/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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a+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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4-1/4/d*b^3/(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))*(2*(a^2+b^2
)^(1/2)+2*a)^(1/2)+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^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+1/d*b/(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))*a^2-1/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^5+1/d*b^3/(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))-3/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-4/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^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))*(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))*(2*(a^2+b^2)^(
1/2)+2*a)^(1/2)*a-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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*a^4+1/4/d*b^3/(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))*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)-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^3-1/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/(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))*a^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^5-1/d*b^3/(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))+3/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+4/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^3

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

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mupad [B]  time = 8.51, size = 2731, normalized size = 26.77 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(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^3*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^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^3*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^4*b^5*d^5)/(a^2*d^4 + b^2*d^4) + (16*
a^6*b^3*d^5)/(a^2*d^4 + b^2*d^4) - (4*a^3*b^3*d^4*(-16*a^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) - (4*a*b^5*d^4*
(-16*a^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) - (32*a^4*b^2*d^2*((-16*a^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^
4)) - (a^3*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^5*d^5)/(a^2*d^4 + b^2*d^
4) + (16*a^6*b^3*d^5)/(a^2*d^4 + b^2*d^4) - (4*a^3*b^3*d^4*(-16*a^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) - (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^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) - (a
^3*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2) - 2*atanh((32*a^2*b^4*d^2*((-16*b^6*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4))
+ (a*b^2*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*a^2*b^7*d^5)/(a^2*d^4 + b^2*d^4
) - 16*a^2*b^5*d - 16*b^7*d + (16*a^4*b^5*d^5)/(a^2*d^4 + b^2*d^4) + (4*a*b^5*d^4*(-16*b^6*d^4)^(1/2))/(a^2*d^
5 + b^2*d^5) + (4*a^3*b^3*d^4*(-16*b^6*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) - (32*b^4*((-16*b^6*d^4)^(1/2)/(16*(a^
2*d^4 + b^2*d^4)) + (a*b^2*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*a^2*b^5*d^3)/(
a^2*d^4 + b^2*d^4) - (16*b^5)/d + (4*a*b^3*d^2*(-16*b^6*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) + (8*a*b^2*((-16*b^6*
d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) + (a*b^2*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2)*(a + b*tan(c + d*x))^(1/2)*(-
16*b^6*d^4)^(1/2))/((16*a^2*b^7*d^5)/(a^2*d^4 + b^2*d^4) - 16*a^2*b^5*d - 16*b^7*d + (16*a^4*b^5*d^5)/(a^2*d^4
+ b^2*d^4) + (4*a*b^5*d^4*(-16*b^6*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (4*a^3*b^3*d^4*(-16*b^6*d^4)^(1/2))/(a^2
*d^5 + b^2*d^5)))*((-16*b^6*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) + (a*b^2*d^2)/(4*(a^2*d^4 + b^2*d^4)))^(1/2) -
2*atanh((32*b^4*((a*b^2*d^2)/(4*(a^2*d^4 + b^2*d^4)) - (-16*b^6*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^5)/d - (16*a^2*b^5*d^3)/(a^2*d^4 + b^2*d^4) + (4*a*b^3*d^2*(-16*b^6*d^4)^(1/2
))/(a^2*d^5 + b^2*d^5)) - (32*a^2*b^4*d^2*((a*b^2*d^2)/(4*(a^2*d^4 + b^2*d^4)) - (-16*b^6*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^7*d + 16*a^2*b^5*d - (16*a^2*b^7*d^5)/(a^2*d^4 + b^2*
d^4) - (16*a^4*b^5*d^5)/(a^2*d^4 + b^2*d^4) + (4*a*b^5*d^4*(-16*b^6*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (4*a^3*b
^3*d^4*(-16*b^6*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)) + (8*a*b^2*((a*b^2*d^2)/(4*(a^2*d^4 + b^2*d^4)) - (-16*b^6*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^6*d^4)^(1/2))/(16*b^7*d + 16*a^2*b^
5*d - (16*a^2*b^7*d^5)/(a^2*d^4 + b^2*d^4) - (16*a^4*b^5*d^5)/(a^2*d^4 + b^2*d^4) + (4*a*b^5*d^4*(-16*b^6*d^4)
^(1/2))/(a^2*d^5 + b^2*d^5) + (4*a^3*b^3*d^4*(-16*b^6*d^4)^(1/2))/(a^2*d^5 + b^2*d^5)))*((a*b^2*d^2)/(4*(a^2*d
^4 + b^2*d^4)) - (-16*b^6*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)))^(1/2) - 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^3*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^4*b^5*d^5)/(a^2*d^4 + b^2*d^4) + (16*a^6*b^3*d^5)/(a^2*d^4 + b^2*d^4) + (4*a^3*b^3*d^4*(
-16*a^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (4*a*b^5*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^3*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^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^4*b^2*d^2*(- (-16*a^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) - (a^3*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^5*d^5)/(a^2*d^4 + b^2*d^4) + (16*a^6*b^3*d^5)/(a^2*d^4 +
b^2*d^4) + (4*a^3*b^3*d^4*(-16*a^4*b^2*d^4)^(1/2))/(a^2*d^5 + b^2*d^5) + (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^4*b^2*d^4)^(1/2)/(16*(a^2*d^4 + b^2*d^4)) - (a^3*d^2)/(4*(a^2*d^4 + b^2*d^4))
)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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