Integrand size = 22, antiderivative size = 364 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=-\frac {a c^2 \sqrt {c+a^2 c x^2}}{2 x}-\frac {1}{6} a^3 c^2 x \sqrt {c+a^2 c x^2}+2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {13}{6} a^2 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 \sqrt {c+a^2 c x^2}} \]
1/3*a^2*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)-13/6*a^2*c^(5/2)*arctanh(a*x*c^( 1/2)/(a^2*c*x^2+c)^(1/2))-5*a^2*c^3*arctan(a*x)*arctanh((1+I*a*x)^(1/2)/(1 -I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+5/2*I*a^2*c^3*polylog (2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2) -5/2*I*a^2*c^3*polylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2 )/(a^2*c*x^2+c)^(1/2)-1/2*a*c^2*(a^2*c*x^2+c)^(1/2)/x-1/6*a^3*c^2*x*(a^2*c *x^2+c)^(1/2)+2*a^2*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)-1/2*c^2*arctan(a*x )*(a^2*c*x^2+c)^(1/2)/x^2
Time = 1.91 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.03 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \left (-6-6 \cot ^2\left (\frac {1}{2} \arctan (a x)\right )-2 a^2 x^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+28 a x \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+4 a^3 x^3 \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-3 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+60 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-e^{i \arctan (a x)}\right )-60 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+e^{i \arctan (a x)}\right )+4 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (-a x+\sqrt {1+a^2 x^2}\right )+48 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )-\sin \left (\frac {1}{2} \arctan (a x)\right )\right )-48 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\cos \left (\frac {1}{2} \arctan (a x)\right )+\sin \left (\frac {1}{2} \arctan (a x)\right )\right )+60 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-60 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+3 \arctan (a x) \csc \left (\frac {1}{2} \arctan (a x)\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )\right ) \tan \left (\frac {1}{2} \arctan (a x)\right )}{24 \sqrt {1+a^2 x^2}} \]
(a^2*c^2*Sqrt[c + a^2*c*x^2]*(-6 - 6*Cot[ArcTan[a*x]/2]^2 - 2*a^2*x^2*Csc[ ArcTan[a*x]/2]^2 + 28*a*x*ArcTan[a*x]*Csc[ArcTan[a*x]/2]^2 + 4*a^3*x^3*Arc Tan[a*x]*Csc[ArcTan[a*x]/2]^2 - 3*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Csc[ArcTa n[a*x]/2]^2 + 60*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 60*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 4*Cot[Ar cTan[a*x]/2]*Log[-(a*x) + Sqrt[1 + a^2*x^2]] + 48*Cot[ArcTan[a*x]/2]*Log[C os[ArcTan[a*x]/2] - Sin[ArcTan[a*x]/2]] - 48*Cot[ArcTan[a*x]/2]*Log[Cos[Ar cTan[a*x]/2] + Sin[ArcTan[a*x]/2]] + (60*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (60*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x ])] + 3*ArcTan[a*x]*Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x] /2])/(24*Sqrt[1 + a^2*x^2])
Time = 2.85 (sec) , antiderivative size = 696, normalized size of antiderivative = 1.91, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {5485, 5485, 5465, 211, 224, 219, 5481, 224, 219, 242, 5493, 5489, 5497, 242, 5493, 5489}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{x^3} \, dx\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle a^2 c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x}dx+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}{x^3}dx\) |
\(\Big \downarrow \) 5485 |
\(\displaystyle a^2 c \left (a^2 c \int x \sqrt {a^2 c x^2+c} \arctan (a x)dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^3}dx\right )\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\int \sqrt {a^2 c x^2+c}dx}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^3}dx\right )\) |
\(\Big \downarrow \) 211 |
\(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^3}dx\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^3}dx\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{x^3}dx\right )\) |
\(\Big \downarrow \) 5481 |
\(\displaystyle a^2 c \left (c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+c \left (a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx+a c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}\right )\right )\) |
\(\Big \downarrow \) 224 |
\(\displaystyle a^2 c \left (c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+c \left (a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-a c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx+a c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}\right )\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle a^2 c \left (c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+c \left (a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx+a c \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}\right )\right )\) |
\(\Big \downarrow \) 242 |
\(\displaystyle a^2 c \left (c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+c \left (a^2 c \left (c \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )\right )\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle a^2 c \left (c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )\right )+c \left (a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )\right )\) |
