Optimal. Leaf size=84 \[ -\frac {a+b \sinh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \sqrt {(c+d x)^2+1}}{6 d e^4 (c+d x)^2}+\frac {b \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{6 d e^4} \]
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Rubi [A] time = 0.07, antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5865, 12, 5661, 266, 51, 63, 207} \[ -\frac {a+b \sinh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \sqrt {(c+d x)^2+1}}{6 d e^4 (c+d x)^2}+\frac {b \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{6 d e^4} \]
Antiderivative was successfully verified.
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Rule 12
Rule 51
Rule 63
Rule 207
Rule 266
Rule 5661
Rule 5865
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{e^4 x^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x^4} \, dx,x,c+d x\right )}{d e^4}\\ &=-\frac {a+b \sinh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x^2}} \, dx,x,c+d x\right )}{3 d e^4}\\ &=-\frac {a+b \sinh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{6 d e^4}\\ &=-\frac {b \sqrt {1+(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \sinh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{12 d e^4}\\ &=-\frac {b \sqrt {1+(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \sinh ^{-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)^2}\right )}{6 d e^4}\\ &=-\frac {b \sqrt {1+(c+d x)^2}}{6 d e^4 (c+d x)^2}-\frac {a+b \sinh ^{-1}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \tanh ^{-1}\left (\sqrt {1+(c+d x)^2}\right )}{6 d e^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 82, normalized size = 0.98 \[ -\frac {2 a+b \sqrt {c^2+2 c d x+d^2 x^2+1} (c+d x)+2 b \sinh ^{-1}(c+d x)-b (c+d x)^3 \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{6 d e^4 (c+d x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 343, normalized size = 4.08 \[ -\frac {2 \, a c^{3} - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} - 1\right ) + {\left (b c^{3} d x + b c^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{6 \, {\left (c^{3} d^{4} e^{4} x^{3} + 3 \, c^{4} d^{3} e^{4} x^{2} + 3 \, c^{5} d^{2} e^{4} x + c^{6} d e^{4}\right )}} \]
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 \[ \int \frac {b \operatorname {arsinh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 74, normalized size = 0.88 \[ \frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\arcsinh \left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {1+\left (d x +c \right )^{2}}}{6 \left (d x +c \right )^{2}}+\frac {\arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{6}\right )}{e^{4}}}{d} \]
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 \[ -\frac {1}{6} \, b {\left (\frac {2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2} + \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )\right )}}{d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}} - \frac {i \, {\left (\log \left (\frac {i \, {\left (d^{2} x + c d\right )}}{d} + 1\right ) - \log \left (-\frac {i \, {\left (d^{2} x + c d\right )}}{d} + 1\right )\right )}}{d e^{4}} - 6 \, \int \frac {1}{3 \, {\left (d^{6} e^{4} x^{6} + 6 \, c d^{5} e^{4} x^{5} + c^{6} e^{4} + c^{4} e^{4} + {\left (15 \, c^{2} d^{4} e^{4} + d^{4} e^{4}\right )} x^{4} + 4 \, {\left (5 \, c^{3} d^{3} e^{4} + c d^{3} e^{4}\right )} x^{3} + 3 \, {\left (5 \, c^{4} d^{2} e^{4} + 2 \, c^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (3 \, c^{5} d e^{4} + 2 \, c^{3} d e^{4}\right )} x + {\left (d^{5} e^{4} x^{5} + 5 \, c d^{4} e^{4} x^{4} + c^{5} e^{4} + c^{3} e^{4} + {\left (10 \, c^{2} d^{3} e^{4} + d^{3} e^{4}\right )} x^{3} + {\left (10 \, c^{3} d^{2} e^{4} + 3 \, c d^{2} e^{4}\right )} x^{2} + {\left (5 \, c^{4} d e^{4} + 3 \, c^{2} d e^{4}\right )} x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}}\,{d x}\right )} - \frac {a}{3 \, {\left (d^{4} e^{4} x^{3} + 3 \, c d^{3} e^{4} x^{2} + 3 \, c^{2} d^{2} e^{4} x + c^{3} d e^{4}\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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