Optimal. Leaf size=597 \[ -\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \text {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \text {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \text {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \text {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \text {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \text {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \text {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \text {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \text {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \text {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )}{d^7}-\frac {315 b^2 \text {PolyLog}\left (7,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 a b \text {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \text {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )}{d^8} \]
[Out]
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Rubi [A]
time = 0.59, antiderivative size = 597, normalized size of antiderivative = 1.00, number of steps
used = 30, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5545, 4275,
4267, 2611, 6744, 2320, 6724, 4269, 3797, 2221} \begin {gather*} \frac {a^2 x^4}{4}-\frac {20160 a b \text {Li}_8\left (-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \text {Li}_8\left (e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \sqrt {x} \text {Li}_7\left (-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \text {Li}_7\left (e^{c+d \sqrt {x}}\right )}{d^7}-\frac {10080 a b x \text {Li}_6\left (-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \text {Li}_6\left (e^{c+d \sqrt {x}}\right )}{d^6}+\frac {3360 a b x^{3/2} \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {315 b^2 \text {Li}_7\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}+\frac {315 b^2 \sqrt {x} \text {Li}_6\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}-\frac {315 b^2 x \text {Li}_5\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}+\frac {210 b^2 x^{3/2} \text {Li}_4\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}-\frac {105 b^2 x^2 \text {Li}_3\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {2 b^2 x^{7/2}}{d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2320
Rule 2611
Rule 3797
Rule 4267
Rule 4269
Rule 4275
Rule 5545
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int x^3 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \, dx &=2 \text {Subst}\left (\int x^7 (a+b \text {csch}(c+d x))^2 \, dx,x,\sqrt {x}\right )\\ &=2 \text {Subst}\left (\int \left (a^2 x^7+2 a b x^7 \text {csch}(c+d x)+b^2 x^7 \text {csch}^2(c+d x)\right ) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^2 x^4}{4}+(4 a b) \text {Subst}\left (\int x^7 \text {csch}(c+d x) \, dx,x,\sqrt {x}\right )+\left (2 b^2\right ) \text {Subst}\left (\int x^7 \text {csch}^2(c+d x) \, dx,x,\sqrt {x}\right )\\ &=\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {(28 a b) \text {Subst}\left (\int x^6 \log \left (1+e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d}+\frac {\left (14 b^2\right ) \text {Subst}\left (\int x^6 \coth (c+d x) \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {(168 a b) \text {Subst}\left (\int x^5 \text {Li}_2\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {(168 a b) \text {Subst}\left (\int x^5 \text {Li}_2\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^2}-\frac {\left (28 b^2\right ) \text {Subst}\left (\int \frac {e^{2 (c+d x)} x^6}{1-e^{2 (c+d x)}} \, dx,x,\sqrt {x}\right )}{d}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {(840 a b) \text {Subst}\left (\int x^4 \text {Li}_3\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}+\frac {(840 a b) \text {Subst}\left (\int x^4 \text {Li}_3\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^3}-\frac {\left (84 b^2\right ) \text {Subst}\left (\int x^5 \log \left (1-e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^2}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {(3360 