Integrand size = 22, antiderivative size = 407 \[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {6 b^2 \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {96 i a b \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d} \]
[Out]
Time = 0.38 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5544, 4275, 4265, 2611, 6744, 2320, 6724, 4269, 3799, 2221} \[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}+\frac {96 i a b \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {48 i a b x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {6 b^2 \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {12 b^2 x \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {8 b^2 x^{3/2} \log \left (e^{2 \left (c+d \sqrt {x}\right )}+1\right )}{d^2}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {2 b^2 x^2}{d} \]
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 4265
Rule 4269
Rule 4275
Rule 5544
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int x^4 (a+b \text {sech}(c+d x))^2 \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (a^2 x^4+2 a b x^4 \text {sech}(c+d x)+b^2 x^4 \text {sech}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{5} a^2 x^{5/2}+(4 a b) \text {Subst}\left (\int x^4 \text {sech}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^4 \text {sech}^2(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(16 i a b) \text {Subst}\left (\int x^3 \log \left (1-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(16 i a b) \text {Subst}\left (\int x^3 \log \left (1+i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}-\frac {\left (8 b^2\right ) \text {Subst}\left (\int x^3 \tanh (c+d x) \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(48 i a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(48 i a b) \text {Subst}\left (\int x^2 \operatorname {PolyLog}\left (2,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (16 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x^3}{1+e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {48 i a b x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {(96 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(96 i a b) \text {Subst}\left (\int x \operatorname {PolyLog}\left (3,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {\left (24 b^2\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2} \\ & = \frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(96 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,-i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(96 i a b) \text {Subst}\left (\int \operatorname {PolyLog}\left (4,i e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}+\frac {\left (24 b^2\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = \frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}+\frac {(96 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {(96 i a b) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,i x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {\left (12 b^2\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4} \\ & = \frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d}-\frac {\left (6 b^2\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5} \\ & = \frac {2 b^2 x^2}{d}+\frac {2}{5} a^2 x^{5/2}+\frac {8 a b x^2 \arctan \left (e^{c+d \sqrt {x}}\right )}{d}-\frac {8 b^2 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )}{d^2}+\frac {16 i a b x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )}{d^2}-\frac {12 b^2 x \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {48 i a b x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )}{d^3}-\frac {48 i a b x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )}{d^3}+\frac {12 b^2 \sqrt {x} \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )}{d^4}+\frac {96 i a b \sqrt {x} \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )}{d^4}-\frac {6 b^2 \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {96 i a b \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )}{d^5}-\frac {96 i a b \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )}{d^5}+\frac {2 b^2 x^2 \tanh \left (c+d \sqrt {x}\right )}{d} \\ \end{align*}
Time = 5.89 (sec) , antiderivative size = 496, normalized size of antiderivative = 1.22 \[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\frac {2 \cosh \left (c+d \sqrt {x}\right ) \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \left (\frac {10 b^2 e^{2 c} x^2 \cosh \left (c+d \sqrt {x}\right )}{d \left (1+e^{2 c}\right )}+a^2 x^{5/2} \cosh \left (c+d \sqrt {x}\right )+\frac {5 i b \cosh \left (c+d \sqrt {x}\right ) \left (2 a d^4 x^2 \log \left (1-i e^{c+d \sqrt {x}}\right )-2 a d^4 x^2 \log \left (1+i e^{c+d \sqrt {x}}\right )+4 i b d^3 x^{3/2} \log \left (1+e^{2 \left (c+d \sqrt {x}\right )}\right )-8 a d^3 x^{3/2} \operatorname {PolyLog}\left (2,-i e^{c+d \sqrt {x}}\right )+8 a d^3 x^{3/2} \operatorname {PolyLog}\left (2,i e^{c+d \sqrt {x}}\right )+6 i b d^2 x \operatorname {PolyLog}\left (2,-e^{2 \left (c+d \sqrt {x}\right )}\right )+24 a d^2 x \operatorname {PolyLog}\left (3,-i e^{c+d \sqrt {x}}\right )-24 a d^2 x \operatorname {PolyLog}\left (3,i e^{c+d \sqrt {x}}\right )-6 i b d \sqrt {x} \operatorname {PolyLog}\left (3,-e^{2 \left (c+d \sqrt {x}\right )}\right )-48 a d \sqrt {x} \operatorname {PolyLog}\left (4,-i e^{c+d \sqrt {x}}\right )+48 a d \sqrt {x} \operatorname {PolyLog}\left (4,i e^{c+d \sqrt {x}}\right )+3 i b \operatorname {PolyLog}\left (4,-e^{2 \left (c+d \sqrt {x}\right )}\right )+48 a \operatorname {PolyLog}\left (5,-i e^{c+d \sqrt {x}}\right )-48 a \operatorname {PolyLog}\left (5,i e^{c+d \sqrt {x}}\right )\right )}{d^5}+\frac {5 b^2 x^2 \text {sech}(c) \sinh \left (d \sqrt {x}\right )}{d}\right )}{5 \left (b+a \cosh \left (c+d \sqrt {x}\right )\right )^2} \]
[In]
[Out]
\[\int x^{\frac {3}{2}} \left (a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )\right )^{2}d x\]
[In]
[Out]
\[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{\frac {3}{2}} \left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \]
[In]
[Out]
\[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int { {\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2} x^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^{3/2} \left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2 \, dx=\int x^{3/2}\,{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \]
[In]
[Out]