Optimal. Leaf size=352 \[ \frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^4 d}-\frac {3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]
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Rubi [A] time = 0.96, antiderivative size = 348, normalized size of antiderivative = 0.99, number of steps used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5866, 12, 5668, 5775, 5666, 3303, 3298, 3301, 5656, 5781} \[ \frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^4 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^4 d}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {3 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
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 )^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}-\frac {\left (2 e^2\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}+\frac {e^2 \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{b d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 \operatorname {Subst}\left (\int \frac {1}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \operatorname {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )} \, dx,x,c+d x\right )}{3 b^3 d}-\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 (a+b x)}-\frac {3 \cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (3 e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {\left (9 e^2\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^3 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (3 e^2 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {\left (9 e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (3 e^2 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {\left (9 e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^3 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^2 (c+d x)}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{24 b^4 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^4 d}-\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{24 b^4 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^4 d}\\ \end {align*}
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Mathematica [A] time = 1.33, size = 272, normalized size = 0.77 \[ \frac {e^2 \left (-\frac {8 b^3 \sqrt {c+d x-1} (c+d x)^2 \sqrt {c+d x+1}}{\left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {4 b^2 \left (2 (c+d x)-3 (c+d x)^3\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+27 \left (3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )-3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )-80 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+80 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (9 (c+d x)^2-2\right )}{a+b \cosh ^{-1}(c+d x)}\right )}{24 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}}{b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}}, x\right ) \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 777, normalized size = 2.21 \[ \frac {\frac {\left (-4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (9 b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+18 a b \,\mathrm {arccosh}\left (d x +c \right )-3 \,\mathrm {arccosh}\left (d x +c \right ) b^{2}+9 a^{2}-3 a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\mathrm {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \Ei \left (1, 3 \,\mathrm {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{4}}+\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )-\mathrm {arccosh}\left (d x +c \right ) b^{2}+a^{2}-a b +2 b^{2}\right )}{48 b^{3} \left (b^{3} \mathrm {arccosh}\left (d x +c \right )^{3}+3 a \,b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+3 a^{2} b \,\mathrm {arccosh}\left (d x +c \right )+a^{3}\right )}-\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \Ei \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{48 b^{3} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \Ei \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{48 b^{4}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{24 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{3} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \Ei \left (1, -3 \,\mathrm {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{4}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \]
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 \[ e^{2} \left (\int \frac {c^{2}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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