Integrand size = 24, antiderivative size = 93 \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\sqrt {\arctan (a x)}}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^2 c \sqrt {c+a^2 c x^2}} \] Output:
-arctan(a*x)^(1/2)/a^2/c/(a^2*c*x^2+c)^(1/2)+1/2*2^(1/2)*Pi^(1/2)*(a^2*x^2 +1)^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a^2/c/(a^2*c*x^2+c) ^(1/2)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.30 \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {-4 \arctan (a x)-i \sqrt {1+a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )+i \sqrt {1+a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )}{4 a^2 c \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \] Input:
Integrate[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2),x]
Output:
(-4*ArcTan[a*x] - I*Sqrt[1 + a^2*x^2]*Sqrt[(-I)*ArcTan[a*x]]*Gamma[1/2, (- I)*ArcTan[a*x]] + I*Sqrt[1 + a^2*x^2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, I*Arc Tan[a*x]])/(4*a^2*c*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
Time = 0.58 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5465, 5440, 5439, 3042, 3785, 3833}
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.
\(\displaystyle \int \frac {x \sqrt {\arctan (a x)}}{\left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle \frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \sqrt {\arctan (a x)}}dx}{2 a}-\frac {\sqrt {\arctan (a x)}}{a^2 c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5440 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}dx}{2 a c \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{a^2 c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 5439 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{a^2 c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )}{\sqrt {\arctan (a x)}}d\arctan (a x)}{2 a^2 c \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{a^2 c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 3785 |
\(\displaystyle \frac {\sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{a^2 c \sqrt {a^2 c x^2+c}}\) |
\(\Big \downarrow \) 3833 |
\(\displaystyle \frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\arctan (a x)}}{a^2 c \sqrt {a^2 c x^2+c}}\) |
Input:
Int[(x*Sqrt[ArcTan[a*x]])/(c + a^2*c*x^2)^(3/2),x]
Output:
-(Sqrt[ArcTan[a*x]]/(a^2*c*Sqrt[c + a^2*c*x^2])) + (Sqrt[Pi/2]*Sqrt[1 + a^ 2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(a^2*c*Sqrt[c + a^2*c*x^2])
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[2/d Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^q/c Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]) Int[(1 + c^2*x^2)^q*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*(q + 1), 0] && !(IntegerQ[q] || GtQ[d, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
\[\int \frac {x \sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}d x\]
Input:
int(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x)
Output:
int(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x)
Exception generated. \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x \sqrt {\operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(x*atan(a*x)**(1/2)/(a**2*c*x**2+c)**(3/2),x)
Output:
Integral(x*sqrt(atan(a*x))/(c*(a**2*x**2 + 1))**(3/2), x)
Exception generated. \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arctan \left (a x\right )}\right ) - \left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\arctan \left (a x\right )}\right ) + 4 \, \sqrt {\arctan \left (a x\right )} e^{\left (i \, \arctan \left (a x\right )\right )} + 4 \, \sqrt {\arctan \left (a x\right )} e^{\left (-i \, \arctan \left (a x\right )\right )}}{8 \, a^{2} c^{\frac {3}{2}}} \] Input:
integrate(x*arctan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")
Output:
-1/8*((I + 1)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arctan(a*x)) ) - (I - 1)*sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arctan(a*x))) + 4*sqrt(arctan(a*x))*e^(I*arctan(a*x)) + 4*sqrt(arctan(a*x))*e^(-I*arcta n(a*x)))/(a^2*c^(3/2))
Timed out. \[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {x\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \] Input:
int((x*atan(a*x)^(1/2))/(c + a^2*c*x^2)^(3/2),x)
Output:
int((x*atan(a*x)^(1/2))/(c + a^2*c*x^2)^(3/2), x)
\[ \int \frac {x \sqrt {\arctan (a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (-2 \sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}+\left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}}{\mathit {atan} \left (a x \right ) a^{4} x^{4}+2 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a^{3} x^{2}+\left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \sqrt {\mathit {atan} \left (a x \right )}}{\mathit {atan} \left (a x \right ) a^{4} x^{4}+2 \mathit {atan} \left (a x \right ) a^{2} x^{2}+\mathit {atan} \left (a x \right )}d x \right ) a \right )}{2 a^{2} c^{2} \left (a^{2} x^{2}+1\right )} \] Input:
int(x*atan(a*x)^(1/2)/(a^2*c*x^2+c)^(3/2),x)
Output:
(sqrt(c)*( - 2*sqrt(a**2*x**2 + 1)*sqrt(atan(a*x)) + int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x)))/(atan(a*x)*a**4*x**4 + 2*atan(a*x)*a**2*x**2 + atan(a *x)),x)*a**3*x**2 + int((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x)))/(atan(a*x)*a **4*x**4 + 2*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a))/(2*a**2*c**2*(a**2*x* *2 + 1))