Optimal. Leaf size=137 \[ -\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1}}{40 d e^6 (c+d x)^2}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{20 d e^6 (c+d x)^4}+\frac {3 b \tan ^{-1}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{40 d e^6} \]
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Rubi [A] time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5866, 12, 5662, 103, 92, 203} \[ -\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1}}{40 d e^6 (c+d x)^2}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{20 d e^6 (c+d x)^4}+\frac {3 b \tan ^{-1}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{40 d e^6} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 103
Rule 203
Rule 5662
Rule 5866
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^6} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e^6 x^6} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{x^6} \, dx,x,c+d x\right )}{d e^6}\\ &=-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^5 \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \operatorname {Subst}\left (\int \frac {3}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{20 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{20 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{40 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{40 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \tan ^{-1}\left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{40 d e^6}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 136, normalized size = 0.99 \[ \frac {-\frac {a+b \cosh ^{-1}(c+d x)}{(c+d x)^5}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{4 (c+d x)^4}+\frac {3 b \left (\frac {(c+d x-1) (c+d x+1)}{(c+d x)^2}+\sqrt {(c+d x)^2-1} \tan ^{-1}\left (\sqrt {(c+d x)^2-1}\right )\right )}{8 \sqrt {c+d x-1} \sqrt {c+d x+1}}}{5 d e^6} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.78, size = 416, normalized size = 3.04 \[ -\frac {8 \, a c^{5} - 6 \, {\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (3 \, b c^{5} d^{3} x^{3} + 9 \, b c^{6} d^{2} x^{2} + 3 \, b c^{8} + 2 \, b c^{6} + {\left (9 \, b c^{7} + 2 \, b c^{5}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{40 \, {\left (c^{5} d^{6} e^{6} x^{5} + 5 \, c^{6} d^{5} e^{6} x^{4} + 10 \, c^{7} d^{4} e^{6} x^{3} + 10 \, c^{8} d^{3} e^{6} x^{2} + 5 \, c^{9} d^{2} e^{6} x + c^{10} d e^{6}\right )}} \]
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 \[ \int \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 152, normalized size = 1.11 \[ -\frac {a}{5 d \,e^{6} \left (d x +c \right )^{5}}-\frac {b \,\mathrm {arccosh}\left (d x +c \right )}{5 d \,e^{6} \left (d x +c \right )^{5}}-\frac {3 b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{40 d \,e^{6} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {3 b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{40 d \,e^{6} \left (d x +c \right )^{2}}+\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{20 d \,e^{6} \left (d x +c \right )^{4}} \]
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 \[ \frac {1}{30} \, b {\left (\frac {6 \, d^{4} x^{4} + 24 \, c d^{3} x^{3} + 6 \, c^{4} + 2 \, {\left (18 \, c^{2} d^{2} + d^{2}\right )} x^{2} + 2 \, c^{2} + 4 \, {\left (6 \, c^{3} d + c d\right )} x - 3 \, {\left (d^{5} x^{5} + 5 \, c d^{4} x^{4} + 10 \, c^{2} d^{3} x^{3} + 10 \, c^{3} d^{2} x^{2} + 5 \, c^{4} d x + c^{5}\right )} \log \left (d x + c + 1\right ) + 3 \, {\left (d^{5} x^{5} + 5 \, c d^{4} x^{4} + 10 \, c^{2} d^{3} x^{3} + 10 \, c^{3} d^{2} x^{2} + 5 \, c^{4} d x + c^{5}\right )} \log \left (d x + c - 1\right ) - 6 \, \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}} - 30 \, \int \frac {1}{5 \, {\left (d^{8} e^{6} x^{8} + 8 \, c d^{7} e^{6} x^{7} + c^{8} e^{6} - c^{6} e^{6} + {\left (28 \, c^{2} d^{6} e^{6} - d^{6} e^{6}\right )} x^{6} + 2 \, {\left (28 \, c^{3} d^{5} e^{6} - 3 \, c d^{5} e^{6}\right )} x^{5} + 5 \, {\left (14 \, c^{4} d^{4} e^{6} - 3 \, c^{2} d^{4} e^{6}\right )} x^{4} + 4 \, {\left (14 \, c^{5} d^{3} e^{6} - 5 \, c^{3} d^{3} e^{6}\right )} x^{3} + {\left (28 \, c^{6} d^{2} e^{6} - 15 \, c^{4} d^{2} e^{6}\right )} x^{2} + 2 \, {\left (4 \, c^{7} d e^{6} - 3 \, c^{5} d e^{6}\right )} x + {\left (d^{7} e^{6} x^{7} + 7 \, c d^{6} e^{6} x^{6} + c^{7} e^{6} - c^{5} e^{6} + {\left (21 \, c^{2} d^{5} e^{6} - d^{5} e^{6}\right )} x^{5} + 5 \, {\left (7 \, c^{3} d^{4} e^{6} - c d^{4} e^{6}\right )} x^{4} + 5 \, {\left (7 \, c^{4} d^{3} e^{6} - 2 \, c^{2} d^{3} e^{6}\right )} x^{3} + {\left (21 \, c^{5} d^{2} e^{6} - 10 \, c^{3} d^{2} e^{6}\right )} x^{2} + {\left (7 \, c^{6} d e^{6} - 5 \, c^{4} d e^{6}\right )} x\right )} e^{\left (\frac {1}{2} \, \log \left (d x + c + 1\right ) + \frac {1}{2} \, \log \left (d x + c - 1\right )\right )}\right )}}\,{d x}\right )} - \frac {a}{5 \, {\left (d^{6} e^{6} x^{5} + 5 \, c d^{5} e^{6} x^{4} + 10 \, c^{2} d^{4} e^{6} x^{3} + 10 \, c^{3} d^{3} e^{6} x^{2} + 5 \, c^{4} d^{2} e^{6} x + c^{5} d e^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^6} \,d x \]
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 \[ \frac {\int \frac {a}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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