Optimal. Leaf size=99 \[ -\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{6 d e^4 (c+d x)^2}+\frac {b \tan ^{-1}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{6 d e^4} \]
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Rubi [A] time = 0.07, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{6 d e^4 (c+d x)^2}+\frac {b \tan ^{-1}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{6 d e^4} \]
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)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e^4}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{6 d e^4}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{6 d e^4}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \tan ^{-1}\left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{6 d e^4}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 101, normalized size = 1.02 \[ \frac {\frac {b \left (\frac {(c+d x-1) (c+d x+1)}{(c+d x)^2}+\sqrt {(c+d x)^2-1} \tan ^{-1}\left (\sqrt {(c+d x)^2-1}\right )\right )}{\sqrt {c+d x-1} \sqrt {c+d x+1}}-\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{(c+d x)^3}}{6 d e^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.98, size = 276, normalized size = 2.79 \[ -\frac {2 \, a c^{3} - 2 \, {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b c^{3} d x + b c^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{6 \, {\left (c^{3} d^{4} e^{4} x^{3} + 3 \, c^{4} d^{3} e^{4} x^{2} + 3 \, c^{5} d^{2} e^{4} x + c^{6} d e^{4}\right )}} \]
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 \[ \int \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 120, normalized size = 1.21 \[ -\frac {a}{3 d \,e^{4} \left (d x +c \right )^{3}}-\frac {b \,\mathrm {arccosh}\left (d x +c \right )}{3 d \,e^{4} \left (d x +c \right )^{3}}-\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{6 d \,e^{4} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{6 d \,e^{4} \left (d x +c \right )^{2}} \]
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 \[ \frac {1}{6} \, b {\left (\frac {2 \, d^{2} x^{2} + 4 \, c d x + 2 \, c^{2} - {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \log \left (d x + c + 1\right ) + {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \log \left (d x + c - 1\right ) - 2 \, \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )}{d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}} - 6 \, \int \frac {1}{3 \, {\left (d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + c^{6} e^{4} - c^{4} e^{4} + {\left (15 \, c^{2} d^{4} e^{4} - d^{4} e^{4}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{4} - c d^{3} e^{4}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{4} - 2 \, c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{5} d e^{4} - 2 \, c^{3} d e^{4}\right )} x + {\left (d^{5} e^{4} x^{5} + 5 \, c d^{4} e^{4} x^{4} + c^{5} e^{4} - c^{3} e^{4} + {\left (10 \, c^{2} d^{3} e^{4} - d^{3} e^{4}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{4} - 3 \, c d^{2} e^{4}\right )} x^{2} + {\left (5 \, c^{4} d e^{4} - 3 \, c^{2} d e^{4}\right )} x\right )} e^{\left (\frac {1}{2} \, \log \left (d x + c + 1\right ) + \frac {1}{2} \, \log \left (d x + c - 1\right )\right )}\right )}}\,{d x}\right )} - \frac {a}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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