Optimal. Leaf size=1041 \[ \frac {x^4}{4 a}+\frac {2 i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}-\frac {2 i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}+\frac {14 b \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}-\frac {14 b \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}+\frac {84 i b \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {84 i b \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {420 b \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}+\frac {420 b \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}-\frac {1680 i b \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {1680 i b \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {5040 b \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}-\frac {5040 b \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}+\frac {10080 i b \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {10080 i b \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {10080 b \text {Li}_8\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8}+\frac {10080 b \text {Li}_8\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8} \]
[Out]
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Rubi [A] time = 1.48, antiderivative size = 1041, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {4204, 4191, 3321, 2264, 2190, 2531, 6609, 2282, 6589} \[ \frac {x^4}{4 a}+\frac {2 i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b-\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}-\frac {2 i b \log \left (\frac {e^{i \left (c+d \sqrt {x}\right )} a}{b+\sqrt {b^2-a^2}}+1\right ) x^{7/2}}{a \sqrt {b^2-a^2} d}+\frac {14 b \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}-\frac {14 b \text {PolyLog}\left (2,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^3}{a \sqrt {b^2-a^2} d^2}+\frac {84 i b \text {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {84 i b \text {PolyLog}\left (3,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{5/2}}{a \sqrt {b^2-a^2} d^3}-\frac {420 b \text {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}+\frac {420 b \text {PolyLog}\left (4,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^2}{a \sqrt {b^2-a^2} d^4}-\frac {1680 i b \text {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {1680 i b \text {PolyLog}\left (5,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x^{3/2}}{a \sqrt {b^2-a^2} d^5}+\frac {5040 b \text {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}-\frac {5040 b \text {PolyLog}\left (6,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) x}{a \sqrt {b^2-a^2} d^6}+\frac {10080 i b \text {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {10080 i b \text {PolyLog}\left (7,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right ) \sqrt {x}}{a \sqrt {b^2-a^2} d^7}-\frac {10080 b \text {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8}+\frac {10080 b \text {PolyLog}\left (8,-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {b^2-a^2}}\right )}{a \sqrt {b^2-a^2} d^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3321
Rule 4191
Rule 4204
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^3}{a+b \sec \left (c+d \sqrt {x}\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^7}{a+b \sec (c+d x)} \, dx,x,\sqrt {x}\right )\\ &=2 \operatorname {Subst}\left (\int \left (\frac {x^7}{a}-\frac {b x^7}{a (b+a \cos (c+d x))}\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {x^4}{4 a}-\frac {(2 b) \operatorname {Subst}\left (\int \frac {x^7}{b+a \cos (c+d x)} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {x^4}{4 a}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{a+2 b e^{i (c+d x)}+a e^{2 i (c+d x)}} \, dx,x,\sqrt {x}\right )}{a}\\ &=\frac {x^4}{4 a}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}+\frac {(4 b) \operatorname {Subst}\left (\int \frac {e^{i (c+d x)} x^7}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (c+d x)}} \, dx,x,\sqrt {x}\right )}{\sqrt {-a^2+b^2}}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {(14 i b) \operatorname {Subst}\left (\int x^6 \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d}+\frac {(14 i b) \operatorname {Subst}\left (\int x^6 \log \left (1+\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {(84 b) \operatorname {Subst}\left (\int x^5 \text {Li}_2\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {(84 b) \operatorname {Subst}\left (\int x^5 \text {Li}_2\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^2}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {(420 i b) \operatorname {Subst}\left (\int x^4 \text {Li}_3\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}+\frac {(420 i b) \operatorname {Subst}\left (\int x^4 \text {Li}_3\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^3}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {(1680 b) \operatorname {Subst}\left (\int x^3 \text {Li}_4\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {(1680 b) \operatorname {Subst}\left (\int x^3 \text {Li}_4\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^4}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {(5040 i b) \operatorname {Subst}\left (\int x^2 \text {Li}_5\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^5}-\frac {(5040 i b) \operatorname {Subst}\left (\int x^2 \text {Li}_5\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^5}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {(10080 b) \operatorname {Subst}\left (\int x \text {Li}_6\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {(10080 b) \operatorname {Subst}\left (\int x \text {Li}_6\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^6}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {(10080 i b) \operatorname {Subst}\left (\int \text {Li}_7\left (-\frac {2 a e^{i (c+d x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^7}+\frac {(10080 i b) \operatorname {Subst}\left (\int \text {Li}_7\left (-\frac {2 a e^{i (c+d x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx,x,\sqrt {x}\right )}{a \sqrt {-a^2+b^2} d^7}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {(10080 b) \operatorname {Subst}\left (\int \frac {\text {Li}_7\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^8}+\frac {(10080 b) \operatorname {Subst}\left (\int \frac {\text {Li}_7\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i \left (c+d \sqrt {x}\right )}\right )}{a \sqrt {-a^2+b^2} d^8}\\ &=\frac {x^4}{4 a}+\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}-\frac {2 i b x^{7/2} \log \left (1+\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d}+\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}-\frac {14 b x^3 \text {Li}_2\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^2}+\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {84 i b x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^3}-\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}+\frac {420 b x^2 \text {Li}_4\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^4}-\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {1680 i b x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^5}+\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}-\frac {5040 b x \text {Li}_6\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^6}+\frac {10080 i b \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 i b \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^7}-\frac {10080 b \text {Li}_8\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b-\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8}+\frac {10080 b \text {Li}_8\left (-\frac {a e^{i \left (c+d \sqrt {x}\right )}}{b+\sqrt {-a^2+b^2}}\right )}{a \sqrt {-a^2+b^2} d^8}\\ \end {align*}
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Mathematica [A] time = 2.67, size = 1122, normalized size = 1.08 \[ \frac {\left (b+a \cos \left (c+d \sqrt {x}\right )\right ) \left (x^4+\frac {8 b e^{i c} \left (7 x^3 \text {Li}_2\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^6-7 x^3 \text {Li}_2\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^6+i \left (x^{7/2} \log \left (\frac {e^{i \left (2 c+d \sqrt {x}\right )} a}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}+1\right ) d^7-x^{7/2} \log \left (\frac {e^{i \left (2 c+d \sqrt {x}\right )} a}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}+1\right ) d^7+42 x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^5-42 x^{5/2} \text {Li}_3\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^5+210 i x^2 \text {Li}_4\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^4-210 i x^2 \text {Li}_4\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^4-840 x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^3+840 x^{3/2} \text {Li}_5\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^3-2520 i x \text {Li}_6\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^2+2520 i x \text {Li}_6\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d^2+5040 \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d-5040 \sqrt {x} \text {Li}_7\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) d+5040 i \text {Li}_8\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{b e^{i c}-\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )-5040 i \text {Li}_8\left (-\frac {a e^{i \left (2 c+d \sqrt {x}\right )}}{e^{i c} b+\sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right )\right )\right )}{d^8 \sqrt {\left (b^2-a^2\right ) e^{2 i c}}}\right ) \sec \left (c+d \sqrt {x}\right )}{4 a \left (a+b \sec \left (c+d \sqrt {x}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3}}{b \sec \left (d \sqrt {x} + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{b \sec \left (d \sqrt {x} + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.13, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{a +b \sec \left (c +d \sqrt {x}\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^3}{a+\frac {b}{\cos \left (c+d\,\sqrt {x}\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{a + b \sec {\left (c + d \sqrt {x} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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