a b) \text {Subst}\left (\int x^3 \text {Li}_4\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {(3360 a b) \text {Subst}\left (\int x^3 \text {Li}_4\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^4}-\frac {\left (210 b^2\right ) \text {Subst}\left (\int x^4 \text {Li}_2\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^3}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \text {Li}_3\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {3360 a b x^{3/2} \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {(10080 a b) \text {Subst}\left (\int x^2 \text {Li}_5\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {(10080 a b) \text {Subst}\left (\int x^2 \text {Li}_5\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^5}+\frac {\left (420 b^2\right ) \text {Subst}\left (\int x^3 \text {Li}_3\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^4}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \text {Li}_3\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \text {Li}_4\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {10080 a b x \text {Li}_6\left (-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \text {Li}_6\left (e^{c+d \sqrt {x}}\right )}{d^6}+\frac {(20160 a b) \text {Subst}\left (\int x \text {Li}_6\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^6}-\frac {(20160 a b) \text {Subst}\left (\int x \text {Li}_6\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^6}-\frac {\left (630 b^2\right ) \text {Subst}\left (\int x^2 \text {Li}_4\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^5}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \text {Li}_3\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \text {Li}_4\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \text {Li}_5\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \text {Li}_6\left (-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \text {Li}_6\left (e^{c+d \sqrt {x}}\right )}{d^6}+\frac {20160 a b \sqrt {x} \text {Li}_7\left (-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \text {Li}_7\left (e^{c+d \sqrt {x}}\right )}{d^7}-\frac {(20160 a b) \text {Subst}\left (\int \text {Li}_7\left (-e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^7}+\frac {(20160 a b) \text {Subst}\left (\int \text {Li}_7\left (e^{c+d x}\right ) \, dx,x,\sqrt {x}\right )}{d^7}+\frac {\left (630 b^2\right ) \text {Subst}\left (\int x \text {Li}_5\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^6}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \text {Li}_3\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \text {Li}_4\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \text {Li}_5\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \text {Li}_6\left (-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \text {Li}_6\left (e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \text {Li}_6\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \text {Li}_7\left (-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \text {Li}_7\left (e^{c+d \sqrt {x}}\right )}{d^7}-\frac {(20160 a b) \text {Subst}\left (\int \frac {\text {Li}_7(-x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^8}+\frac {(20160 a b) \text {Subst}\left (\int \frac {\text {Li}_7(x)}{x} \, dx,x,e^{c+d \sqrt {x}}\right )}{d^8}-\frac {\left (315 b^2\right ) \text {Subst}\left (\int \text {Li}_6\left (e^{2 (c+d x)}\right ) \, dx,x,\sqrt {x}\right )}{d^7}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \text {Li}_3\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \text {Li}_4\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \text {Li}_5\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \text {Li}_6\left (-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \text {Li}_6\left (e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \text {Li}_6\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \text {Li}_7\left (-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \text {Li}_7\left (e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \text {Li}_8\left (-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \text {Li}_8\left (e^{c+d \sqrt {x}}\right )}{d^8}-\frac {\left (315 b^2\right ) \text {Subst}\left (\int \frac {\text {Li}_6(x)}{x} \, dx,x,e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}\\ &=-\frac {2 b^2 x^{7/2}}{d}+\frac {a^2 x^4}{4}-\frac {8 a b x^{7/2} \tanh ^{-1}\left (e^{c+d \sqrt {x}}\right )}{d}-\frac {2 b^2 x^{7/2} \coth \left (c+d \sqrt {x}\right )}{d}+\frac {14 b^2 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^2}-\frac {28 a b x^3 \text {Li}_2\left (-e^{c+d \sqrt {x}}\right )}{d^2}+\frac {28 a b x^3 \text {Li}_2\left (e^{c+d \sqrt {x}}\right )}{d^2}+\frac {42 b^2 x^{5/2} \text {Li}_2\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^3}+\frac {168 a b x^{5/2} \text {Li}_3\left (-e^{c+d \sqrt {x}}\right )}{d^3}-\frac {168 a b x^{5/2} \text {Li}_3\left (e^{c+d \sqrt {x}}\right )}{d^3}-\frac {105 b^2 x^2 \text {Li}_3\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^4}-\frac {840 a b x^2 \text {Li}_4\left (-e^{c+d \sqrt {x}}\right )}{d^4}+\frac {840 a b x^2 \text {Li}_4\left (e^{c+d \sqrt {x}}\right )}{d^4}+\frac {210 b^2 x^{3/2} \text {Li}_4\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^5}+\frac {3360 a b x^{3/2} \text {Li}_5\left (-e^{c+d \sqrt {x}}\right )}{d^5}-\frac {3360 a b x^{3/2} \text {Li}_5\left (e^{c+d \sqrt {x}}\right )}{d^5}-\frac {315 b^2 x \text {Li}_5\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^6}-\frac {10080 a b x \text {Li}_6\left (-e^{c+d \sqrt {x}}\right )}{d^6}+\frac {10080 a b x \text {Li}_6\left (e^{c+d \sqrt {x}}\right )}{d^6}+\frac {315 b^2 \sqrt {x} \text {Li}_6\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{d^7}+\frac {20160 a b \sqrt {x} \text {Li}_7\left (-e^{c+d \sqrt {x}}\right )}{d^7}-\frac {20160 a b \sqrt {x} \text {Li}_7\left (e^{c+d \sqrt {x}}\right )}{d^7}-\frac {315 b^2 \text {Li}_7\left (e^{2 \left (c+d \sqrt {x}\right )}\right )}{2 d^8}-\frac {20160 a b \text {Li}_8\left (-e^{c+d \sqrt {x}}\right )}{d^8}+\frac {20160 a b \text {Li}_8\left (e^{c+d \sqrt {x}}\right )}{d^8}\\ \end {align*}
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Mathematica [A]
time = 9.98, size = 847, normalized size = 1.42 \begin {gather*} \frac {a^2 x^4 \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \sinh ^2\left (c+d \sqrt {x}\right )}{4 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}+\frac {b \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \left (-\frac {8 b d^7 e^{2 c} x^{7/2}}{-1+e^{2 c}}+8 a d^7 x^{7/2} \log \left (1-e^{c+d \sqrt {x}}\right )-8 a d^7 x^{7/2} \log \left (1+e^{c+d \sqrt {x}}\right )+28 b d^6 x^3 \log \left (1-e^{2 \left (c+d \sqrt {x}\right )}\right )-56 a d^6 x^3 \text {PolyLog}\left (2,-e^{c+d \sqrt {x}}\right )+56 a d^6 x^3 \text {PolyLog}\left (2,e^{c+d \sqrt {x}}\right )+84 b d^5 x^{5/2} \text {PolyLog}\left (2,e^{2 \left (c+d \sqrt {x}\right )}\right )+336 a d^5 x^{5/2} \text {PolyLog}\left (3,-e^{c+d \sqrt {x}}\right )-336 a d^5 x^{5/2} \text {PolyLog}\left (3,e^{c+d \sqrt {x}}\right )-210 b d^4 x^2 \text {PolyLog}\left (3,e^{2 \left (c+d \sqrt {x}\right )}\right )-1680 a d^4 x^2 \text {PolyLog}\left (4,-e^{c+d \sqrt {x}}\right )+1680 a d^4 x^2 \text {PolyLog}\left (4,e^{c+d \sqrt {x}}\right )+420 b d^3 x^{3/2} \text {PolyLog}\left (4,e^{2 \left (c+d \sqrt {x}\right )}\right )+6720 a d^3 x^{3/2} \text {PolyLog}\left (5,-e^{c+d \sqrt {x}}\right )-6720 a d^3 x^{3/2} \text {PolyLog}\left (5,e^{c+d \sqrt {x}}\right )-630 b d^2 x \text {PolyLog}\left (5,e^{2 \left (c+d \sqrt {x}\right )}\right )-20160 a d^2 x \text {PolyLog}\left (6,-e^{c+d \sqrt {x}}\right )+20160 a d^2 x \text {PolyLog}\left (6,e^{c+d \sqrt {x}}\right )+630 b d \sqrt {x} \text {PolyLog}\left (6,e^{2 \left (c+d \sqrt {x}\right )}\right )+40320 a d \sqrt {x} \text {PolyLog}\left (7,-e^{c+d \sqrt {x}}\right )-40320 a d \sqrt {x} \text {PolyLog}\left (7,e^{c+d \sqrt {x}}\right )-315 b \text {PolyLog}\left (7,e^{2 \left (c+d \sqrt {x}\right )}\right )-40320 a \text {PolyLog}\left (8,-e^{c+d \sqrt {x}}\right )+40320 a \text {PolyLog}\left (8,e^{c+d \sqrt {x}}\right )\right ) \sinh ^2\left (c+d \sqrt {x}\right )}{2 d^8 \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}+\frac {b^2 x^{7/2} \text {csch}\left (\frac {c}{2}\right ) \text {csch}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \sinh ^2\left (c+d \sqrt {x}\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2}-\frac {b^2 x^{7/2} \left (a+b \text {csch}\left (c+d \sqrt {x}\right )\right )^2 \text {sech}\left (\frac {c}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d \sqrt {x}}{2}\right ) \sinh ^2\left (c+d \sqrt {x}\right ) \sinh \left (\frac {d \sqrt {x}}{2}\right )}{d \left (b+a \sinh \left (c+d \sqrt {x}\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{3} \left (a +b \,\mathrm {csch}\left (c +d \sqrt {x}\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.43, size = 648, normalized size = 1.09 \begin {gather*} \frac {1}{4} \, a^{2} x^{4} - \frac {4 \, b^{2} x^{\frac {7}{2}}}{d e^{\left (2 \, d \sqrt {x} + 2 \, c\right )} - d} - \frac {4 \, {\left (d^{7} x^{\frac {7}{2}} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 7 \, d^{6} x^{3} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 42 \, d^{5} x^{\frac {5}{2}} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 210 \, d^{4} x^{2} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 840 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )}) + 2520 \, d^{2} x {\rm Li}_{6}(-e^{\left (d \sqrt {x} + c\right )}) - 5040 \, d \sqrt {x} {\rm Li}_{7}(-e^{\left (d \sqrt {x} + c\right )}) + 5040 \, {\rm Li}_{8}(-e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{8}} + \frac {4 \, {\left (d^{7} x^{\frac {7}{2}} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 7 \, d^{6} x^{3} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 42 \, d^{5} x^{\frac {5}{2}} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 210 \, d^{4} x^{2} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 840 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )}) + 2520 \, d^{2} x {\rm Li}_{6}(e^{\left (d \sqrt {x} + c\right )}) - 5040 \, d \sqrt {x} {\rm Li}_{7}(e^{\left (d \sqrt {x} + c\right )}) + 5040 \, {\rm Li}_{8}(e^{\left (d \sqrt {x} + c\right )})\right )} a b}{d^{8}} + \frac {14 \, {\left (d^{6} x^{3} \log \left (e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 6 \, d^{5} x^{\frac {5}{2}} {\rm Li}_2\left (-e^{\left (d \sqrt {x} + c\right )}\right ) - 30 \, d^{4} x^{2} {\rm Li}_{3}(-e^{\left (d \sqrt {x} + c\right )}) + 120 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{4}(-e^{\left (d \sqrt {x} + c\right )}) - 360 \, d^{2} x {\rm Li}_{5}(-e^{\left (d \sqrt {x} + c\right )}) + 720 \, d \sqrt {x} {\rm Li}_{6}(-e^{\left (d \sqrt {x} + c\right )}) - 720 \, {\rm Li}_{7}(-e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{8}} + \frac {14 \, {\left (d^{6} x^{3} \log \left (-e^{\left (d \sqrt {x} + c\right )} + 1\right ) + 6 \, d^{5} x^{\frac {5}{2}} {\rm Li}_2\left (e^{\left (d \sqrt {x} + c\right )}\right ) - 30 \, d^{4} x^{2} {\rm Li}_{3}(e^{\left (d \sqrt {x} + c\right )}) + 120 \, d^{3} x^{\frac {3}{2}} {\rm Li}_{4}(e^{\left (d \sqrt {x} + c\right )}) - 360 \, d^{2} x {\rm Li}_{5}(e^{\left (d \sqrt {x} + c\right )}) + 720 \, d \sqrt {x} {\rm Li}_{6}(e^{\left (d \sqrt {x} + c\right )}) - 720 \, {\rm Li}_{7}(e^{\left (d \sqrt {x} + c\right )})\right )} b^{2}}{d^{8}} - \frac {a b d^{8} x^{4} + 4 \, b^{2} d^{7} x^{\frac {7}{2}}}{2 \, d^{8}} + \frac {a b d^{8} x^{4} - 4 \, b^{2} d^{7} x^{\frac {7}{2}}}{2 \, d^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{3} \left (a + b \operatorname {csch}{\left (c + d \sqrt {x} \right )}\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^3\,{\left (a+\frac {b}{\mathrm {sinh}\left (c+d\,\sqrt {x}\right )}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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