Integrand size = 18, antiderivative size = 355 \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=-\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 i a b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d} \]
[Out]
Time = 0.52 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {4289, 4275, 4266, 2611, 6744, 2320, 6724, 4269, 3800, 2221} \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {24 i a b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {2 i b^2 x^{3/2}}{d} \]
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3800
Rule 4266
Rule 4269
Rule 4275
Rule 4289
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^3 (a+b \sec (c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^3+2 a b x^3 \sec (c+d x)+b^2 x^3 \sec ^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^2}{2}+(4 a b) \text {Subst}\left (\int x^3 \sec (c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^3 \sec ^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(12 a b) \text {Subst}\left (\int x^2 \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(12 a b) \text {Subst}\left (\int x^2 \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int x^2 \tan (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(24 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {(24 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}+\frac {\left (12 i b^2\right ) \text {Subst}\left (\int \frac {e^{2 i (c+d x)} x^2}{1+e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {(24 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(24 a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int x \log \left (1+e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}-\frac {(24 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {(24 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {\left (6 i b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{2 i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 i a b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4} \\ & = -\frac {2 i b^2 x^{3/2}}{d}+\frac {a^2 x^2}{2}-\frac {8 i a b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 b^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^2}+\frac {12 i a b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 i a b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b^2 \sqrt {x} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {24 a b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {3 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {24 i a b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {24 i a b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {2 b^2 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{d} \\ \end{align*}
Time = 0.75 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.98 \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {-4 i b^2 d^3 x^{3/2}+a^2 d^4 x^2-16 i a b d^3 x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )+12 b^2 d^2 x \log \left (1+e^{2 i \left (c+d \sqrt {x}\right )}\right )+24 i a b d^2 x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )-24 i a b d^2 x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )-12 i b^2 d \sqrt {x} \operatorname {PolyLog}\left (2,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-48 a b d \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )+48 a b d \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )+6 b^2 \operatorname {PolyLog}\left (3,-e^{2 i \left (c+d \sqrt {x}\right )}\right )-48 i a b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )+48 i a b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )+4 b^2 d^3 x^{3/2} \tan \left (c+d \sqrt {x}\right )}{2 d^4} \]
[In]
[Out]
\[\int x \left (a +b \sec \left (c +d \sqrt {x}\right )\right )^{2}d x\]
[In]
[Out]
\[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2} x \,d x } \]
[In]
[Out]
\[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x \left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1991 vs. \(2 (270) = 540\).
Time = 0.45 (sec) , antiderivative size = 1991, normalized size of antiderivative = 5.61 \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )}^{2} x \,d x } \]
[In]
[Out]
Timed out. \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x\,{\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
[In]
[Out]