3.328 \(\int \frac {e^{5 i \tan ^{-1}(a x)}}{(c+a^2 c x^2)^{3/2}} \, dx\)

Optimal. Leaf size=95 \[ -\frac {i \sqrt {a^2 x^2+1}}{2 a c (a x+i)^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{3 a c (a x+i)^3 \sqrt {a^2 c x^2+c}} \]

[Out]

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

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Rubi [A]  time = 0.08, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {5076, 5073, 43} \[ -\frac {i \sqrt {a^2 x^2+1}}{2 a c (a x+i)^2 \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{3 a c (a x+i)^3 \sqrt {a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Int[E^((5*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]

[Out]

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 5073

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

Rule 5076

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 56, normalized size = 0.59 \[ -\frac {i (3 a x-i) \sqrt {a^2 x^2+1}}{6 a c (a x+i)^3 \sqrt {a^2 c x^2+c}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^((5*I)*ArcTan[a*x])/(c + a^2*c*x^2)^(3/2),x]

[Out]

((-1/6*I)*(-I + 3*a*x)*Sqrt[1 + a^2*x^2])/(a*c*(I + a*x)^3*Sqrt[c + a^2*c*x^2])

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fricas [A]  time = 0.49, size = 101, normalized size = 1.06 \[ \frac {\sqrt {a^{2} c x^{2} + c} {\left (i \, a^{2} x^{3} - 3 \, a x^{2} - 6 i \, x\right )} \sqrt {a^{2} x^{2} + 1}}{6 \, a^{5} c^{2} x^{5} + 18 i \, a^{4} c^{2} x^{4} - 12 \, a^{3} c^{2} x^{3} + 12 i \, a^{2} c^{2} x^{2} - 18 \, a c^{2} x - 6 i \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="fricas")

[Out]

sqrt(a^2*c*x^2 + c)*(I*a^2*x^3 - 3*a*x^2 - 6*I*x)*sqrt(a^2*x^2 + 1)/(6*a^5*c^2*x^5 + 18*I*a^4*c^2*x^4 - 12*a^3
*c^2*x^3 + 12*I*a^2*c^2*x^2 - 18*a*c^2*x - 6*I*c^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a x + 1\right )}^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.16, size = 48, normalized size = 0.51 \[ -\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (3 i a x +1\right )}{6 \sqrt {a^{2} x^{2}+1}\, c^{2} a \left (a x +i\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6/(a^2*x^2+1)^(1/2)*(c*(a^2*x^2+1))^(1/2)*(3*I*a*x+1)/c^2/a/(I+a*x)^3

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (i \, a x + 1\right )}^{5}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} + 1\right )}^{\frac {5}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)^5/(a^2*x^2+1)^(5/2)/(a^2*c*x^2+c)^(3/2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.62, size = 48, normalized size = 0.51 \[ -\frac {\sqrt {c\,\left (a^2\,x^2+1\right )}\,\left (3\,a\,x-\mathrm {i}\right )}{6\,a\,c^2\,\sqrt {a^2\,x^2+1}\,{\left (-1+a\,x\,1{}\mathrm {i}\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x*1i + 1)^5/((c + a^2*c*x^2)^(3/2)*(a^2*x^2 + 1)^(5/2)),x)

[Out]

-((c*(a^2*x^2 + 1))^(1/2)*(3*a*x - 1i))/(6*a*c^2*(a^2*x^2 + 1)^(1/2)*(a*x*1i - 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ i \left (\int \left (- \frac {i}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {5 a x}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {10 a^{3} x^{3}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx + \int \frac {a^{5} x^{5}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \frac {10 i a^{2} x^{2}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\, dx + \int \left (- \frac {5 i a^{4} x^{4}}{a^{6} c x^{6} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{4} c x^{4} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + 3 a^{2} c x^{2} \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c} + c \sqrt {a^{2} x^{2} + 1} \sqrt {a^{2} c x^{2} + c}}\right )\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+I*a*x)**5/(a**2*x**2+1)**(5/2)/(a**2*c*x**2+c)**(3/2),x)

[Out]

I*(Integral(-I/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)*sqrt
(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*
c*x**2 + c)), x) + Integral(5*a*x/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4*sqrt(
a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x
**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(-10*a**3*x**3/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2
+ c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*
c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(a**5*x**5/(a**6*c*x**6*sqrt(a**2*x**
2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a*
*2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)), x) + Integral(10*I*a**2*x**
2/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2
+ c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c)),
x) + Integral(-5*I*a**4*x**4/(a**6*c*x**6*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**4*c*x**4*sqrt(a**2
*x**2 + 1)*sqrt(a**2*c*x**2 + c) + 3*a**2*c*x**2*sqrt(a**2*x**2 + 1)*sqrt(a**2*c*x**2 + c) + c*sqrt(a**2*x**2
+ 1)*sqrt(a**2*c*x**2 + c)), x))

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