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[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)

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Rubi [A]  time = 0.149244, antiderivative size = 102, normalized size of antiderivative = 1., 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 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]

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 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]

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*}

Mathematica [A]  time = 0.167018, 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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Maple [B]  time = 0.139, size = 1905, normalized size = 18.7 \begin{align*} \text{result too large to display} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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

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Fricas [B]  time = 1.97254, size = 7656, normalized size = 75.06 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \tan \left (d x + c\right ) - a}{\sqrt{b \tan \left (d x + c\right ) + a}}\,{d x} \end{align*}

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]

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