Integrand size = 28, antiderivative size = 102 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {(1+i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}}{d}-\frac {2 \sqrt {\cot (c+d x)} \sqrt {a+i a \tan (c+d x)}}{d} \] Output:
(1+I)*a^(1/2)*arctanh((1+I)*a^(1/2)*tan(d*x+c)^(1/2)/(a+I*a*tan(d*x+c))^(1 /2))*cot(d*x+c)^(1/2)*tan(d*x+c)^(1/2)/d-2*cot(d*x+c)^(1/2)*(a+I*a*tan(d*x +c))^(1/2)/d
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\frac {\sqrt {\cot (c+d x)} \left (\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {i a \tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right ) \sqrt {i a \tan (c+d x)}-2 \sqrt {a+i a \tan (c+d x)}\right )}{d} \] Input:
Integrate[Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]],x]
Output:
(Sqrt[Cot[c + d*x]]*(Sqrt[2]*ArcTanh[(Sqrt[2]*Sqrt[I*a*Tan[c + d*x]])/Sqrt [a + I*a*Tan[c + d*x]]]*Sqrt[I*a*Tan[c + d*x]] - 2*Sqrt[a + I*a*Tan[c + d* x]]))/d
Time = 0.49 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 4729, 3042, 4031, 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.
\(\displaystyle \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cot (c+d x)^{3/2} \sqrt {a+i a \tan (c+d x)}dx\) |
\(\Big \downarrow \) 4729 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan ^{\frac {3}{2}}(c+d x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \int \frac {\sqrt {i \tan (c+d x) a+a}}{\tan (c+d x)^{3/2}}dx\) |
\(\Big \downarrow \) 4031 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (i \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (i \int \frac {\sqrt {i \tan (c+d x) a+a}}{\sqrt {\tan (c+d x)}}dx-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 4027 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {2 a^2 \int \frac {1}{-\frac {2 \tan (c+d x) a^2}{i \tan (c+d x) a+a}-i a}d\frac {\sqrt {\tan (c+d x)}}{\sqrt {i \tan (c+d x) a+a}}}{d}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {(1+i) \sqrt {a} \text {arctanh}\left (\frac {(1+i) \sqrt {a} \sqrt {\tan (c+d x)}}{\sqrt {a+i a \tan (c+d x)}}\right )}{d}-\frac {2 \sqrt {a+i a \tan (c+d x)}}{d \sqrt {\tan (c+d x)}}\right )\) |
Input:
Int[Cot[c + d*x]^(3/2)*Sqrt[a + I*a*Tan[c + d*x]],x]
Output:
Sqrt[Cot[c + d*x]]*Sqrt[Tan[c + d*x]]*(((1 + I)*Sqrt[a]*ArcTanh[((1 + I)*S qrt[a]*Sqrt[Tan[c + d*x]])/Sqrt[a + I*a*Tan[c + d*x]]])/d - (2*Sqrt[a + I* a*Tan[c + d*x]])/(d*Sqrt[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[(-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]
Int[(cot[(a_.) + (b_.)*(x_)]*(c_.))^(m_.)*(u_), x_Symbol] :> Simp[(c*Cot[a + b*x])^m*(c*Tan[a + b*x])^m Int[ActivateTrig[u]/(c*Tan[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownTangentIntegrandQ[u, x]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 528 vs. \(2 (84 ) = 168\).
Time = 1.97 (sec) , antiderivative size = 529, normalized size of antiderivative = 5.19
method | result | size |
default | \(\frac {\csc \left (d x +c \right ) \cot \left (d x +c \right )^{\frac {3}{2}} \left (-\cos \left (d x +c \right )+1\right ) \sqrt {-\left (-1-i \tan \left (d x +c \right )\right ) a}\, \left (2 i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )+2 i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )+i \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}\, \sin \left (d x +c \right )-\sin \left (d x +c \right )-\cos \left (d x +c \right )+1}{\sqrt {-2 \csc \left (d x +c \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, \sin \left (d x +c \right )+\sin \left (d x +c \right )+\cos \left (d x +c \right )-1}\right )-\sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \ln \left (-\frac {\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}\, \sin \left (d x +c \right )+\sin \left (d x +c \right )+\cos \left (d x +c \right )-1}{\sqrt {-2 \csc \left (d x +c \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {2}\, \sin \left (d x +c \right )-\sin \left (d x +c \right )-\cos \left (d x +c \right )+1}\right )-2 \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}+1\right )-2 \sqrt {2}\, \sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \arctan \left (\sqrt {\cot \left (d x +c \right )-\csc \left (d x +c \right )}\, \sqrt {2}-1\right )-4 \csc \left (d x +c \right )+4 \cot \left (d x +c \right )+4 i\right )}{d \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )+i\right ) \left (\csc \left (d x +c \right )^{2} \left (-\cos \left (d x +c \right )+1\right )^{2}-1\right )}\) | \(529\) |
Input:
int(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
