Integrand size = 40, antiderivative size = 268 \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) d \sqrt {\sqrt {3}-2 i x^2}}{\left (2 i c^2-\sqrt {3} d^2\right ) (c+d x)}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) d \sqrt {\sqrt {3}+2 i x^2}}{\left (2 i c^2+\sqrt {3} d^2\right ) (c+d x)}+\frac {(1+i) c \arctan \left (\frac {\sqrt {3} d+2 i c x}{\sqrt {2 i c^2-\sqrt {3} d^2} \sqrt {\sqrt {3}-2 i x^2}}\right )}{\left (2 i c^2-\sqrt {3} d^2\right )^{3/2}}+\frac {(1-i) c \text {arctanh}\left (\frac {\sqrt {3} d-2 i c x}{\sqrt {2 i c^2+\sqrt {3} d^2} \sqrt {\sqrt {3}+2 i x^2}}\right )}{\left (2 i c^2+\sqrt {3} d^2\right )^{3/2}} \]
(1+I)*c*arctan((2*I*c*x+d*3^(1/2))/(-2*I*x^2+3^(1/2))^(1/2)/(2*I*c^2-d^2*3 ^(1/2))^(1/2))/(2*I*c^2-d^2*3^(1/2))^(3/2)+(1-I)*c*arctanh((-2*I*c*x+d*3^( 1/2))/(2*I*x^2+3^(1/2))^(1/2)/(2*I*c^2+d^2*3^(1/2))^(1/2))/(2*I*c^2+d^2*3^ (1/2))^(3/2)+(1/2-1/2*I)*d*(-2*I*x^2+3^(1/2))^(1/2)/(d*x+c)/(2*I*c^2-d^2*3 ^(1/2))-(1/2+1/2*I)*d*(2*I*x^2+3^(1/2))^(1/2)/(d*x+c)/(2*I*c^2+d^2*3^(1/2) )
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 3.42 (sec) , antiderivative size = 1464, normalized size of antiderivative = 5.46 \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=-\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}} \left (3 d^2 \left (2 x^2+\sqrt {3+4 x^4}\right )+2 c^2 \left (3+8 x^4+4 x^2 \sqrt {3+4 x^4}\right )\right )}{\left (4 c^4+3 d^4\right ) (c+d x) \left (3+8 x^4+4 x^2 \sqrt {3+4 x^4}\right )}+\frac {8 c^5 \arctan \left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}\right )}{\left (4 c^4+3 d^4\right )^{3/2} \sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}-\frac {6 c d^4 \arctan \left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}\right )}{\left (4 c^4+3 d^4\right )^{3/2} \sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}+\frac {4 c^3 \arctan \left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}\right )}{\left (4 c^4+3 d^4\right ) \sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}-\frac {8 c^5 \arctan \left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}\right )}{\left (4 c^4+3 d^4\right )^{3/2} \sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}+\frac {6 c d^4 \arctan \left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}\right )}{\left (4 c^4+3 d^4\right )^{3/2} \sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}+\frac {4 c^3 \arctan \left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}\right )}{\left (4 c^4+3 d^4\right ) \sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}-\frac {\text {RootSum}\left [9 d^2-24 c^2 \text {$\#$1}-6 d^2 \text {$\#$1}^2-8 c^2 \text {$\#$1}^3+d^2 \text {$\#$1}^4\&,\frac {128 c^4 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right )+3 d^4 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right )+16 c^2 d^2 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right ) \text {$\#$1}+d^4 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{6 c^2+3 d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-d^2 \text {$\#$1}^3}\&\right ]}{d^4}+\frac {\text {RootSum}\left [9 d^2-24 c^2 \text {$\#$1}-6 d^2 \text {$\#$1}^2-8 c^2 \text {$\#$1}^3+d^2 \text {$\#$1}^4\&,\frac {512 c^8 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right )+408 c^4 d^4 