Integrand size = 25, antiderivative size = 115 \[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=-\frac {i \sqrt {i a-b} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i \sqrt {i a+b} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d} \] Output:
-I*(I*a-b)^(1/2)*arctan((I*a-b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x+c))^(1 /2))/d-I*(I*a+b)^(1/2)*arctanh((I*a+b)^(1/2)*tan(d*x+c)^(1/2)/(a+b*tan(d*x +c))^(1/2))/d
Time = 0.06 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=\frac {(-1)^{3/4} \left (\sqrt {-a+i b} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-\sqrt {a+i b} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )\right )}{d} \] Input:
Integrate[Sqrt[a + b*Tan[c + d*x]]/Sqrt[Tan[c + d*x]],x]
Output:
((-1)^(3/4)*(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]*ArcTan[((-1)^(1/4)*Sqrt[ a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]))/d
Time = 0.34 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 4058, 610, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}}dx\) |
\(\Big \downarrow \) 4058 |
\(\displaystyle \frac {\int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 610 |
\(\displaystyle \frac {\int \left (\frac {i a-b}{2 (i-\tan (c+d x)) \sqrt {\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}+\frac {i a+b}{2 \sqrt {\tan (c+d x)} (\tan (c+d x)+i) \sqrt {a+b \tan (c+d x)}}\right )d\tan (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-i \sqrt {-b+i a} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )-i \sqrt {b+i a} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}\) |
Input:
Int[Sqrt[a + b*Tan[c + d*x]]/Sqrt[Tan[c + d*x]],x]
Output:
((-I)*Sqrt[I*a - b]*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*T an[c + d*x]]] - I*Sqrt[I*a + b]*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]]) /Sqrt[a + b*Tan[c + d*x]]])/d
Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_))/((a_) + (b_.)*(x_)^2), x_S ymbol] :> Simp[e^(m + 1/2) Int[ExpandIntegrand[1/(Sqrt[e*x]*Sqrt[c + d*x] ), x^(m + 1/2)*((c + d*x)^(n + 1/2)/(a + b*x^2)), x], x], x] /; FreeQ[{a, b , c, d, e}, x] && IGtQ[n + 1/2, 0] && ILtQ[m - 1/2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[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]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 0.61 (sec) , antiderivative size = 1085174, normalized size of antiderivative = 9436.30
\[\text {output too large to display}\]
Input:
int((a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x)
Output:
result too large to display
Leaf count of result is larger than twice the leaf count of optimal. 2625 vs. \(2 (87) = 174\).
Time = 0.42 (sec) , antiderivative size = 2625, normalized size of antiderivative = 22.83 \[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=\text {Too large to display} \] Input:
integrate((a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="fricas")
Output:
-1/8*sqrt(-(d^2*sqrt(-a^2/d^4) + b)/d^2)*log((2*(2*a^4*b + 4*a^2*b^3 + (a^ 5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-a^2/d^4))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) + ((a^4*b + 4*a^2*b^3)*d*tan(d*x + c)^2 - 2*(a^5 + 3*a^3*b^2 + 4*a*b^4)*d*tan(d*x + c) - (3*a^4*b + 4*a^2*b^3)*d + (a^4*d^3 - (a^4 + 6*a^2*b^2 + 8*b^4)*d^3*tan(d*x + c)^2 - 4*(a^3*b + 2*a*b^3)*d^3*ta n(d*x + c))*sqrt(-a^2/d^4))*sqrt(-(d^2*sqrt(-a^2/d^4) + b)/d^2))/(tan(d*x + c)^2 + 1)) - 1/8*sqrt(-(d^2*sqrt(-a^2/d^4) + b)/d^2)*log(-(2*(2*a^4*b + 4*a^2*b^3 + (a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3 )*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-a^2/d^4))*sqrt(b *tan(d*x + c) + a)*sqrt(tan(d*x + c)) + ((a^4*b + 4*a^2*b^3)*d*tan(d*x + c )^2 - 2*(a^5 + 3*a^3*b^2 + 4*a*b^4)*d*tan(d*x + c) - (3*a^4*b + 4*a^2*b^3) *d + (a^4*d^3 - (a^4 + 6*a^2*b^2 + 8*b^4)*d^3*tan(d*x + c)^2 - 4*(a^3*b + 2*a*b^3)*d^3*tan(d*x + c))*sqrt(-a^2/d^4))*sqrt(-(d^2*sqrt(-a^2/d^4) + b)/ d^2))/(tan(d*x + c)^2 + 1)) + 1/8*sqrt(-(d^2*sqrt(-a^2/d^4) + b)/d^2)*log( (2*(2*a^4*b + 4*a^2*b^3 + (a^5 + 3*a^3*b^2 + 4*a*b^4)*tan(d*x + c) + (2*(a ^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-a^ 2/d^4))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - ((a^4*b + 4*a^2*b^3) *d*tan(d*x + c)^2 - 2*(a^5 + 3*a^3*b^2 + 4*a*b^4)*d*tan(d*x + c) - (3*a^4* b + 4*a^2*b^3)*d + (a^4*d^3 - (a^4 + 6*a^2*b^2 + 8*b^4)*d^3*tan(d*x + c...
