Integrand size = 28, antiderivative size = 85 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {a} d}+\frac {\sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}} \] Output:
(1/2-1/2*I)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2 ))/a^(1/2)/d+tan(d*x+c)^(1/2)/d/(a+I*a*tan(d*x+c))^(1/2)
Time = 0.38 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\tan (c+d x)} \left (\frac {\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {i a \tan (c+d x)}}+\frac {2}{\sqrt {a+i a \tan (c+d x)}}\right )}{2 d} \] Input:
Integrate[1/(Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]
Output:
(Sqrt[Tan[c + d*x]]*((Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqr t[a + I*a*Tan[c + d*x]]])/Sqrt[I*a*Tan[c + d*x]] + 2/Sqrt[a + I*a*Tan[c + d*x]]))/(2*d)
Time = 0.53 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.47, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4031, 3042, 4029, 3042, 4027, 218}
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.
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}dx\) |
| \(\Big \downarrow \) 4031 |
| \(\displaystyle i \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \int \frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}dx+\frac {2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4029 |
| \(\displaystyle i \left (\frac {i \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}-\frac {i \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )+\frac {2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle i \left (\frac {i \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}-\frac {i \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx}{2 a}\right )+\frac {2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 4027 |
| \(\displaystyle i \left (\frac {i \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}-\frac {a \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}\right )+\frac {2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle i \left (\frac {i \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{\sqrt {a} d}\right )+\frac {2 \sqrt {\tan (c+d x)}}{d \sqrt {a+i a \tan (c+d x)}}\) |
Input:
Int[1/(Sqrt[Tan[c + d*x]]*Sqrt[a + I*a*Tan[c + d*x]]),x]
Output:
(2*Sqrt[Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]]) + I*(((-1/2 - I/2)*A rcTanh[((1 + I)*Sqrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/( Sqrt[a]*d) + (I*Sqrt[Tan[c + d*x]])/(d*Sqrt[a + I*a*Tan[c + d*x]]))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Simp[-2*a*(b/f) Subst[Int[1/(a*c - b*d - 2* a^2*x^2), x], x, Sqrt[c + d*Tan[e + f*x]]/Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && N eQ[c^2 + d^2, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*b*f*m)), x] - Simp[(a*c - b*d)/(2*b^2) Int[(a + b*Tan[e + f *x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Eq Q[m + n, 0] && LeQ[m, -2^(-1)]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-d)*(a + b*Tan[e + f*x])^m*((c + d*Ta n[e + f*x])^(n + 1)/(f*m*(c^2 + d^2))), x] + Simp[a/(a*c - b*d) Int[(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d , e, f, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^ 2, 0] && EqQ[m + n + 1, 0] && !LtQ[m, -1]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (67 ) = 134\).
Time = 1.82 (sec) , antiderivative size = 349, normalized size of antiderivative = 4.11
| method | result | size |
| derivativedivides | \(-\frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{2}-i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +2 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+4 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )+4 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{4 d a \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{2} \sqrt {-i a}}\) | \(349\) |
| default | \(-\frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \left (i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )^{2}-i \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a +2 \sqrt {2}\, \ln \left (\frac {2 \sqrt {2}\, \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}-i a +3 a \tan \left (d x +c \right )}{\tan \left (d x +c \right )+i}\right ) a \tan \left (d x +c \right )+4 i \sqrt {-i a}\, \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \tan \left (d x +c \right )+4 \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-i a}\right )}{4 d a \sqrt {a \tan \left (d x +c \right ) \left (1+i \tan \left (d x +c \right )\right )}\, \left (-\tan \left (d x +c \right )+i\right )^{2} \sqrt {-i a}}\) | \(349\) |
Input:
int(1/tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/4/d*tan(d*x+c)^(1/2)*(a*(1+I*tan(d*x+c)))^(1/2)/a*(I*2^(1/2)*ln((2*2^(1 /2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c)) /(tan(d*x+c)+I))*a*tan(d*x+c)^2-I*2^(1/2)*ln((2*2^(1/2)*(-I*a)^(1/2)*(a*ta n(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-I*a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*a+2*2 ^(1/2)*ln((2*2^(1/2)*(-I*a)^(1/2)*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)-I* a+3*a*tan(d*x+c))/(tan(d*x+c)+I))*a*tan(d*x+c)+4*I*(-I*a)^(1/2)*(a*tan(d*x +c)*(1+I*tan(d*x+c)))^(1/2)*tan(d*x+c)+4*(a*tan(d*x+c)*(1+I*tan(d*x+c)))^( 1/2)*(-I*a)^(1/2))/(a*tan(d*x+c)*(1+I*tan(d*x+c)))^(1/2)/(-tan(d*x+c)+I)^2 /(-I*a)^(1/2)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 299 vs. \(2 (61) = 122\).