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right )+9 d^8 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right )+64 c^6 d^2 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right ) \text {$\#$1}+36 c^2 d^6 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right ) \text {$\#$1}+8 c^4 d^4 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2+3 d^8 \log \left (2 x^2+\sqrt {3+4 x^4}+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-\text {$\#$1}\right ) \text {$\#$1}^2}{6 c^2+3 d^2 \text {$\#$1}+6 c^2 \text {$\#$1}^2-d^2 \text {$\#$1}^3}\&\right ]}{4 c^4 d^4+3 d^8} \]
-((d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]]*(3*d^2*(2*x^2 + Sqrt[3 + 4*x^4]) + 2*c^ 2*(3 + 8*x^4 + 4*x^2*Sqrt[3 + 4*x^4])))/((4*c^4 + 3*d^4)*(c + d*x)*(3 + 8* x^4 + 4*x^2*Sqrt[3 + 4*x^4]))) + (8*c^5*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4* x^4]])/Sqrt[-2*c^2 - Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2 *c^2 - Sqrt[4*c^4 + 3*d^4]]) - (6*c*d^4*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4* x^4]])/Sqrt[-2*c^2 - Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2 *c^2 - Sqrt[4*c^4 + 3*d^4]]) + (4*c^3*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^ 4]])/Sqrt[-2*c^2 - Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)*Sqrt[-2*c^2 - S qrt[4*c^4 + 3*d^4]]) - (8*c^5*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]])/Sqr t[-2*c^2 + Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2*c^2 + Sqr t[4*c^4 + 3*d^4]]) + (6*c*d^4*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]])/Sqr t[-2*c^2 + Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)^(3/2)*Sqrt[-2*c^2 + Sqr t[4*c^4 + 3*d^4]]) + (4*c^3*ArcTan[(d*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]])/Sqrt[ -2*c^2 + Sqrt[4*c^4 + 3*d^4]]])/((4*c^4 + 3*d^4)*Sqrt[-2*c^2 + Sqrt[4*c^4 + 3*d^4]]) - RootSum[9*d^2 - 24*c^2*#1 - 6*d^2*#1^2 - 8*c^2*#1^3 + d^2*#1^ 4 & , (128*c^4*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2*x^2 + Sqrt[3 + 4*x ^4]] - #1] + 3*d^4*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]] - #1] + 16*c^2*d^2*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2*x^2 + Sqrt[3 + 4*x^4]] - #1]*#1 + d^4*Log[2*x^2 + Sqrt[3 + 4*x^4] + 2*x*Sqrt[2* x^2 + Sqrt[3 + 4*x^4]] - #1]*#1^2)/(6*c^2 + 3*d^2*#1 + 6*c^2*#1^2 - d^2...
Time = 0.50 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.15, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2558, 491, 488, 217, 219}
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.
| \(\displaystyle \int \frac {\sqrt {\sqrt {4 x^4+3}+2 x^2}}{\sqrt {4 x^4+3} (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 2558 |
| \(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(c+d x)^2 \sqrt {\sqrt {3}-2 i x^2}}dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(c+d x)^2 \sqrt {2 i x^2+\sqrt {3}}}dx\) |
| \(\Big \downarrow \) 491 |
