Optimal. Leaf size=673 \[ \frac {2 i \sqrt {i+a} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}-\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}+\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )}{d^2}+\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{d^2}-\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}-\frac {i c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.73, antiderivative size = 673, normalized size of antiderivative = 1.00, number
of steps used = 31, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules
used = {5159, 2455, 2516, 2498, 327, 211, 2512, 266, 2463, 2441, 2440, 2438, 214}
\begin {gather*} \frac {2 i \sqrt {a+i} \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+i}}\right )}{\sqrt {b} d}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}-\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c+\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (\frac {d \left (-\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c+\sqrt {-a+i} d}\right )}{d^2}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a-i}\right )}{\sqrt {b} c-\sqrt {-a-i} d}\right )}{d^2}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log \left (-\frac {d \left (\sqrt {b} \sqrt {x}+\sqrt {-a+i}\right )}{\sqrt {b} c-\sqrt {-a+i} d}\right )}{d^2}-\frac {i c \log (-i a-i b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log (i a+i b x+1) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log (-i a-i b x+1)}{d}-\frac {i \sqrt {x} \log (i a+i b x+1)}{d}-\frac {2 i \sqrt {-a+i} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {-a+i}}\right )}{\sqrt {b} d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 214
Rule 266
Rule 327
Rule 2438
Rule 2440
Rule 2441
Rule 2455
Rule 2463
Rule 2498
Rule 2512
Rule 2516
Rule 5159
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a+b x)}{c+d \sqrt {x}} \, dx &=\frac {1}{2} i \int \frac {\log (1-i a-i b x)}{c+d \sqrt {x}} \, dx-\frac {1}{2} i \int \frac {\log (1+i a+i b x)}{c+d \sqrt {x}} \, dx\\ &=i \text {Subst}\left (\int \frac {x \log \left (1-i a-i b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )-i \text {Subst}\left (\int \frac {x \log \left (1+i a+i b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )\\ &=i \text {Subst}\left (\int \left (\frac {\log \left (1-i a-i b x^2\right )}{d}-\frac {c \log \left (1-i a-i b x^2\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right )-i \text {Subst}\left (\int \left (\frac {\log \left (1+i a+i b x^2\right )}{d}-\frac {c \log \left (1+i a+i b x^2\right )}{d (c+d x)}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {i \text {Subst}\left (\int \log \left (1-i a-i b x^2\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {i \text {Subst}\left (\int \log \left (1+i a+i b x^2\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {(i c) \text {Subst}\left (\int \frac {\log \left (1-i a-i b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}+\frac {(i c) \text {Subst}\left (\int \frac {\log \left (1+i a+i b x^2\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}+\frac {(2 b c) \text {Subst}\left (\int \frac {x \log (c+d x)}{1-i a-i b x^2} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 b c) \text {Subst}\left (\int \frac {x \log (c+d x)}{1+i a+i b x^2} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{1-i a-i b x^2} \, dx,x,\sqrt {x}\right )}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x^2}{1+i a+i b x^2} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}+\frac {(2 b c) \text {Subst}\left (\int \left (-\frac {i \log (c+d x)}{2 \sqrt {b} \left (\sqrt {-i-a}-\sqrt {b} x\right )}+\frac {i \log (c+d x)}{2 \sqrt {b} \left (\sqrt {-i-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(2 b c) \text {Subst}\left (\int \left (\frac {i \log (c+d x)}{2 \sqrt {b} \left (\sqrt {i-a}-\sqrt {b} x\right )}-\frac {i \log (c+d x)}{2 \sqrt {b} \left (\sqrt {i-a}+\sqrt {b} x\right )}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(2 (i-a)) \text {Subst}\left (\int \frac {1}{1+i a+i b x^2} \, dx,x,\sqrt {x}\right )}{d}+\frac {(2 (i+a)) \text {Subst}\left (\int \frac {1}{1-i a-i b x^2} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}-\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}-\frac {\left (i \sqrt {b} c\right ) \text {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {-i-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (i \sqrt {b} c\right ) \text {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {i-a}-\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (i \sqrt {b} c\right ) \text {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {-i-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (i \sqrt {b} c\right ) \text {Subst}\left (\int \frac {\log (c+d x)}{\sqrt {i-a}+\sqrt {b} x} \, dx,x,\sqrt {x}\right )}{d^2}\\ &=\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}-\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}-\frac {(i c) \text {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}+\frac {(i c) \text {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} x\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}-\frac {(i c) \text {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {-i-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {-i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}+\frac {(i c) \text {Subst}\left (\int \frac {\log \left (\frac {d \left (\sqrt {i-a}+\sqrt {b} x\right )}{-\sqrt {b} c+\sqrt {i-a} d}\right )}{c+d x} \, dx,x,\sqrt {x}\right )}{d}\\ &=\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}-\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}-\frac {(i c) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {-i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}-\frac {(i c) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}+\frac {(i c) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {b} x}{-\sqrt {b} c+\sqrt {i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}+\frac {(i c) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {b} x}{\sqrt {b} c+\sqrt {i-a} d}\right )}{x} \, dx,x,c+d \sqrt {x}\right )}{d^2}\\ &=\frac {2 i \sqrt {i+a} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b} d}-\frac {2 i \sqrt {i-a} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b} d}+\frac {i c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i c \log \left (-\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}-\frac {i c \log \left (-\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )}{d^2}+\frac {i \sqrt {x} \log (1-i a-i b x)}{d}-\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1-i a-i b x)}{d^2}-\frac {i \sqrt {x} \log (1+i a+i b x)}{d}+\frac {i c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)}{d^2}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )}{d^2}+\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )}{d^2}-\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )}{d^2}-\frac {i c \text {Li}_2\left (\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )}{d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.42, size = 604, normalized size = 0.90 \begin {gather*} \frac {i \left (\frac {2 \sqrt {i+a} d \text {ArcTan}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i+a}}\right )}{\sqrt {b}}-\frac {2 \sqrt {i-a} d \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {i-a}}\right )}{\sqrt {b}}+c \log \left (\frac {d \left (\sqrt {-i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}-\sqrt {b} \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )+c \log \left (\frac {d \left (\sqrt {-i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {-i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-c \log \left (\frac {d \left (\sqrt {i-a}+\sqrt {b} \sqrt {x}\right )}{-\sqrt {b} c+\sqrt {i-a} d}\right ) \log \left (c+d \sqrt {x}\right )-d \sqrt {x} \log (1+i a+i b x)+c \log \left (c+d \sqrt {x}\right ) \log (1+i a+i b x)+d \sqrt {x} \log (-i (i+a+b x))-c \log \left (c+d \sqrt {x}\right ) \log (-i (i+a+b x))+c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {-i-a} d}\right )+c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {-i-a} d}\right )-c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c-\sqrt {i-a} d}\right )-c \text {PolyLog}\left (2,\frac {\sqrt {b} \left (c+d \sqrt {x}\right )}{\sqrt {b} c+\sqrt {i-a} d}\right )\right )}{d^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.07, size = 364, normalized size = 0.54
method | result | size |
derivativedivides | \(\frac {2 \arctan \left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \arctan \left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}-\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{b \,\textit {\_R}^{3}-3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\) | \(364\) |
default | \(\frac {2 \arctan \left (b x +a \right ) \sqrt {x}}{d}-\frac {2 \arctan \left (b x +a \right ) c \ln \left (c +d \sqrt {x}\right )}{d^{2}}-\frac {4 b \left (\frac {d^{2} \left (\munderset {\textit {\_R} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\left (\textit {\_R}^{2}-2 \textit {\_R} c +c^{2}\right ) \ln \left (d \sqrt {x}-\textit {\_R} +c \right )}{b \,\textit {\_R}^{3}-3 \textit {\_R}^{2} b c +\textit {\_R} a \,d^{2}+3 \textit {\_R} b \,c^{2}-a c \,d^{2}-b \,c^{3}}\right )}{4 b}-\frac {c \,d^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (b^{2} \textit {\_Z}^{4}-4 b^{2} c \,\textit {\_Z}^{3}+\left (2 a b \,d^{2}+6 b^{2} c^{2}\right ) \textit {\_Z}^{2}+\left (-4 a b c \,d^{2}-4 b^{2} c^{3}\right ) \textit {\_Z} +a^{2} d^{4}+2 a b \,c^{2} d^{2}+b^{2} c^{4}+d^{4}\right )}{\sum }\frac {\ln \left (c +d \sqrt {x}\right ) \ln \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d \sqrt {x}+\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2} b -2 \textit {\_R1} b c +a \,d^{2}+b \,c^{2}}\right )}{4 b}\right )}{d^{2}}\) | \(364\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {atan}\left (a+b\,x\right )}{c+d\,\sqrt {x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________