Time = 0.10 (sec) , antiderivative size = 299, normalized size of antiderivative = 3.52 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {{\left (a d \sqrt {-\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (\frac {1}{4} \, a d \sqrt {-\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) - a d \sqrt {-\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} \log \left (-\frac {1}{4} \, a d \sqrt {-\frac {2 i}{a d^{2}}} e^{\left (i \, d x + i \, c\right )} + \frac {1}{4} \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right ) + 2 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a d} \] Input:
integrate(1/tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas ")
Output:
1/4*(a*d*sqrt(-2*I/(a*d^2))*e^(I*d*x + I*c)*log(1/4*a*d*sqrt(-2*I/(a*d^2)) *e^(I*d*x + I*c) + 1/4*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I* e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1)) - a*d*sqrt(-2*I/(a*d^2))*e^(I*d*x + I*c)*log(-1/4*a*d*sqrt(-2*I/(a*d^ 2))*e^(I*d*x + I*c) + 1/4*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(( -I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c ) + 1)) + 2*sqrt(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*(e^(2*I*d*x + 2*I*c) + 1))*e^(-I* d*x - I*c)/(a*d)
\[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \sqrt {\tan {\left (c + d x \right )}}}\, dx \] Input:
integrate(1/tan(d*x+c)**(1/2)/(a+I*a*tan(d*x+c))**(1/2),x)
Output:
Integral(1/(sqrt(I*a*(tan(c + d*x) - I))*sqrt(tan(c + d*x))), x)
\[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\int { \frac {1}{\sqrt {i \, a \tan \left (d x + c\right ) + a} \sqrt {\tan \left (d x + c\right )}} \,d x } \] Input:
integrate(1/tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima ")
Output:
integrate(1/(sqrt(I*a*tan(d*x + c) + a)*sqrt(tan(d*x + c))), x)
Exception generated. \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(1/tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2),x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument TypeError: Bad Argument TypeDone
Time = 3.52 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.01 \[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,2{}\mathrm {i}}{\left (\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}-\sqrt {a}\right )\,\left (d\,1{}\mathrm {i}-\frac {a\,d\,\mathrm {tan}\left (c+d\,x\right )}{{\left (\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}-\sqrt {a}\right )}^2}\right )}+\frac {2\,\sqrt {\frac {1}{8}{}\mathrm {i}}\,\mathrm {atanh}\left (\frac {32\,\sqrt {\frac {1}{8}{}\mathrm {i}}\,{\left (-a\right )}^{9/2}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{\left (a^4\,4{}\mathrm {i}-\frac {4\,a^5\,\mathrm {tan}\left (c+d\,x\right )}{{\left (\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}-\sqrt {a}\right )}^2}\right )\,\left (\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}-\sqrt {a}\right )}\right )}{\sqrt {-a}\,d} \] Input:
int(1/(tan(c + d*x)^(1/2)*(a + a*tan(c + d*x)*1i)^(1/2)),x)
Output:
(tan(c + d*x)^(1/2)*2i)/(((a + a*tan(c + d*x)*1i)^(1/2) - a^(1/2))*(d*1i - (a*d*tan(c + d*x))/((a + a*tan(c + d*x)*1i)^(1/2) - a^(1/2))^2)) + (2*(1i /8)^(1/2)*atanh((32*(1i/8)^(1/2)*(-a)^(9/2)*tan(c + d*x)^(1/2))/((a^4*4i - (4*a^5*tan(c + d*x))/((a + a*tan(c + d*x)*1i)^(1/2) - a^(1/2))^2)*((a + a *tan(c + d*x)*1i)^(1/2) - a^(1/2)))))/((-a)^(1/2)*d)
\[ \int \frac {1}{\sqrt {\tan (c+d x)} \sqrt {a+i a \tan (c+d x)}} \, dx=\frac {2 \sqrt {a}\, \left (-2 \sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}\, \tan \left (d x +c \right ) i +\sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}+2 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{2}+1}d x \right ) \tan \left (d x +c \right )^{2} d i +2 \left (\int \frac {\sqrt {\tan \left (d x +c \right )}\, \sqrt {\tan \left (d x +c \right ) i +1}}{\tan \left (d x +c \right )^{2}+1}d x \right ) d i \right )}{a d \left (\tan \left (d x +c \right )^{2}+1\right )} \] Input:
int(1/tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1/2),x)
Output:
(2*sqrt(a)*( - 2*sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*i + 1)*tan(c + d*x)* i + sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*i + 1) + 2*int((sqrt(tan(c + d*x) )*sqrt(tan(c + d*x)*i + 1))/(tan(c + d*x)**2 + 1),x)*tan(c + d*x)**2*d*i + 2*int((sqrt(tan(c + d*x))*sqrt(tan(c + d*x)*i + 1))/(tan(c + d*x)**2 + 1) ,x)*d*i))/(a*d*(tan(c + d*x)**2 + 1))