Optimal. Leaf size=43 \[ -\frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 43, 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 = {5076, 5073, 31} \[ -\frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 5073
Rule 5076
Rubi steps
\begin {align*} \int \frac {e^{-i \tan ^{-1}(a x)}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {e^{-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}{1+i a x} \, dx}{\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 = 0.02, size = 43, normalized size = 1.00 \[ -\frac {i \sqrt {a^2 x^2+1} \log (-a x+i)}{a \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.54, size = 255, normalized size = 5.93 \[ \frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, a^{3} x^{3} - 8 i \, a^{2} x^{2} + 8 \, a x - 8 i}\right ) - \frac {1}{2} i \, \sqrt {\frac {1}{a^{2} c}} \log \left (\frac {{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1} + {\left (-i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt {\frac {1}{a^{2} c}}}{8 \, a^{3} x^{3} - 8 i \, a^{2} x^{2} + 8 \, a x - 8 i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 42, normalized size = 0.98 \[ -\frac {i \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}\, c a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 15, normalized size = 0.35 \[ -\frac {i \, \log \left (i \, a x + 1\right )}{a \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a^2\,x^2+1}}{\sqrt {c\,a^2\,x^2+c}\,\left (1+a\,x\,1{}\mathrm {i}\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - i \int \frac {\sqrt {a^{2} x^{2} + 1}}{a x \sqrt {a^{2} c x^{2} + c} - i \sqrt {a^{2} c x^{2} + c}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________