\(\int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx\) [508]

Optimal result

Integrand size = 21, antiderivative size = 116 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {\sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3653, 3620, 3618, 65, 214, 3715} \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {\sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{d}+\frac {\sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.96 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\frac {-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a}}\right )+\sqrt {a-i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )+\sqrt {a+i b} \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{d} \]

[In]

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

[Out]

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

Maple [F(-1)]

Timed out.

hanged

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (90) = 180\).

Time = 0.27 (sec) , antiderivative size = 614, normalized size of antiderivative = 5.29 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\left [\frac {d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) - d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (-d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) + d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) - d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (-d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) + 2 \, \sqrt {a} \log \left (\frac {b \tan \left (d x + c\right ) - 2 \, \sqrt {b \tan \left (d x + c\right ) + a} \sqrt {a} + 2 \, a}{\tan \left (d x + c\right )}\right )}{2 \, d}, \frac {d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) - d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} \log \left (-d \sqrt {\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} + a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) + d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) - d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} \log \left (-d \sqrt {-\frac {d^{2} \sqrt {-\frac {b^{2}}{d^{4}}} - a}{d^{2}}} + \sqrt {b \tan \left (d x + c\right ) + a}\right ) + 4 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right ) + a} \sqrt {-a}}{a}\right )}{2 \, d}\right ] \]

[In]

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

[Out]

[1/2*(d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a)
) - d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2)*log(-d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a))
+ d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*log(d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a))
- d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2)*log(-d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a)
) + 2*sqrt(a)*log((b*tan(d*x + c) - 2*sqrt(b*tan(d*x + c) + a)*sqrt(a) + 2*a)/tan(d*x + c)))/d, 1/2*(d*sqrt((d
^2*sqrt(-b^2/d^4) + a)/d^2)*log(d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a)) - d*sqrt((d^2
*sqrt(-b^2/d^4) + a)/d^2)*log(-d*sqrt((d^2*sqrt(-b^2/d^4) + a)/d^2) + sqrt(b*tan(d*x + c) + a)) + d*sqrt(-(d^2
*sqrt(-b^2/d^4) - a)/d^2)*log(d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a)) - d*sqrt(-(d^2
*sqrt(-b^2/d^4) - a)/d^2)*log(-d*sqrt(-(d^2*sqrt(-b^2/d^4) - a)/d^2) + sqrt(b*tan(d*x + c) + a)) + 4*sqrt(-a)*
arctan(sqrt(b*tan(d*x + c) + a)*sqrt(-a)/a))/d]

Sympy [F]

\[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int \sqrt {a + b \tan {\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\int { \sqrt {b \tan \left (d x + c\right ) + a} \cot \left (d x + c\right ) \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(cot(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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(c
onst gen &

Mupad [B] (verification not implemented)