\[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=\int \frac {\sqrt {a + b \tan {\left (c + d x \right )}}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx \] Input:
integrate((a+b*tan(d*x+c))**(1/2)/tan(d*x+c)**(1/2),x)
Output:
Integral(sqrt(a + b*tan(c + d*x))/sqrt(tan(c + d*x)), x)
\[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=\int { \frac {\sqrt {b \tan \left (d x + c\right ) + a}}{\sqrt {\tan \left (d x + c\right )}} \,d x } \] Input:
integrate((a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*tan(d*x + c) + a)/sqrt(tan(d*x + c)), x)
Timed out. \[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate((a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x, algorithm="giac")
Output:
Timed out
Time = 3.46 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=-\mathrm {atan}\left (\frac {\sqrt {a}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {b+a\,1{}\mathrm {i}}{d^2}}-d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {b+a\,1{}\mathrm {i}}{d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}{a+b\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {a}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}\right )\,\sqrt {-\frac {b+a\,1{}\mathrm {i}}{d^2}}\,1{}\mathrm {i}-\mathrm {atan}\left (\frac {d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-b+a\,1{}\mathrm {i}}{d^2}}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}-\sqrt {a}\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {-b+a\,1{}\mathrm {i}}{d^2}}}{a+b\,\mathrm {tan}\left (c+d\,x\right )-\sqrt {a}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}}\right )\,\sqrt {\frac {-b+a\,1{}\mathrm {i}}{d^2}}\,1{}\mathrm {i} \] Input:
int((a + b*tan(c + d*x))^(1/2)/tan(c + d*x)^(1/2),x)
Output:
- atan((a^(1/2)*d*tan(c + d*x)^(1/2)*(-(a*1i + b)/d^2)^(1/2) - d*tan(c + d *x)^(1/2)*(-(a*1i + b)/d^2)^(1/2)*(a + b*tan(c + d*x))^(1/2))/(a + b*tan(c + d*x) - a^(1/2)*(a + b*tan(c + d*x))^(1/2)))*(-(a*1i + b)/d^2)^(1/2)*1i - atan((d*tan(c + d*x)^(1/2)*((a*1i - b)/d^2)^(1/2)*(a + b*tan(c + d*x))^( 1/2) - a^(1/2)*d*tan(c + d*x)^(1/2)*((a*1i - b)/d^2)^(1/2))/(a + b*tan(c + d*x) - a^(1/2)*(a + b*tan(c + d*x))^(1/2)))*((a*1i - b)/d^2)^(1/2)*1i
\[ \int \frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {\tan (c+d x)}} \, dx=\frac {2 \sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}-2 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )^{2}}{a +\tan \left (d x +c \right ) b}d x \right ) b d -\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}\, \tan \left (d x +c \right )}{a +\tan \left (d x +c \right ) b}d x \right ) a d +\left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{2} b +\tan \left (d x +c \right ) a}d x \right ) a d}{2 d} \] Input:
int((a+b*tan(d*x+c))^(1/2)/tan(d*x+c)^(1/2),x)
Output:
(2*sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a) - 2*int((sqrt(tan(c + d*x)) *sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**2)/(tan(c + d*x)*b + a),x)*b*d - i nt((sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a)*tan(c + d*x))/(tan(c + d*x )*b + a),x)*a*d + int((sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*b + a))/(tan(c + d*x)**2*b + tan(c + d*x)*a),x)*a*d)/(2*d)