3.329 \(\int \frac {e^{4 i \tan ^{-1}(a x)}}{(c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=69 \[ \frac {i c (1+i a x)^5}{15 a \left (a^2 c x^2+c\right )^{5/2}}-\frac {i c (1+i a x)^4}{3 a \left (a^2 c x^2+c\right )^{5/2}} \]

[Out]

-1/3*I*c*(1+I*a*x)^4/a/(a^2*c*x^2+c)^(5/2)+1/15*I*c*(1+I*a*x)^5/a/(a^2*c*x^2+c)^(5/2)

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Rubi [A]  time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5075, 659, 651} \[ \frac {i c (1+i a x)^5}{15 a \left (a^2 c x^2+c\right )^{5/2}}-\frac {i c (1+i a x)^4}{3 a \left (a^2 c x^2+c\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[E^((4*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]

[Out]

((-I/3)*c*(1 + I*a*x)^4)/(a*(c + a^2*c*x^2)^(5/2)) + ((I/15)*c*(1 + I*a*x)^5)/(a*(c + a^2*c*x^2)^(5/2))

Rule 651

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1))
/(2*c*d*(p + 1)), x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && EqQ[m + 2*p
+ 2, 0]

Rule 659

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[(e*(d + e*x)^m*(a + c*x^2)^(p + 1)
)/(2*c*d*(m + p + 1)), x] + Dist[Simplify[m + 2*p + 2]/(2*d*(m + p + 1)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p,
x], x] /; FreeQ[{a, c, d, e, m, p}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && ILtQ[Simplify[m + 2*p + 2
], 0]

Rule 5075

Int[E^(ArcTan[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/c^((I*n)/2), Int[(c + d*x^2)^(
p + (I*n)/2)/(1 + I*a*x)^(I*n), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[d, a^2*c] &&  !(IntegerQ[p] || GtQ[c,
0]) && ILtQ[(I*n)/2, 0]

Rubi steps

\begin {align*} \int \frac {e^{4 i \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=c^2 \int \frac {(1+i a x)^4}{\left (c+a^2 c x^2\right )^{7/2}} \, dx\\ &=-\frac {i c (1+i a x)^4}{3 a \left (c+a^2 c x^2\right )^{5/2}}-\frac {1}{3} c^2 \int \frac {(1+i a x)^5}{\left (c+a^2 c x^2\right )^{7/2}} \, dx\\ &=-\frac {i c (1+i a x)^4}{3 a \left (c+a^2 c x^2\right )^{5/2}}+\frac {i c (1+i a x)^5}{15 a \left (c+a^2 c x^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 77, normalized size = 1.12 \[ \frac {(1+i a x)^{3/2} (a x+4 i) \sqrt {a^2 x^2+1}}{15 a c \sqrt {1-i a x} (a x+i)^2 \sqrt {a^2 c x^2+c}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^((4*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]

[Out]

((1 + I*a*x)^(3/2)*(4*I + a*x)*Sqrt[1 + a^2*x^2])/(15*a*c*Sqrt[1 - I*a*x]*(I + a*x)^2*Sqrt[c + a^2*c*x^2])

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fricas [A]  time = 0.52, size = 67, normalized size = 0.97 \[ -\frac {\sqrt {a^{2} c x^{2} + c} {\left (a^{2} x^{2} + 3 i \, a x + 4\right )}}{15 \, a^{4} c^{2} x^{3} + 45 i \, a^{3} c^{2} x^{2} - 45 \, a^{2} c^{2} x - 15 i \, a c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

-sqrt(a^2*c*x^2 + c)*(a^2*x^2 + 3*I*a*x + 4)/(15*a^4*c^2*x^3 + 45*I*a^3*c^2*x^2 - 45*a^2*c^2*x - 15*I*a*c^2)

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giac [B]  time = 0.16, size = 135, normalized size = 1.96 \[ -\frac {2 \, {\left (5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{2} c i - 15 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{3} \sqrt {c} + c^{2} i + 5 \, {\left (\sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )} c^{\frac {3}{2}}\right )}}{15 \, {\left (\sqrt {c} i + \sqrt {a^{2} c} x - \sqrt {a^{2} c x^{2} + c}\right )}^{5} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

-2/15*(5*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 + c))^2*c*i - 15*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 + c))^3*sqrt(c) + c^
2*i + 5*(sqrt(a^2*c)*x - sqrt(a^2*c*x^2 + c))*c^(3/2))/((sqrt(c)*i + sqrt(a^2*c)*x - sqrt(a^2*c*x^2 + c))^5*a*
c)

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maple [B]  time = 0.19, size = 940, normalized size = 13.62 \[ \frac {x}{c \sqrt {a^{2} c \,x^{2}+c}}-\frac {2 \left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {1}{5 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {3 a^{2} \left (-\frac {1}{3 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {2 \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c +2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{5 \sqrt {-a^{2}}}\right )}{a^{3}}-\frac {2 \left (i \sqrt {-a^{2}}+a \right ) \left (-\frac {1}{3 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}-\frac {2 \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c +2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {\left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c +2 c \sqrt {-a^{2}}\, \left (x -\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}}+\frac {2 \left (i \sqrt {-a^{2}}-a \right ) \left (\frac {1}{5 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {3 a^{2} \left (\frac {1}{3 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {2 \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c -2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{5 \sqrt {-a^{2}}}\right )}{a^{3}}-\frac {2 \left (i \sqrt {-a^{2}}-a \right ) \left (\frac {1}{3 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}+\frac {2 \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right ) a^{2} c -2 c \sqrt {-a^{2}}}{3 c^{2} \sqrt {-a^{2}}\, \sqrt {\left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )^{2} a^{2} c -2 c \sqrt {-a^{2}}\, \left (x +\frac {\sqrt {-a^{2}}}{a^{2}}\right )}}\right )}{a \sqrt {-a^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+I*a*x)^4/(a^2*x^2+1)^2/(a^2*c*x^2+c)^(3/2),x)

