Optimal. Leaf size=86 \[ \frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{a (-a x+i) \sqrt {a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.08, antiderivative size = 86, 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} \[ \frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1}}{a (-a x+i) \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 43
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{-3 i \tan ^{-1}(a x)}}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \frac {1-i a x}{(1+i a x)^2} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {\sqrt {1+a^2 x^2} \int \left (-\frac {2}{(-i+a x)^2}+\frac {i}{-i+a x}\right ) \, dx}{\sqrt {c+a^2 c x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2}}{a (i-a x) \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \log (i-a x)}{a \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 60, normalized size = 0.70 \[ \frac {\sqrt {a^2 x^2+1} \left (\frac {i \log (-a x+i)}{a}-\frac {2}{a (-a x+i)}\right )}{\sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.62, size = 360, normalized size = 4.19 \[ \frac {{\left (-i \, a^{3} c x^{3} - a^{2} c x^{2} - i \, a c x - c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, a^{3} x^{3} - 8 i \, a^{2} x^{2} + 8 \, a x - 8 i}\right ) + {\left (i \, a^{3} c x^{3} + a^{2} c x^{2} + i \, a c x + c\right )} \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, a^{3} x^{3} - 8 i \, a^{2} x^{2} + 8 \, a x - 8 i}\right ) - 4 i \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} x}{2 \, a^{3} c x^{3} - 2 i \, a^{2} c x^{2} + 2 \, a c x - 2 i \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{\sqrt {a^{2} c x^{2} + c} {\left (i \, a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.16, size = 66, normalized size = 0.77 \[ \frac {\left (-i \ln \left (-a x +i\right ) x a -\ln \left (-a x +i\right )-2\right ) \sqrt {c \left (a^{2} x^{2}+1\right )}}{\sqrt {a^{2} x^{2}+1}\, c a \left (-a x +i\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 35, normalized size = 0.41 \[ \frac {i \, \log \left (i \, a x + 1\right )}{a \sqrt {c}} + \frac {2}{a^{2} \sqrt {c} x - i \, a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a^2\,x^2+1\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}\,{\left (1+a\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ i \left (\int \frac {\sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} \sqrt {a^{2} c x^{2} + c} - 3 i a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 3 a x \sqrt {a^{2} c x^{2} + c} + i \sqrt {a^{2} c x^{2} + c}}\, dx + \int \frac {a^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{a^{3} x^{3} \sqrt {a^{2} c x^{2} + c} - 3 i a^{2} x^{2} \sqrt {a^{2} c x^{2} + c} - 3 a x \sqrt {a^{2} c x^{2} + c} + i \sqrt {a^{2} c x^{2} + c}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________