Time = 4.81 (sec) , antiderivative size = 682, normalized size of antiderivative = 5.88 \[ \int \cot (c+d x) \sqrt {a+b \tan (c+d x)} \, dx=-\frac {2\,\sqrt {a}\,\mathrm {atanh}\left (\frac {64\,\sqrt {a}\,b^{12}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{576\,a^5\,b^8+640\,a^3\,b^{10}+64\,a\,b^{12}}+\frac {640\,a^{5/2}\,b^{10}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{576\,a^5\,b^8+640\,a^3\,b^{10}+64\,a\,b^{12}}+\frac {576\,a^{9/2}\,b^8\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{576\,a^5\,b^8+640\,a^3\,b^{10}+64\,a\,b^{12}}\right )}{d}-\mathrm {atan}\left (-\frac {32\,a\,b^{11}\,\sqrt {\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,a\,b^{12}}{d}-\frac {a^2\,b^{11}\,48{}\mathrm {i}}{d}+\frac {16\,a^3\,b^{10}}{d}-\frac {a^4\,b^9\,48{}\mathrm {i}}{d}}+\frac {a^2\,b^{10}\,\sqrt {\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{\frac {16\,a\,b^{12}}{d}-\frac {a^2\,b^{11}\,48{}\mathrm {i}}{d}+\frac {16\,a^3\,b^{10}}{d}-\frac {a^4\,b^9\,48{}\mathrm {i}}{d}}+\frac {96\,a^3\,b^9\,\sqrt {\frac {a}{4\,d^2}-\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,a\,b^{12}}{d}-\frac {a^2\,b^{11}\,48{}\mathrm {i}}{d}+\frac {16\,a^3\,b^{10}}{d}-\frac {a^4\,b^9\,48{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a-b\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {32\,a\,b^{11}\,\sqrt {\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,a\,b^{12}}{d}+\frac {a^2\,b^{11}\,48{}\mathrm {i}}{d}+\frac {16\,a^3\,b^{10}}{d}+\frac {a^4\,b^9\,48{}\mathrm {i}}{d}}+\frac {a^2\,b^{10}\,\sqrt {\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}\,128{}\mathrm {i}}{\frac {16\,a\,b^{12}}{d}+\frac {a^2\,b^{11}\,48{}\mathrm {i}}{d}+\frac {16\,a^3\,b^{10}}{d}+\frac {a^4\,b^9\,48{}\mathrm {i}}{d}}-\frac {96\,a^3\,b^9\,\sqrt {\frac {a}{4\,d^2}+\frac {b\,1{}\mathrm {i}}{4\,d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{\frac {16\,a\,b^{12}}{d}+\frac {a^2\,b^{11}\,48{}\mathrm {i}}{d}+\frac {16\,a^3\,b^{10}}{d}+\frac {a^4\,b^9\,48{}\mathrm {i}}{d}}\right )\,\sqrt {\frac {a+b\,1{}\mathrm {i}}{4\,d^2}}\,2{}\mathrm {i} \]

[In]

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

[Out]

- atan((a^2*b^10*(a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*128i)/((16*a*b^12)/d - (a^2*b^1
1*48i)/d + (16*a^3*b^10)/d - (a^4*b^9*48i)/d) - (32*a*b^11*(a/(4*d^2) - (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d
*x))^(1/2))/((16*a*b^12)/d - (a^2*b^11*48i)/d + (16*a^3*b^10)/d - (a^4*b^9*48i)/d) + (96*a^3*b^9*(a/(4*d^2) -
(b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*a*b^12)/d - (a^2*b^11*48i)/d + (16*a^3*b^10)/d - (a^4*b
^9*48i)/d))*((a - b*1i)/(4*d^2))^(1/2)*2i - atan((32*a*b^11*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c +
d*x))^(1/2))/((16*a*b^12)/d + (a^2*b^11*48i)/d + (16*a^3*b^10)/d + (a^4*b^9*48i)/d) + (a^2*b^10*(a/(4*d^2) + (
b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2)*128i)/((16*a*b^12)/d + (a^2*b^11*48i)/d + (16*a^3*b^10)/d + (a
^4*b^9*48i)/d) - (96*a^3*b^9*(a/(4*d^2) + (b*1i)/(4*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2))/((16*a*b^12)/d + (
a^2*b^11*48i)/d + (16*a^3*b^10)/d + (a^4*b^9*48i)/d))*((a + b*1i)/(4*d^2))^(1/2)*2i - (2*a^(1/2)*atanh((64*a^(
1/2)*b^12*(a + b*tan(c + d*x))^(1/2))/(64*a*b^12 + 640*a^3*b^10 + 576*a^5*b^8) + (640*a^(5/2)*b^10*(a + b*tan(
c + d*x))^(1/2))/(64*a*b^12 + 640*a^3*b^10 + 576*a^5*b^8) + (576*a^(9/2)*b^8*(a + b*tan(c + d*x))^(1/2))/(64*a
*b^12 + 640*a^3*b^10 + 576*a^5*b^8)))/d