Integrand size = 22, antiderivative size = 86 \[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\frac {e \text {arctanh}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \sqrt {c d^2+a e^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
[Out]
Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {975, 272, 65, 214, 739, 212} \[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\frac {e \text {arctanh}\left (\frac {a e-c d x}{\sqrt {a+c x^2} \sqrt {a e^2+c d^2}}\right )}{d \sqrt {a e^2+c d^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
[In]
[Out]
Rule 65
Rule 212
Rule 214
Rule 272
Rule 739
Rule 975
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{d x \sqrt {a+c x^2}}-\frac {e}{d (d+e x) \sqrt {a+c x^2}}\right ) \, dx \\ & = \frac {\int \frac {1}{x \sqrt {a+c x^2}} \, dx}{d}-\frac {e \int \frac {1}{(d+e x) \sqrt {a+c x^2}} \, dx}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+c x}} \, dx,x,x^2\right )}{2 d}+\frac {e \text {Subst}\left (\int \frac {1}{c d^2+a e^2-x^2} \, dx,x,\frac {a e-c d x}{\sqrt {a+c x^2}}\right )}{d} \\ & = \frac {e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \sqrt {c d^2+a e^2}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{c}+\frac {x^2}{c}} \, dx,x,\sqrt {a+c x^2}\right )}{c d} \\ & = \frac {e \tanh ^{-1}\left (\frac {a e-c d x}{\sqrt {c d^2+a e^2} \sqrt {a+c x^2}}\right )}{d \sqrt {c d^2+a e^2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\frac {2 \left (\frac {e \arctan \left (\frac {\sqrt {c} (d+e x)-e \sqrt {a+c x^2}}{\sqrt {-c d^2-a e^2}}\right )}{\sqrt {-c d^2-a e^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {c} x-\sqrt {a+c x^2}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(157\) vs. \(2(74)=148\).
Time = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.84
method | result | size |
default | \(-\frac {\ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {c \,x^{2}+a}}{x}\right )}{d \sqrt {a}}+\frac {\ln \left (\frac {\frac {2 e^{2} a +2 c \,d^{2}}{e^{2}}-\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+2 \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}\, \sqrt {\left (x +\frac {d}{e}\right )^{2} c -\frac {2 c d \left (x +\frac {d}{e}\right )}{e}+\frac {e^{2} a +c \,d^{2}}{e^{2}}}}{x +\frac {d}{e}}\right )}{d \sqrt {\frac {e^{2} a +c \,d^{2}}{e^{2}}}}\) | \(158\) |
[In]
[Out]
none
Time = 0.41 (sec) , antiderivative size = 634, normalized size of antiderivative = 7.37 \[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\left [\frac {\sqrt {c d^{2} + a e^{2}} a e \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + {\left (c d^{2} + a e^{2}\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right )}{2 \, {\left (a c d^{3} + a^{2} d e^{2}\right )}}, \frac {2 \, \sqrt {-c d^{2} - a e^{2}} a e \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (c d^{2} + a e^{2}\right )} \sqrt {a} \log \left (-\frac {c x^{2} - 2 \, \sqrt {c x^{2} + a} \sqrt {a} + 2 \, a}{x^{2}}\right )}{2 \, {\left (a c d^{3} + a^{2} d e^{2}\right )}}, \frac {\sqrt {c d^{2} + a e^{2}} a e \log \left (\frac {2 \, a c d e x - a c d^{2} - 2 \, a^{2} e^{2} - {\left (2 \, c^{2} d^{2} + a c e^{2}\right )} x^{2} + 2 \, \sqrt {c d^{2} + a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, {\left (c d^{2} + a e^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right )}{2 \, {\left (a c d^{3} + a^{2} d e^{2}\right )}}, \frac {\sqrt {-c d^{2} - a e^{2}} a e \arctan \left (\frac {\sqrt {-c d^{2} - a e^{2}} {\left (c d x - a e\right )} \sqrt {c x^{2} + a}}{a c d^{2} + a^{2} e^{2} + {\left (c^{2} d^{2} + a c e^{2}\right )} x^{2}}\right ) + {\left (c d^{2} + a e^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {-a}}{\sqrt {c x^{2} + a}}\right )}{a c d^{3} + a^{2} d e^{2}}\right ] \]
[In]
[Out]
\[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{x \sqrt {a + c x^{2}} \left (d + e x\right )}\, dx \]
[In]
[Out]
\[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + a} {\left (e x + d\right )} x} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x (d+e x) \sqrt {a+c x^2}} \, dx=\int \frac {1}{x\,\sqrt {c\,x^2+a}\,\left (d+e\,x\right )} \,d x \]
[In]
[Out]