\(\Big \downarrow \) 5489 |
\(\displaystyle c \left (c \left (-c \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )+a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
\(\Big \downarrow \) 5497 |
\(\displaystyle c \left (c \left (-c \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )+a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
\(\Big \downarrow \) 242 |
\(\displaystyle c \left (c \left (-c \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )+a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle c \left (c \left (-c \left (-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )+a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
\(\Big \downarrow \) 5489 |
\(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}}{3 a}\right )+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )\right )+c \left (a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x) \sqrt {a^2 c x^2+c}-\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )\right )+c \left (-c \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{x^2}-\frac {a \sqrt {a^2 c x^2+c}}{x}\right )\right )\) |
a^2*c*(a^2*c*(((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/(3*a^2*c) - ((x*Sqrt[c + a^2*c*x^2])/2 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a ))/(3*a)) + c*(Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - Sqrt[c]*ArcTanh[(a*Sqrt[c ]*x)/Sqrt[c + a^2*c*x^2]] + (c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[S qrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/Sqrt[c + a^2*c* x^2])) + c*(c*(-((a*Sqrt[c + a^2*c*x^2])/x) - (Sqrt[c + a^2*c*x^2]*ArcTan[ a*x])/x^2 - c*(-1/2*(a*Sqrt[c + a^2*c*x^2])/(c*x) - (Sqrt[c + a^2*c*x^2]*A rcTan[a*x])/(2*c*x^2) - (a^2*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqr t[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I* a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/(2*Sqrt[c + a^2*c *x^2]))) + a^2*c*(Sqrt[c + a^2*c*x^2]*ArcTan[a*x] - Sqrt[c]*ArcTanh[(a*Sqr t[c]*x)/Sqrt[c + a^2*c*x^2]] + (c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTan h[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/Sqrt[c + a^2 *c*x^2]))
3.3.22.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x ])/(f*(m + 2))), x] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTan[c*x])/Sq rt[d + e*x^2]), x], x] - Simp[b*c*(d/(f*(m + 2))) Int[(f*x)^(m + 1)/Sqrt[ d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan [c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[ e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Ar cTan[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1))) Int[(f*x)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[c^2*((m + 2)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x ^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
Time = 1.93 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.70
method | result | size |
default | \(-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{4} x^{4}+\sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+15 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) \arctan \left (a x \right ) a^{2} x^{2}-14 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}-15 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right ) a^{2} x^{2}-26 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{2} x^{2}-15 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) a^{2} x^{2}+3 \sqrt {a^{2} x^{2}+1}\, a x +3 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\right ) c^{2}}{6 \sqrt {a^{2} x^{2}+1}\, x^{2}}\) | \(256\) |
-1/6/(a^2*x^2+1)^(1/2)*(c*(a*x-I)*(I+a*x))^(1/2)*(-2*arctan(a*x)*(a^2*x^2+ 1)^(1/2)*a^4*x^4+(a^2*x^2+1)^(1/2)*a^3*x^3+15*ln((1+I*a*x)/(a^2*x^2+1)^(1/ 2)+1)*arctan(a*x)*a^2*x^2-14*arctan(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2-15*I*di log((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)*a^2*x^2-26*I*arctan((1+I*a*x)/(a^2*x^2+ 1)^(1/2))*a^2*x^2-15*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2))*a^2*x^2+3*(a^2*x ^2+1)^(1/2)*a*x+3*arctan(a*x)*(a^2*x^2+1)^(1/2))*c^2/x^2
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}{\left (a x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )}{x^{3}} \,d x } \]
1/3*(a^4*c^2*x^2 + a^2*c^2)*sqrt(a^2*x^2 + 1)*sqrt(c)*arctan(a*x) - 1/6*(a ^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*(a^3*c^2*x*cos(1/2*arctan2(4*a*x, -a^2*x^2 + 3)) + 2*a^2*c^2*sin(1/2*arctan2(4*a*x, -a^2*x^2 + 3)))*sqrt(c) + 1/12*(a ^2*c^2*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin(1/2*arctan2(4*a*x, a^2 *x^2 - 3)) + 2, a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*cos(1/2*arctan2(4*a *x, a^2*x^2 - 3))) + a^2*c^2*arctan2((a^4*x^4 + 10*a^2*x^2 + 9)^(1/4)*sin( 1/2*arctan2(4*a*x, a^2*x^2 - 3)) - 2, -a*x + (a^4*x^4 + 10*a^2*x^2 + 9)^(1 /4)*cos(1/2*arctan2(4*a*x, a^2*x^2 - 3))) + 24*a^2*c^2*integrate(sqrt(a^2* x^2 + 1)*arctan(a*x)/x, x) + 12*c^2*integrate(sqrt(a^2*x^2 + 1)*arctan(a*x )/x^3, x))*sqrt(c)
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{x^3} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^3} \,d x \]