Optimal. Leaf size=171 \[ \frac{\left (-2 i a^2-2 a+i\right ) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^3}-\frac{\left (-2 a^2+2 i a+1\right ) \sinh ^{-1}(a+b x)}{2 b^3}+\frac{x \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{3 b^2}+\frac{(-4 a+i) \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.125868, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5095, 90, 80, 50, 53, 619, 215} \[ \frac{\left (-2 i a^2-2 a+i\right ) \sqrt{i a+i b x+1} \sqrt{-i a-i b x+1}}{2 b^3}-\frac{\left (-2 a^2+2 i a+1\right ) \sinh ^{-1}(a+b x)}{2 b^3}+\frac{x \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{3 b^2}+\frac{(-4 a+i) \sqrt{i a+i b x+1} (-i a-i b x+1)^{3/2}}{6 b^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5095
Rule 90
Rule 80
Rule 50
Rule 53
Rule 619
Rule 215
Rubi steps
\begin{align*} \int e^{-i \tan ^{-1}(a+b x)} x^2 \, dx &=\int \frac{x^2 \sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx\\ &=\frac{x (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{3 b^2}+\frac{\int \frac{\sqrt{1-i a-i b x} \left (-1-a^2+(i-4 a) b x\right )}{\sqrt{1+i a+i b x}} \, dx}{3 b^2}\\ &=\frac{(i-4 a) (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{6 b^3}+\frac{x (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{3 b^2}-\frac{\left (1+2 i a-2 a^2\right ) \int \frac{\sqrt{1-i a-i b x}}{\sqrt{1+i a+i b x}} \, dx}{2 b^2}\\ &=-\frac{\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{(i-4 a) (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{6 b^3}+\frac{x (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{3 b^2}-\frac{\left (1+2 i a-2 a^2\right ) \int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{2 b^2}\\ &=-\frac{\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{(i-4 a) (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{6 b^3}+\frac{x (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{3 b^2}-\frac{\left (1+2 i a-2 a^2\right ) \int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{2 b^2}\\ &=-\frac{\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{(i-4 a) (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{6 b^3}+\frac{x (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{3 b^2}-\frac{\left (1+2 i a-2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{4 b^4}\\ &=-\frac{\left (2 a-i \left (1-2 a^2\right )\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{2 b^3}+\frac{(i-4 a) (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{6 b^3}+\frac{x (1-i a-i b x)^{3/2} \sqrt{1+i a+i b x}}{3 b^2}-\frac{\left (1+2 i a-2 a^2\right ) \sinh ^{-1}(a+b x)}{2 b^3}\\ \end{align*}
Mathematica [A] time = 0.28138, size = 162, normalized size = 0.95 \[ \frac{i \sqrt{i a+i b x+1} \left (2 i a^3+7 a^2+a (8 b x+5 i)+2 i b^3 x^3-5 b^2 x^2-7 i b x+4\right )}{6 b^3 \sqrt{-i (a+b x+i)}}+\frac{\sqrt [4]{-1} \left (2 a^2-2 i a-1\right ) \sqrt{-i b} \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{b^{7/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.124, size = 605, normalized size = 3.5 \begin{align*}{\frac{-{\frac{i}{3}}}{{b}^{3}} \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) ^{{\frac{3}{2}}}}+{\frac{iax}{{b}^{2}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{i{a}^{2}}{{b}^{3}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{ia}{{b}^{2}}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+{\frac{x}{2\,{b}^{2}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{a}{2\,{b}^{3}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{1}{2\,{b}^{2}}\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{i{a}^{2}}{{b}^{3}}\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b}}+{\frac{i}{{b}^{3}}\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b}}-2\,{\frac{a}{{b}^{3}}\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b}}+{\frac{{a}^{2}}{{b}^{2}}\ln \left ({ \left ( ib+ \left ( x-{\frac{i-a}{b}} \right ){b}^{2} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{1}{{b}^{2}}\ln \left ({ \left ( ib+ \left ( x-{\frac{i-a}{b}} \right ){b}^{2} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{\frac{2\,ia}{{b}^{2}}\ln \left ({ \left ( ib+ \left ( x-{\frac{i-a}{b}} \right ){b}^{2} \right ){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{ \left ( x-{\frac{i-a}{b}} \right ) ^{2}{b}^{2}+2\,i \left ( x-{\frac{i-a}{b}} \right ) b} \right ){\frac{1}{\sqrt{{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51059, size = 217, normalized size = 1.27 \begin{align*} \frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a x}{b^{2}} + \frac{a^{2} \operatorname{arsinh}\left (b x + a\right )}{b^{3}} + \frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} x}{2 \, b^{2}} - \frac{i \, a \operatorname{arsinh}\left (b x + a\right )}{b^{3}} - \frac{i \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}}}{3 \, b^{3}} - \frac{3 \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} a}{2 \, b^{3}} - \frac{\operatorname{arsinh}\left (b x + a\right )}{2 \, b^{3}} + \frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.28399, size = 282, normalized size = 1.65 \begin{align*} \frac{-7 i \, a^{3} - 21 \, a^{2} - 12 \,{\left (2 \, a^{2} - 2 i \, a - 1\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{\left (-8 i \, b^{2} x^{2} - 4 \,{\left (-2 i \, a - 3\right )} b x - 8 i \, a^{2} - 36 \, a + 16 i\right )} + 9 i \, a}{24 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}{i a + i b x + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.12717, size = 158, normalized size = 0.92 \begin{align*} -\frac{1}{6} \, \sqrt{{\left (b x + a\right )}^{2} + 1}{\left ({\left (\frac{2 \, i x}{b} - \frac{2 \, a b^{3} i + 3 \, b^{3}}{b^{5}}\right )} x + \frac{2 \, a^{2} b^{2} i + 9 \, a b^{2} - 4 \, b^{2} i}{b^{5}}\right )} - \frac{{\left (2 \, a^{2} - 2 \, a i - 1\right )} \log \left (-a b -{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}{\left | b \right |}\right )}{2 \, b^{2}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]