Optimal. Leaf size=82 \[ -\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5801, 725, 206} \[ -\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{e \sqrt {c^2 d^2+e^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 5801
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{(d+e x)^2} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}+\frac {(b c) \int \frac {1}{(d+e x) \sqrt {1+c^2 x^2}} \, dx}{e}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{c^2 d^2+e^2-x^2} \, dx,x,\frac {e-c^2 d x}{\sqrt {1+c^2 x^2}}\right )}{e}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{e (d+e x)}-\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{e \sqrt {c^2 d^2+e^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 79, normalized size = 0.96 \[ -\frac {\frac {a+b \sinh ^{-1}(c x)}{d+e x}+\frac {b c \tanh ^{-1}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{\sqrt {c^2 d^2+e^2}}}{e} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 253, normalized size = 3.09 \[ -\frac {a c^{2} d^{3} + a d e^{2} - {\left (b c^{2} d^{2} e + b e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (b c d e x + b c d^{2}\right )} \sqrt {c^{2} d^{2} + e^{2}} \log \left (-\frac {c^{3} d^{2} x - c d e + \sqrt {c^{2} d^{2} + e^{2}} {\left (c^{2} d x - e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} + e^{2}} c d + e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{e x + d}\right ) - {\left (b c^{2} d^{3} + b d e^{2} + {\left (b c^{2} d^{2} e + b e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} d^{4} e + d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} + d e^{4}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.86, size = 214, normalized size = 2.61 \[ {\left (\frac {c e^{\left (-1\right )} \log \left (-c^{2} d + \sqrt {c^{2} d^{2} + e^{2}} {\left | c \right |}\right ) \mathrm {sgn}\left (\frac {1}{x e + d}\right )}{\sqrt {c^{2} d^{2} + e^{2}}} - \frac {c e^{\left (-1\right )} \log \left (-c^{2} d + \sqrt {c^{2} d^{2} + e^{2}} {\left (\sqrt {c^{2} - \frac {2 \, c^{2} d}{x e + d} + \frac {c^{2} d^{2}}{{\left (x e + d\right )}^{2}} + \frac {e^{2}}{{\left (x e + d\right )}^{2}}} + \frac {\sqrt {c^{2} d^{2} e^{2} + e^{4}} e^{\left (-1\right )}}{x e + d}\right )}\right )}{\sqrt {c^{2} d^{2} + e^{2}} \mathrm {sgn}\left (\frac {1}{x e + d}\right )} - \frac {e^{\left (-1\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{x e + d}\right )} b - \frac {a e^{\left (-1\right )}}{x e + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 178, normalized size = 2.17 \[ -\frac {c a}{\left (c e x +c d \right ) e}-\frac {c b \arcsinh \left (c x \right )}{\left (c e x +c d \right ) e}-\frac {c b \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {c d}{e}\right )^{2}-\frac {2 c d \left (c x +\frac {c d}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {c d}{e}}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 94, normalized size = 1.15 \[ -b {\left (\frac {\operatorname {arsinh}\left (c x\right )}{e^{2} x + d e} - \frac {c \operatorname {arsinh}\left (\frac {c d e x}{{\left | e^{2} x + d e \right |}} - \frac {e^{2}}{c {\left | e^{2} x + d e \right |}}\right )}{\sqrt {\frac {c^{2} d^{2}}{e^{2}} + 1} e^{2}}\right )} - \frac {a}{e^{2} x + d e} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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