Optimal. Leaf size=509 \[ -\frac {175 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {105 b^{7/2} e^2 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}-\frac {35 b^{7/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3456 d}+\frac {105 b^{7/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {35 b^{7/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3456 d} \]
[Out]
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Rubi [A]
time = 1.37, antiderivative size = 509, normalized size of antiderivative = 1.00, number
of steps used = 35, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules
used = {5996, 12, 5884, 5939, 5915, 5879, 5881, 3389, 2211, 2236, 2235, 5887, 5556}
\begin {gather*} -\frac {105 \sqrt {\pi } b^{7/2} e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}-\frac {35 \sqrt {\frac {\pi }{3}} b^{7/2} e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3456 d}+\frac {105 \sqrt {\pi } b^{7/2} e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {35 \sqrt {\frac {\pi }{3}} b^{7/2} e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3456 d}-\frac {35 b^3 e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}-\frac {175 b^3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}-\frac {7 b e^2 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}-\frac {7 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5879
Rule 5881
Rule 5884
Rule 5887
Rule 5915
Rule 5939
Rule 5996
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (7 b e^2\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{6 d}\\ &=-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (7 b e^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}+\frac {\left (35 b^2 e^2\right ) \text {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{36 d}\\ &=\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^2 e^2\right ) \text {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{18 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3 \sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{108 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {x \sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{12 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{432 d}\\ &=-\frac {175 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{432 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{216 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{24 d}\\ &=-\frac {175 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{216 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{24 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 \sqrt {a+b x}}+\frac {\sinh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{432 d}\\ &=-\frac {175 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{432 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{432 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{48 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{48 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1728 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1728 d}\\ &=-\frac {175 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{216 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{216 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{24 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{24 d}-\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}-\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}+\frac {\left (35 b^4 e^2\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3456 d}\\ &=-\frac {175 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {175 b^{7/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{216 d}+\frac {175 b^{7/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{216 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}-\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}+\frac {\left (35 b^3 e^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{1728 d}\\ &=-\frac {175 b^3 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{54 d}-\frac {35 b^3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{216 d}+\frac {35 b^2 e^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {35 b^2 e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{108 d}-\frac {7 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{9 d}-\frac {7 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{3 d}-\frac {105 b^{7/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}-\frac {35 b^{7/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3456 d}+\frac {105 b^{7/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{128 d}+\frac {35 b^{7/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3456 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1523\) vs. \(2(509)=1018\).
time = 11.92, size = 1523, normalized size = 2.99 \begin {gather*} e^2 \left (\frac {a^3 e^{-\frac {3 a}{b}} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (9 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+9 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {3}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {3}{2},\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )}{72 d \sqrt {-\frac {\left (a+b \cosh ^{-1}(c+d x)\right )^2}{b^2}}}+\frac {a^2 \sqrt {b} \left (9 \left (-12 \sqrt {b} \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}+8 \sqrt {b} (c+d x) \cosh ^{-1}(c+d x) \sqrt {a+b \cosh ^{-1}(c+d x)}+(2 a+3 b) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+(2 a-3 b) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+(2 a+b) \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )+(2 a-b) \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 \sqrt {b} \sqrt {a+b \cosh ^{-1}(c+d x)} \left (2 \cosh ^{-1}(c+d x) \cosh \left (3 \cosh ^{-1}(c+d x)\right )-\sinh \left (3 \cosh ^{-1}(c+d x)\right )\right )\right )}{96 d}+\frac {a \left (-27 \left (-4 b \sqrt {a+b \cosh ^{-1}(c+d x)} \left (2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \left (a-5 b \cosh ^{-1}(c+d x)\right )+b (c+d x) \left (15+4 \cosh ^{-1}(c+d x)^2\right )\right )+\sqrt {b} \left (4 a^2+12 a b+15 b^2\right ) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )-\sinh \left (\frac {a}{b}\right )\right )+\sqrt {b} \left (4 a^2-12 a b+15 b^2\right ) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )-\sqrt {b} \left (12 a^2+12 a b+5 b^2\right ) \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )-\sqrt {b} \left (12 a^2-12 a b+5 b^2\right ) \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )+12 b \sqrt {a+b \cosh ^{-1}(c+d x)} \left (b \left (5+12 \cosh ^{-1}(c+d x)^2\right ) \cosh \left (3 \cosh ^{-1}(c+d x)\right )+2 \left (a-5 b \cosh ^{-1}(c+d x)\right ) \sinh \left (3 \cosh ^{-1}(c+d x)\right )\right )\right )}{576 d}+\frac {-81 \left (4 b \sqrt {a+b \cosh ^{-1}(c+d x)} \left (\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \left (4 a^2-4 a b \cosh ^{-1}(c+d x)+7 b^2 \left (15+4 \cosh ^{-1}(c+d x)^2\right )\right )-2 b (c+d x) \left (-10 a+b \cosh ^{-1}(c+d x) \left (35+4 \cosh ^{-1}(c+d x)^2\right )\right )\right )+\sqrt {b} \left (8 a^3+36 a^2 b+90 a b^2+105 b^3\right ) \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (-\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+\sqrt {b} \left (-8 a^3+36 a^2 b-90 a b^2+105 b^3\right ) \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )\right )+\sqrt {b} \left (72 a^3+108 a^2 b+90 a b^2+35 b^3\right ) \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )-\sinh \left (\frac {3 a}{b}\right )\right )-\sqrt {b} \left (-72 a^3+108 a^2 b-90 a b^2+35 b^3\right ) \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {3 a}{b}\right )+\sinh \left (\frac {3 a}{b}\right )\right )-12 b \sqrt {a+b \cosh ^{-1}(c+d x)} \left (-2 b \left (-10 a+b \cosh ^{-1}(c+d x) \left (35+36 \cosh ^{-1}(c+d x)^2\right )\right ) \cosh \left (3 \cosh ^{-1}(c+d x)\right )+\left (12 a^2-12 a b \cosh ^{-1}(c+d x)+7 b^2 \left (5+12 \cosh ^{-1}(c+d x)^2\right )\right ) \sinh \left (3 \cosh ^{-1}(c+d x)\right )\right )}{10368 d}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (d e x +c e \right )^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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