Optimal. Leaf size=431 \[ \frac {\sqrt {\pi } e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 \sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 \sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {24 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
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Rubi [A] time = 1.47, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5866, 12, 5668, 5775, 5666, 3307, 2180, 2204, 2205, 5656, 5781} \[ \frac {\sqrt {\pi } e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 \sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 \sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {24 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5656
Rule 5666
Rule 5668
Rule 5775
Rule 5781
Rule 5866
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}-\frac {\left (4 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\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 (6 e^2\right ) \operatorname {Subst}\left (\int \frac {x^3}{\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^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {\left (8 e^2\right ) \operatorname {Subst}\left (\int \frac {1}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}+\frac {\left (12 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{15 b^3 d}-\frac {\left (24 e^2\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (6 e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (18 e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (8 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}-\frac {\left (8 e^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (9 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (9 e^2\right ) \operatorname {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}-\frac {\left (16 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (6 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac {\left (6 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac {\left (18 e^2\right ) \operatorname {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac {\left (18 e^2\right ) \operatorname {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}\\ \end {align*}
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Mathematica [A] time = 2.92, size = 452, normalized size = 1.05 \[ \frac {e^2 \left (-2 e^{-\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (2 e^{\frac {a}{b}+\cosh ^{-1}(c+d x)} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-2 a-2 b \cosh ^{-1}(c+d x)+b\right )-2 e^{-\frac {a}{b}} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (e^{\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (2 a+2 b \cosh ^{-1}(c+d x)+b\right )+2 b \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )\right )-3 \left (a+b \cosh ^{-1}(c+d x)\right ) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+2 e^{-3 \cosh ^{-1}(c+d x)} \left (6 \sqrt {3} e^{3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+6 a \left (e^{6 \cosh ^{-1}(c+d x)}-1\right )-6 b \cosh ^{-1}(c+d x)+b e^{6 \cosh ^{-1}(c+d x)} \left (6 \cosh ^{-1}(c+d x)+1\right )+b\right )\right )-6 b^2 \sqrt {\frac {c+d x-1}{c+d x+1}} (c+d x+1)-6 b^2 \sinh \left (3 \cosh ^{-1}(c+d x)\right )\right )}{60 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}\, dx \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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 \[ e^{2} \left (\int \frac {c^{2}}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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