Optimal. Leaf size=319 \[ -\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}-\frac {105 b^3 e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {7 b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d} \]
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Rubi [A] time = 1.28, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5866, 12, 5664, 5759, 5676, 5670, 5448, 3308, 2180, 2204, 2205} \[ -\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}+\frac {105 \sqrt {\frac {\pi }{2}} b^{7/2} e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}-\frac {105 b^3 e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}-\frac {7 b e \sqrt {c+d x-1} (c+d x) \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3308
Rule 5448
Rule 5664
Rule 5670
Rule 5676
Rule 5759
Rule 5866
Rubi steps
\begin {align*} \int (c e+d e x) \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2} \, dx &=\frac {\operatorname {Subst}\left (\int e x \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^{7/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {(7 b e) \operatorname {Subst}\left (\int \frac {x^2 \left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {(7 b e) \operatorname {Subst}\left (\int \frac {\left (a+b \cosh ^{-1}(x)\right )^{5/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{8 d}+\frac {\left (35 b^2 e\right ) \operatorname {Subst}\left (\int x \left (a+b \cosh ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{16 d}\\ &=\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac {105 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b \cosh ^{-1}(x)}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{128 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{256 d}\\ &=-\frac {105 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {105 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{256 d}\\ &=-\frac {105 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{512 d}\\ &=-\frac {105 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (105 b^4 e\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{1024 d}\\ &=-\frac {105 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {\left (105 b^3 e\right ) \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{512 d}+\frac {\left (105 b^3 e\right ) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{512 d}\\ &=-\frac {105 b^3 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \sqrt {a+b \cosh ^{-1}(c+d x)}}{128 d}-\frac {35 b^2 e \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {35 b^2 e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {7 b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}{8 d}-\frac {e \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{4 d}+\frac {e (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}}{2 d}-\frac {105 b^{7/2} e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}+\frac {105 b^{7/2} e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{1024 d}\\ \end {align*}
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Mathematica [A] time = 5.75, size = 288, normalized size = 0.90 \[ \frac {e \left (8 \sqrt {a+b \cosh ^{-1}(c+d x)} \left (4 a \left (16 a^2+35 b^2\right ) \cosh \left (2 \cosh ^{-1}(c+d x)\right )+4 b \cosh ^{-1}(c+d x) \left (\left (48 a^2+35 b^2\right ) \cosh \left (2 \cosh ^{-1}(c+d x)\right )-56 a b \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )-7 b \left (16 a^2+15 b^2\right ) \sinh \left (2 \cosh ^{-1}(c+d x)\right )+16 b^2 \cosh ^{-1}(c+d x)^2 \left (12 a \cosh \left (2 \cosh ^{-1}(c+d x)\right )-7 b \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )+64 b^3 \cosh \left (2 \cosh ^{-1}(c+d x)\right ) \cosh ^{-1}(c+d x)^3\right )-105 \sqrt {2 \pi } b^{7/2} \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+105 \sqrt {2 \pi } b^{7/2} \left (\cosh \left (\frac {2 a}{b}\right )-\sinh \left (\frac {2 a}{b}\right )\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )\right )}{2048 d} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \left (d e x +c e \right ) \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}\, dx \]
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 \[ \int {\left (d e x + c e\right )} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}\,{d x} \]
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 \[ \int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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