Optimal. Leaf size=201 \[ -\frac{\sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-18 a^2+2 (6 a+i) b x-10 i a+7\right )}{24 b^4}-\frac{\left (8 i a^3-12 a^2-12 i a+3\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}+\frac{\left (-8 a^3-12 i a^2+12 a+3 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}+\frac{x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.192415, antiderivative size = 201, 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, 100, 147, 50, 53, 619, 215} \[ -\frac{\sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2} \left (-18 a^2+2 (6 a+i) b x-10 i a+7\right )}{24 b^4}-\frac{\left (8 i a^3-12 a^2-12 i a+3\right ) \sqrt{-i a-i b x+1} \sqrt{i a+i b x+1}}{8 b^4}+\frac{\left (-8 a^3-12 i a^2+12 a+3 i\right ) \sinh ^{-1}(a+b x)}{8 b^4}+\frac{x^2 \sqrt{-i a-i b x+1} (i a+i b x+1)^{3/2}}{4 b^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5095
Rule 100
Rule 147
Rule 50
Rule 53
Rule 619
Rule 215
Rubi steps
\begin{align*} \int e^{i \tan ^{-1}(a+b x)} x^3 \, dx &=\int \frac{x^3 \sqrt{1+i a+i b x}}{\sqrt{1-i a-i b x}} \, dx\\ &=\frac{x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}+\frac{\int \frac{x \sqrt{1+i a+i b x} \left (-2 \left (1+a^2\right )-(i+6 a) b x\right )}{\sqrt{1-i a-i b x}} \, dx}{4 b^2}\\ &=\frac{x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac{\sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac{\left (3 i+12 a-12 i a^2-8 a^3\right ) \int \frac{\sqrt{1+i a+i b x}}{\sqrt{1-i a-i b x}} \, dx}{8 b^3}\\ &=-\frac{\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}+\frac{x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac{\sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac{\left (3 i+12 a-12 i a^2-8 a^3\right ) \int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx}{8 b^3}\\ &=-\frac{\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}+\frac{x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac{\sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac{\left (3 i+12 a-12 i a^2-8 a^3\right ) \int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx}{8 b^3}\\ &=-\frac{\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}+\frac{x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac{\sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac{\left (3 i+12 a-12 i a^2-8 a^3\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 )}{16 b^5}\\ &=-\frac{\left (3-12 i a-12 a^2+8 i a^3\right ) \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}}{8 b^4}+\frac{x^2 \sqrt{1-i a-i b x} (1+i a+i b x)^{3/2}}{4 b^2}-\frac{\sqrt{1-i a-i b x} (1+i a+i b x)^{3/2} \left (7-10 i a-18 a^2+2 (i+6 a) b x\right )}{24 b^4}+\frac{\left (3 i+12 a-12 i a^2-8 a^3\right ) \sinh ^{-1}(a+b x)}{8 b^4}\\ \end{align*}
Mathematica [A] time = 0.272815, size = 176, normalized size = 0.88 \[ \frac{\sqrt{b} \sqrt{a^2+2 a b x+b^2 x^2+1} \left (a^2 (44+6 i b x)-6 i a^3+a \left (-6 i b^2 x^2-20 b x+39 i\right )+6 i b^3 x^3+8 b^2 x^2-9 i b x-16\right )-6 \sqrt [4]{-1} \left (8 a^3+12 i a^2-12 a-3 i\right ) \sqrt{-i b} \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{24 b^{9/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.117, size = 465, normalized size = 2.3 \begin{align*}{\frac{{\frac{3\,i}{8}}}{{b}^{3}}\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{{\frac{i}{4}}{a}^{2}x}{{b}^{3}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{\frac{i}{4}}a{x}^{2}}{{b}^{2}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{\frac{3\,i}{8}}x}{{b}^{3}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{{\frac{13\,i}{8}}a}{{b}^{4}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{\frac{i}{4}}{a}^{3}}{{b}^{4}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{\frac{3\,i}{2}}{a}^{2}}{{b}^{3}}\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{{\frac{i}{4}}{x}^{3}}{b}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{{x}^{2}}{3\,{b}^{2}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{5\,ax}{6\,{b}^{3}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}+{\frac{11\,{a}^{2}}{6\,{b}^{4}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}}-{\frac{{a}^{3}}{{b}^{3}}\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{3\,a}{2\,{b}^{3}}\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{2}{3\,{b}^{4}}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.71922, size = 386, normalized size = 1.92 \begin{align*} \frac{-45 i \, a^{4} + 224 \, a^{3} + 192 i \, a^{2} +{\left (192 \, a^{3} + 288 i \, a^{2} - 288 \, a - 72 i\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) +{\left (48 i \, b^{3} x^{3} - 16 \,{\left (3 i \, a - 4\right )} b^{2} x^{2} - 48 i \, a^{3} +{\left (48 i \, a^{2} - 160 \, a - 72 i\right )} b x + 352 \, a^{2} + 312 i \, a - 128\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 72 \, a}{192 \, b^{4}} \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^{3} \left (i a + i b x + 1\right )}{\sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17295, size = 220, normalized size = 1.09 \begin{align*} \frac{1}{24} \, \sqrt{{\left (b x + a\right )}^{2} + 1}{\left ({\left (2 \,{\left (\frac{3 \, i x}{b} - \frac{3 \, a b^{5} i - 4 \, b^{5}}{b^{7}}\right )} x + \frac{6 \, a^{2} b^{4} i - 20 \, a b^{4} - 9 \, b^{4} i}{b^{7}}\right )} x - \frac{6 \, a^{3} b^{3} i - 44 \, a^{2} b^{3} - 39 \, a b^{3} i + 16 \, b^{3}}{b^{7}}\right )} + \frac{{\left (8 \, a^{3} + 12 \, a^{2} i - 12 \, a - 3 \, i\right )} \log \left (-a b -{\left (x{\left | b \right |} - \sqrt{{\left (b x + a\right )}^{2} + 1}\right )}{\left | b \right |}\right )}{8 \, b^{3}{\left | b \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]