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

Optimal. Leaf size=106 \[ \frac{2 \sqrt{a+b \tan (c+d x)}}{d}-\frac{\sqrt{a-i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{\sqrt{a+i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.1693, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3528, 3539, 3537, 63, 208} \[ \frac{2 \sqrt{a+b \tan (c+d x)}}{d}-\frac{\sqrt{a-i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{\sqrt{a+i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3528

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

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

Mathematica [A]  time = 0.10283, size = 100, normalized size = 0.94 \[ -\frac{-2 \sqrt{a+b \tan (c+d x)}+\sqrt{a-i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )+\sqrt{a+i b} \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.048, size = 478, normalized size = 4.5 \begin{align*} 2\,{\frac{\sqrt{a+b\tan \left ( dx+c \right ) }}{d}}-{\frac{1}{4\,d}\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}\ln \left ( b\tan \left ( dx+c \right ) +a+\sqrt{a+b\tan \left ( dx+c \right ) }\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}+\sqrt{{a}^{2}+{b}^{2}} \right ) }+{\frac{a}{d}\arctan \left ({ \left ( 2\,\sqrt{a+b\tan \left ( dx+c \right ) }+\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}}-{\frac{1}{d}\arctan \left ({ \left ( 2\,\sqrt{a+b\tan \left ( dx+c \right ) }+\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \right ) \sqrt{{a}^{2}+{b}^{2}}{\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}}+{\frac{1}{4\,d}\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}\ln \left ( \sqrt{a+b\tan \left ( dx+c \right ) }\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}-b\tan \left ( dx+c \right ) -a-\sqrt{{a}^{2}+{b}^{2}} \right ) }-{\frac{a}{d}\arctan \left ({ \left ( \sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}-2\,\sqrt{a+b\tan \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}}+{\frac{1}{d}\arctan \left ({ \left ( \sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}+2\,a}-2\,\sqrt{a+b\tan \left ( dx+c \right ) } \right ){\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \right ) \sqrt{{a}^{2}+{b}^{2}}{\frac{1}{\sqrt{2\,\sqrt{{a}^{2}+{b}^{2}}-2\,a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(a+b*tan(d*x+c))^(1/2)/d-1/4/d*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)*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))+1/d/(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/(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+b^2)^(1/2)+1/4/d*(2*(a^2
+b^2)^(1/2)+2*a)^(1/2)*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))
-1/d/(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/(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+b^2)^(1/2)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.21933, size = 3951, normalized size = 37.27 \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))^(1/2)*tan(d*x+c),x, algorithm="fricas")

[Out]

1/4*(4*sqrt(2)*d^5*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*sqrt(b^2/d^4)*((a^2 + b^2)/d^4)^(3/4)*
arctan(-((a^2 + b^2)*d^4*sqrt(b^2/d^4)*sqrt((a^2 + b^2)/d^4) + (a^3 + a*b^2)*d^2*sqrt(b^2/d^4) + sqrt(2)*(d^7*
sqrt(b^2/d^4)*sqrt((a^2 + b^2)/d^4) + a*d^5*sqrt(b^2/d^4))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)
)*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*((a^2 + b^2)/d^4)^(3/4) - sqrt(2)*(d^7*sqrt(b^2/d^4)*sq
rt((a^2 + b^2)/d^4) + a*d^5*sqrt(b^2/d^4))*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*sqrt(((a^2 + b
^2)*d^2*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + sqrt(2)*(a*d^3*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + (a^2 + b^2)*d
*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 -
b^2)/b^2)*((a^2 + b^2)/d^4)^(1/4) + (a^3 + a*b^2)*cos(d*x + c) + (a^2*b + b^3)*sin(d*x + c))/((a^2 + b^2)*cos
(d*x + c)))*((a^2 + b^2)/d^4)^(3/4))/(a^2*b^2 + b^4)) + 4*sqrt(2)*d^5*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2
- b^2)/b^2)*sqrt(b^2/d^4)*((a^2 + b^2)/d^4)^(3/4)*arctan(((a^2 + b^2)*d^4*sqrt(b^2/d^4)*sqrt((a^2 + b^2)/d^4)
+ (a^3 + a*b^2)*d^2*sqrt(b^2/d^4) - sqrt(2)*(d^7*sqrt(b^2/d^4)*sqrt((a^2 + b^2)/d^4) + a*d^5*sqrt(b^2/d^4))*s
qrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*((a^2
+ b^2)/d^4)^(3/4) + sqrt(2)*(d^7*sqrt(b^2/d^4)*sqrt((a^2 + b^2)/d^4) + a*d^5*sqrt(b^2/d^4))*sqrt(-(a*d^2*sqrt
((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*sqrt(((a^2 + b^2)*d^2*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) - sqrt(2)*(a*d^3*
sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + (a^2 + b^2)*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*
x + c))*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*((a^2 + b^2)/d^4)^(1/4) + (a^3 + a*b^2)*cos(d*x +
c) + (a^2*b + b^3)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c)))*((a^2 + b^2)/d^4)^(3/4))/(a^2*b^2 + b^4)) - sqrt
(2)*(a*d^3*sqrt((a^2 + b^2)/d^4) + (a^2 + b^2)*d)*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*((a^2 +
b^2)/d^4)^(1/4)*log(((a^2 + b^2)*d^2*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + sqrt(2)*(a*d^3*sqrt((a^2 + b^2)/d^4
)*cos(d*x + c) + (a^2 + b^2)*d*cos(d*x + c))*sqrt((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-(a*d^2
*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*((a^2 + b^2)/d^4)^(1/4) + (a^3 + a*b^2)*cos(d*x + c) + (a^2*b + b^3)*
sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) + sqrt(2)*(a*d^3*sqrt((a^2 + b^2)/d^4) + (a^2 + b^2)*d)*sqrt(-(a*d^2
*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*((a^2 + b^2)/d^4)^(1/4)*log(((a^2 + b^2)*d^2*sqrt((a^2 + b^2)/d^4)*co
s(d*x + c) - sqrt(2)*(a*d^3*sqrt((a^2 + b^2)/d^4)*cos(d*x + c) + (a^2 + b^2)*d*cos(d*x + c))*sqrt((a*cos(d*x +
c) + b*sin(d*x + c))/cos(d*x + c))*sqrt(-(a*d^2*sqrt((a^2 + b^2)/d^4) - a^2 - b^2)/b^2)*((a^2 + b^2)/d^4)^(1/
4) + (a^3 + a*b^2)*cos(d*x + c) + (a^2*b + b^3)*sin(d*x + c))/((a^2 + b^2)*cos(d*x + c))) + 8*(a^2 + b^2)*sqrt
((a*cos(d*x + c) + b*sin(d*x + c))/cos(d*x + c)))/((a^2 + b^2)*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \tan{\left (c + d x \right )}} \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))**(1/2)*tan(d*x+c),x)

[Out]

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

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out