| \(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {2 c \int \frac {1}{(c+d x) \sqrt {\sqrt {3}-2 i x^2}}dx}{2 c^2+i \sqrt {3} d^2}+\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {2 c \int \frac {1}{(c+d x) \sqrt {2 i x^2+\sqrt {3}}}dx}{2 c^2-i \sqrt {3} d^2}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}-\frac {2 c \int \frac {1}{-2 i c^2+\sqrt {3} d^2-\frac {\left (\sqrt {3} d+2 i c x\right )^2}{\sqrt {3}-2 i x^2}}d\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2}}}{2 c^2+i \sqrt {3} d^2}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {2 c \int \frac {1}{2 i c^2+\sqrt {3} d^2-\frac {\left (\sqrt {3} d-2 i c x\right )^2}{2 i x^2+\sqrt {3}}}d\frac {\sqrt {3} d-2 i c x}{\sqrt {2 i x^2+\sqrt {3}}}}{2 c^2-i \sqrt {3} d^2}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {2 c \int \frac {1}{2 i c^2+\sqrt {3} d^2-\frac {\left (\sqrt {3} d-2 i c x\right )^2}{2 i x^2+\sqrt {3}}}d\frac {\sqrt {3} d-2 i c x}{\sqrt {2 i x^2+\sqrt {3}}}}{2 c^2-i \sqrt {3} d^2}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )+\left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {2 c \arctan \left (\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2} \sqrt {-\sqrt {3} d^2+2 i c^2}}\right )}{\sqrt {-\sqrt {3} d^2+2 i c^2} \left (2 c^2+i \sqrt {3} d^2\right )}+\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \left (\frac {1}{2}-\frac {i}{2}\right ) \left (\frac {2 c \arctan \left (\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2} \sqrt {-\sqrt {3} d^2+2 i c^2}}\right )}{\sqrt {-\sqrt {3} d^2+2 i c^2} \left (2 c^2+i \sqrt {3} d^2\right )}+\frac {d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-\frac {2 c \text {arctanh}\left (\frac {\sqrt {3} d-2 i c x}{\sqrt {\sqrt {3}+2 i x^2} \sqrt {\sqrt {3} d^2+2 i c^2}}\right )}{\left (2 c^2-i \sqrt {3} d^2\right ) \sqrt {\sqrt {3} d^2+2 i c^2}}-\frac {d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}\right )\) |
(1/2 - I/2)*((d*Sqrt[Sqrt[3] - (2*I)*x^2])/(((2*I)*c^2 - Sqrt[3]*d^2)*(c + d*x)) + (2*c*ArcTan[(Sqrt[3]*d + (2*I)*c*x)/(Sqrt[(2*I)*c^2 - Sqrt[3]*d^2 ]*Sqrt[Sqrt[3] - (2*I)*x^2])])/(Sqrt[(2*I)*c^2 - Sqrt[3]*d^2]*(2*c^2 + I*S qrt[3]*d^2))) + (1/2 + I/2)*(-((d*Sqrt[Sqrt[3] + (2*I)*x^2])/(((2*I)*c^2 + Sqrt[3]*d^2)*(c + d*x))) - (2*c*ArcTanh[(Sqrt[3]*d - (2*I)*c*x)/(Sqrt[(2* I)*c^2 + Sqrt[3]*d^2]*Sqrt[Sqrt[3] + (2*I)*x^2])])/((2*c^2 - I*Sqrt[3]*d^2 )*Sqrt[(2*I)*c^2 + Sqrt[3]*d^2]))
3.10.14.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
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/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b*(c/(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n + 2*p + 3, 0]
Int[(((c_.) + (d_.)*(x_))^(m_.)*Sqrt[(b_.)*(x_)^2 + Sqrt[(a_) + (e_.)*(x_)^ 4]])/Sqrt[(a_) + (e_.)*(x_)^4], x_Symbol] :> Simp[(1 - I)/2 Int[(c + d*x) ^m/Sqrt[Sqrt[a] - I*b*x^2], x], x] + Simp[(1 + I)/2 Int[(c + d*x)^m/Sqrt[ Sqrt[a] + I*b*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[e, b^2] && G tQ[a, 0]
\[\int \frac {\sqrt {2 x^{2}+\sqrt {4 x^{4}+3}}}{\left (d x +c \right )^{2} \sqrt {4 x^{4}+3}}d x\]
Timed out. \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int \frac {\sqrt {2 x^{2} + \sqrt {4 x^{4} + 3}}}{\left (c + d x\right )^{2} \sqrt {4 x^{4} + 3}}\, dx \]
\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int { \frac {\sqrt {2 \, x^{2} + \sqrt {4 \, x^{4} + 3}}}{\sqrt {4 \, x^{4} + 3} {\left (d x + c\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int { \frac {\sqrt {2 \, x^{2} + \sqrt {4 \, x^{4} + 3}}}{\sqrt {4 \, x^{4} + 3} {\left (d x + c\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx=\int \frac {\sqrt {2\,x^2+\sqrt {4\,x^4+3}}}{\sqrt {4\,x^4+3}\,{\left (c+d\,x\right )}^2} \,d x \]