[Out]

x/c/(a^2*c*x^2+c)^(1/2)-2/a^3*(I*(-a^2)^(1/2)+a)*(-1/5/c/(-a^2)^(1/2)/(x-(-a^2)^(1/2)/a^2)^2/((x-(-a^2)^(1/2)/
a^2)^2*a^2*c+2*c*(-a^2)^(1/2)*(x-(-a^2)^(1/2)/a^2))^(1/2)-3/5*a^2/(-a^2)^(1/2)*(-1/3/c/(-a^2)^(1/2)/(x-(-a^2)^
(1/2)/a^2)/((x-(-a^2)^(1/2)/a^2)^2*a^2*c+2*c*(-a^2)^(1/2)*(x-(-a^2)^(1/2)/a^2))^(1/2)-1/3/c^2/(-a^2)^(1/2)*(2*
(x-(-a^2)^(1/2)/a^2)*a^2*c+2*c*(-a^2)^(1/2))/((x-(-a^2)^(1/2)/a^2)^2*a^2*c+2*c*(-a^2)^(1/2)*(x-(-a^2)^(1/2)/a^
2))^(1/2)))-2/a*(I*(-a^2)^(1/2)+a)/(-a^2)^(1/2)*(-1/3/c/(-a^2)^(1/2)/(x-(-a^2)^(1/2)/a^2)/((x-(-a^2)^(1/2)/a^2
)^2*a^2*c+2*c*(-a^2)^(1/2)*(x-(-a^2)^(1/2)/a^2))^(1/2)-1/3/c^2/(-a^2)^(1/2)*(2*(x-(-a^2)^(1/2)/a^2)*a^2*c+2*c*
(-a^2)^(1/2))/((x-(-a^2)^(1/2)/a^2)^2*a^2*c+2*c*(-a^2)^(1/2)*(x-(-a^2)^(1/2)/a^2))^(1/2))+2/a^3*(I*(-a^2)^(1/2
)-a)*(1/5/c/(-a^2)^(1/2)/(x+(-a^2)^(1/2)/a^2)^2/((x+(-a^2)^(1/2)/a^2)^2*a^2*c-2*c*(-a^2)^(1/2)*(x+(-a^2)^(1/2)
/a^2))^(1/2)+3/5*a^2/(-a^2)^(1/2)*(1/3/c/(-a^2)^(1/2)/(x+(-a^2)^(1/2)/a^2)/((x+(-a^2)^(1/2)/a^2)^2*a^2*c-2*c*(
-a^2)^(1/2)*(x+(-a^2)^(1/2)/a^2))^(1/2)+1/3/c^2/(-a^2)^(1/2)*(2*(x+(-a^2)^(1/2)/a^2)*a^2*c-2*c*(-a^2)^(1/2))/(
(x+(-a^2)^(1/2)/a^2)^2*a^2*c-2*c*(-a^2)^(1/2)*(x+(-a^2)^(1/2)/a^2))^(1/2)))-2/a*(I*(-a^2)^(1/2)-a)/(-a^2)^(1/2
)*(1/3/c/(-a^2)^(1/2)/(x+(-a^2)^(1/2)/a^2)/((x+(-a^2)^(1/2)/a^2)^2*a^2*c-2*c*(-a^2)^(1/2)*(x+(-a^2)^(1/2)/a^2)
)^(1/2)+1/3/c^2/(-a^2)^(1/2)*(2*(x+(-a^2)^(1/2)/a^2)*a^2*c-2*c*(-a^2)^(1/2))/((x+(-a^2)^(1/2)/a^2)^2*a^2*c-2*c
*(-a^2)^(1/2)*(x+(-a^2)^(1/2)/a^2))^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a x + 1\right )}^{4}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^4/(a^2*x^2+1)^2/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

integrate((I*a*x + 1)^4/((a^2*c*x^2 + c)^(3/2)*(a^2*x^2 + 1)^2), x)

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mupad [B]  time = 0.98, size = 46, normalized size = 0.67 \[ \frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,\left (a^2\,x^2\,1{}\mathrm {i}-3\,a\,x+4{}\mathrm {i}\right )}{15\,a\,c^2\,{\left (-1+a\,x\,1{}\mathrm {i}\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x*1i + 1)^4/((c + a^2*c*x^2)^(3/2)*(a^2*x^2 + 1)^2),x)

[Out]

((c*(a^2*x^2 + 1))^(1/2)*(a^2*x^2*1i - 3*a*x + 4i))/(15*a*c^2*(a*x*1i - 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a x - i\right )^{4}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a^{2} x^{2} + 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)**4/(a**2*x**2+1)**2/(a**2*c*x**2+c)**(3/2),x)

[Out]

Integral((a*x - I)**4/((c*(a**2*x**2 + 1))**(3/2)*(a**2*x**2 + 1)**2), x)

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