Integrand size = 33, antiderivative size = 383 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=-\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 c^{3/2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c e} \]
[Out]
Time = 0.80 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3781, 6857, 654, 635, 212, 1050, 1044, 214} \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=-\frac {\sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {-\sqrt {a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt {2} \sqrt {-\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}+\frac {\sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \text {arctanh}\left (\frac {\sqrt {a^2-2 a c+b^2+c^2}+a+b \tan (d+e x)-c}{\sqrt {2} \sqrt {\sqrt {a^2-2 a c+b^2+c^2}+a-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} e \sqrt {a^2-2 a c+b^2+c^2}}-\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 c^{3/2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c e} \]
[In]
[Out]
Rule 212
Rule 214
Rule 635
Rule 654
Rule 1044
Rule 1050
Rule 3781
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^3}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \left (\frac {x}{\sqrt {a+b x+c x^2}}-\frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}}\right ) \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\text {Subst}\left (\int \frac {x}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}-\frac {\text {Subst}\left (\int \frac {x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c e}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 c e}+\frac {\text {Subst}\left (\int \frac {-b+\left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\text {Subst}\left (\int \frac {-b+\left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right ) x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e} \\ & = \frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c e}-\frac {b \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{c e}+\frac {\left (b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e}-\frac {\left (b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b x^2} \, dx,x,\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a^2+b^2-2 a c+c^2} e} \\ & = -\frac {\sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c-\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c-\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \text {arctanh}\left (\frac {a-c+\sqrt {a^2+b^2-2 a c+c^2}+b \tan (d+e x)}{\sqrt {2} \sqrt {a-c+\sqrt {a^2+b^2-2 a c+c^2}} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt {a^2+b^2-2 a c+c^2} e}-\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{2 c^{3/2} e}+\frac {\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c e} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.91 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.66 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\frac {\frac {\text {arctanh}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a-i b-c}}+\frac {\text {arctanh}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {a+i b-c}}-\frac {b \text {arctanh}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{c^{3/2}}+\frac {2 \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c}}{2 e} \]
[In]
[Out]
result has leaf size over 500,000. Avoiding possible recursion issues.
Time = 1.57 (sec) , antiderivative size = 9581103, normalized size of antiderivative = 25015.93
\[\text {output too large to display}\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 5168 vs. \(2 (338) = 676\).
Time = 1.05 (sec) , antiderivative size = 10337, normalized size of antiderivative = 26.99 \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {\tan ^{3}{\left (d + e x \right )}}{\sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}}}\, dx \]
[In]
[Out]
\[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int { \frac {\tan \left (e x + d\right )^{3}}{\sqrt {c \tan \left (e x + d\right )^{2} + b \tan \left (e x + d\right ) + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \frac {\tan ^3(d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}} \, dx=\int \frac {{\mathrm {tan}\left (d+e\,x\right )}^3}{\sqrt {c\,{\mathrm {tan}\left (d+e\,x\right )}^2+b\,\mathrm {tan}\left (d+e\,x\right )+a}} \,d x \]
[In]
[Out]