Integrand size = 16, antiderivative size = 220 \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {12 i b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {12 i b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4} \]
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Time = 0.23 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {14, 4289, 4266, 2611, 6744, 2320, 6724} \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {12 i b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {12 i b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2} \]
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Rule 14
Rule 2320
Rule 2611
Rule 4266
Rule 4289
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = \int \left (a x+b x \sec \left (c+d \sqrt {x}\right )\right ) \, dx \\ & = \frac {a x^2}{2}+b \int x \sec \left (c+d \sqrt {x}\right ) \, dx \\ & = \frac {a x^2}{2}+(2 b) \text {Subst}\left (\int x^3 \sec (c+d x) \, dx,x,\sqrt {x}\right ) \\ & = \frac {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}-\frac {(6 b) \text {Subst}\left (\int x^2 \log \left (1-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(6 b) \text {Subst}\left (\int x^2 \log \left (1+i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d} \\ & = \frac {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {(12 i 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 {(12 i 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 {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {(12 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {(12 b) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3} \\ & = \frac {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {(12 i 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 {(12 i 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 {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i \left (c+d \sqrt {x}\right )}\right )}{d}+\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {12 i b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {12 i b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.01 \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {a x^2}{2}-\frac {4 i b x^{3/2} \arctan \left (e^{i c+i d \sqrt {x}}\right )}{d}+\frac {6 i b x \operatorname {PolyLog}\left (2,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {6 i b x \operatorname {PolyLog}\left (2,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {12 b \sqrt {x} \operatorname {PolyLog}\left (3,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^3}-\frac {12 i b \operatorname {PolyLog}\left (4,-i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {12 i b \operatorname {PolyLog}\left (4,i e^{i \left (c+d \sqrt {x}\right )}\right )}{d^4} \]
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\[\int x \left (a +b \sec \left (c +d \sqrt {x}\right )\right )d x\]
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\[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )} x \,d x } \]
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\[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\int x \left (a + b \sec {\left (c + d \sqrt {x} \right )}\right )\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (160) = 320\).
Time = 0.41 (sec) , antiderivative size = 540, normalized size of antiderivative = 2.45 \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\frac {{\left (d \sqrt {x} + c\right )}^{4} a - 4 \, {\left (d \sqrt {x} + c\right )}^{3} a c + 6 \, {\left (d \sqrt {x} + c\right )}^{2} a c^{2} - 4 \, {\left (d \sqrt {x} + c\right )} a c^{3} - 4 \, b c^{3} \log \left (\sec \left (d \sqrt {x} + c\right ) + \tan \left (d \sqrt {x} + c\right )\right ) - 4 \, {\left (i \, {\left (d \sqrt {x} + c\right )}^{3} b - 3 i \, {\left (d \sqrt {x} + c\right )}^{2} b c + 3 i \, {\left (d \sqrt {x} + c\right )} b c^{2}\right )} \arctan \left (\cos \left (d \sqrt {x} + c\right ), \sin \left (d \sqrt {x} + c\right ) + 1\right ) - 4 \, {\left (i \, {\left (d \sqrt {x} + c\right )}^{3} b - 3 i \, {\left (d \sqrt {x} + c\right )}^{2} b c + 3 i \, {\left (d \sqrt {x} + c\right )} b c^{2}\right )} \arctan \left (\cos \left (d \sqrt {x} + c\right ), -\sin \left (d \sqrt {x} + c\right ) + 1\right ) - 12 \, {\left (i \, {\left (d \sqrt {x} + c\right )}^{2} b - 2 i \, {\left (d \sqrt {x} + c\right )} b c + i \, b c^{2}\right )} {\rm Li}_2\left (i \, e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) - 12 \, {\left (-i \, {\left (d \sqrt {x} + c\right )}^{2} b + 2 i \, {\left (d \sqrt {x} + c\right )} b c - i \, b c^{2}\right )} {\rm Li}_2\left (-i \, e^{\left (i \, d \sqrt {x} + i \, c\right )}\right ) + 2 \, {\left ({\left (d \sqrt {x} + c\right )}^{3} b - 3 \, {\left (d \sqrt {x} + c\right )}^{2} b c + 3 \, {\left (d \sqrt {x} + c\right )} b c^{2}\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} + 2 \, \sin \left (d \sqrt {x} + c\right ) + 1\right ) - 2 \, {\left ({\left (d \sqrt {x} + c\right )}^{3} b - 3 \, {\left (d \sqrt {x} + c\right )}^{2} b c + 3 \, {\left (d \sqrt {x} + c\right )} b c^{2}\right )} \log \left (\cos \left (d \sqrt {x} + c\right )^{2} + \sin \left (d \sqrt {x} + c\right )^{2} - 2 \, \sin \left (d \sqrt {x} + c\right ) + 1\right ) + 24 i \, b {\rm Li}_{4}(i \, e^{\left (i \, d \sqrt {x} + i \, c\right )}) - 24 i \, b {\rm Li}_{4}(-i \, e^{\left (i \, d \sqrt {x} + i \, c\right )}) + 24 \, {\left ({\left (d \sqrt {x} + c\right )} b - b c\right )} {\rm Li}_{3}(i \, e^{\left (i \, d \sqrt {x} + i \, c\right )}) - 24 \, {\left ({\left (d \sqrt {x} + c\right )} b - b c\right )} {\rm Li}_{3}(-i \, e^{\left (i \, d \sqrt {x} + i \, c\right )})}{2 \, d^{4}} \]
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\[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\int { {\left (b \sec \left (d \sqrt {x} + c\right ) + a\right )} x \,d x } \]
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Timed out. \[ \int x \left (a+b \sec \left (c+d \sqrt {x}\right )\right ) \, dx=\int x\,\left (a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}\right ) \,d x \]
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