1/d*csc(d*x+c)*cot(d*x+c)^(3/2)*(-cos(d*x+c)+1)*(-(-1-I*tan(d*x+c))*a)^(1/ 2)*(2*I*2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d*x+c)-csc(d*x+c ))^(1/2)*2^(1/2)+1)+2*I*2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot( d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)-1)+I*2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2 )*ln(-((cot(d*x+c)-csc(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)-sin(d*x+c)-cos(d*x +c)+1)/((-2*csc(d*x+c)*sin(1/2*d*x+1/2*c)^2)^(1/2)*2^(1/2)*sin(d*x+c)+sin( d*x+c)+cos(d*x+c)-1))-2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*ln(-((cot(d*x+ c)-csc(d*x+c))^(1/2)*2^(1/2)*sin(d*x+c)+sin(d*x+c)+cos(d*x+c)-1)/((-2*csc( d*x+c)*sin(1/2*d*x+1/2*c)^2)^(1/2)*2^(1/2)*sin(d*x+c)-sin(d*x+c)-cos(d*x+c )+1))-2*2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d*x+c)-csc(d*x+c ))^(1/2)*2^(1/2)+1)-2*2^(1/2)*(cot(d*x+c)-csc(d*x+c))^(1/2)*arctan((cot(d* x+c)-csc(d*x+c))^(1/2)*2^(1/2)-1)-4*csc(d*x+c)+4*cot(d*x+c)+4*I)/(cot(d*x+ c)-csc(d*x+c)+I)/(csc(d*x+c)^2*(-cos(d*x+c)+1)^2-1)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (78) = 156\).
Time = 0.13 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.76 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=-\frac {8 \, \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} e^{\left (i \, d x + i \, c\right )} - d \sqrt {\frac {8 i \, a}{d^{2}}} \log \left ({\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} + 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + d \sqrt {\frac {8 i \, a}{d^{2}}} \log \left (-{\left (\sqrt {2} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {8 i \, a}{d^{2}}} - 4 i \, a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{4 \, d} \] Input:
integrate(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="fricas")
Output:
-1/4*(8*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^(I*d*x + I*c) - d*sqrt(8*I*a/d^2)*lo g((sqrt(2)*(d*e^(2*I*d*x + 2*I*c) - d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*s qrt((I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) - 1))*sqrt(8*I*a/d^2) + 4*I*a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + d*sqrt(8*I*a/d^2)*log(-(sqrt (2)*(d*e^(2*I*d*x + 2*I*c) - d)*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))*sqrt(8*I*a/d^2) - 4*I* a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)))/d
\[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \] Input:
integrate(cot(d*x+c)**(3/2)*(a+I*a*tan(d*x+c))**(1/2),x)
Output:
Integral(sqrt(I*a*(tan(c + d*x) - I))*cot(c + d*x)**(3/2), x)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (78) = 156\).
Time = 0.43 (sec) , antiderivative size = 540, normalized size of antiderivative = 5.29 \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx =\text {Too large to display} \] Input:
integrate(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2),x, algorithm="maxima")
Output:
1/2*((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1 /4)*sqrt(a)*((2*I - 2)*arctan2(2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d* x + 2*c) - 1)) + 2*sin(d*x + c), 2*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^ 2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2* d*x + 2*c) - 1)) + 2*cos(d*x + c)) + (I + 1)*log(4*cos(d*x + c)^2 + 4*sin( d*x + c)^2 + 4*sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)*(cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))^2) + 8*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*(cos(d*x + c)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)) + sin(d*x + c) *sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1))))) - 4*(((I + 1) *cos(d*x + c) + (I - 1)*sin(d*x + c))*cos(1/2*arctan2(sin(2*d*x + 2*c), co s(2*d*x + 2*c) - 1)) + (-(I - 1)*cos(d*x + c) + (I + 1)*sin(d*x + c))*sin( 1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) - 1)))*sqrt(a))/((cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 - 2*cos(2*d*x + 2*c) + 1)^(1/4)*d)
Exception generated. \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(cot(d*x+c)^(3/2)*(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
Timed out. \[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\int {\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}} \,d x \] Input:
int(cot(c + d*x)^(3/2)*(a + a*tan(c + d*x)*1i)^(1/2),x)
Output:
int(cot(c + d*x)^(3/2)*(a + a*tan(c + d*x)*1i)^(1/2), x)
\[ \int \cot ^{\frac {3}{2}}(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx=\sqrt {a}\, \left (\int \sqrt {\tan \left (d x +c \right ) i +1}\, \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )d x \right ) \] Input:
int(cot(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^(1/2),x)
Output:
sqrt(a)*int(sqrt(tan(c + d*x)*i + 1)*sqrt(cot(c + d*x))*cot(c + d*x),x)