Optimal. Leaf size=441 \[ \frac {16 \sqrt {\pi } e^3 e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 \sqrt {2 \pi } e^3 e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {16 \sqrt {\pi } e^3 e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 \sqrt {2 \pi } e^3 e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {128 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
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Rubi [A] time = 1.41, antiderivative size = 441, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5866, 12, 5668, 5775, 5666, 3307, 2180, 2204, 2205} \[ \frac {16 \sqrt {\pi } e^3 e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 \sqrt {2 \pi } e^3 e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {16 \sqrt {\pi } e^3 e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 \sqrt {2 \pi } e^3 e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {128 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5666
Rule 5668
Rule 5775
Rule 5866
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^3}{\left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}-\frac {\left (6 e^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}+\frac {\left (8 e^3\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {\left (8 e^3\right ) \operatorname {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}+\frac {\left (64 e^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}-\frac {\left (128 e^3\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 \sqrt {a+b x}}-\frac {\cosh (4 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (8 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}-\frac {\left (8 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (64 e^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (64 e^3\right ) \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^3\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 )}{5 b^4 d}-\frac {\left (16 e^3\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 )}{5 b^4 d}+\frac {\left (32 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (32 e^3\right ) \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^3 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {4 e^3 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {\left (64 e^3\right ) \operatorname {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (64 e^3\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 )}{15 b^4 d}+\frac {\left (64 e^3\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 )}{15 b^4 d}+\frac {\left (64 e^3\right ) \operatorname {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}\\ &=-\frac {2 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {4 e^3 (c+d x)^2}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {16 e^3 (c+d x)^4}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {128 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^3 e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 e^3 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {16 e^3 e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {4 e^3 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}\\ \end {align*}
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Mathematica [A] time = 3.10, size = 445, normalized size = 1.01 \[ \frac {e^3 \left (-2 \left (\left (a+b \cosh ^{-1}(c+d x)\right ) \left (2 e^{-2 \cosh ^{-1}(c+d x)} \left (4 a \left (e^{4 \cosh ^{-1}(c+d x)}-1\right )-4 b \cosh ^{-1}(c+d x)+b e^{4 \cosh ^{-1}(c+d x)} \left (4 \cosh ^{-1}(c+d x)+1\right )+b\right )+8 \sqrt {2} b e^{-\frac {2 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+8 \sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )+3 b^2 \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )-4 e^{-4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (16 b e^{4 \cosh ^{-1}(c+d x)} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {4 a}{b}} \left (16 e^{4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+8 a \left (e^{8 \cosh ^{-1}(c+d x)}-1\right )-8 b \cosh ^{-1}(c+d x)+b e^{8 \cosh ^{-1}(c+d x)} \left (8 \cosh ^{-1}(c+d x)+1\right )+b\right )\right )-3 b^2 \sinh \left (4 \cosh ^{-1}(c+d x)\right )\right )}{60 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (d e x +c e \right )^{3}}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{3}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ e^{3} \left (\int \frac {c^{3}}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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