Optimal. Leaf size=49 \[ \frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \tanh ^{-1}\left (\sqrt {1+\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {5870, 6449,
379, 272, 65, 213} \begin {gather*} \frac {c \tanh ^{-1}\left (\sqrt {\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}+1}\right )}{b}+\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 213
Rule 272
Rule 379
Rule 5870
Rule 6449
Rubi steps
\begin {align*} \int \sinh ^{-1}\left (\frac {c}{a+b x}\right ) \, dx &=\int \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right ) \, dx\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\int \frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right ) \sqrt {1+\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}}} \, dx\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {1}{x^2}} x} \, dx,x,\frac {a}{c}+\frac {b x}{c}\right )}{b}\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,\frac {1}{\left (\frac {a}{c}+\frac {b x}{c}\right )^2}\right )}{2 b}\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {c \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+\frac {c^2}{(a+b x)^2}}\right )}{b}\\ &=\frac {(a+b x) \text {csch}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\frac {c \tanh ^{-1}\left (\sqrt {1+\frac {c^2}{(a+b x)^2}}\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(330\) vs. \(2(49)=98\).
time = 0.67, size = 330, normalized size = 6.73 \begin {gather*} x \sinh ^{-1}\left (\frac {c}{a+b x}\right )-\frac {(a+b x) \sqrt {\frac {a^2+c^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (2 a \left (b+\sqrt {b^2}\right ) \tanh ^{-1}\left (\frac {a+\sqrt {b^2} x-\sqrt {a^2+c^2+2 a b x+b^2 x^2}}{c}\right )+2 a \left (-b+\sqrt {b^2}\right ) \tanh ^{-1}\left (\frac {a-\sqrt {b^2} x+\sqrt {a^2+c^2+2 a b x+b^2 x^2}}{c}\right )+c \left (\sqrt {b^2} \log \left (-a-\sqrt {b^2} x+\sqrt {a^2+c^2+2 a b x+b^2 x^2}\right )+\left (-b+\sqrt {b^2}\right ) \log \left (a-\sqrt {b^2} x+\sqrt {a^2+c^2+2 a b x+b^2 x^2}\right )+b \log \left (a b^2+\left (b^2\right )^{3/2} x-b^2 \sqrt {a^2+c^2+2 a b x+b^2 x^2}\right )\right )\right )}{2 b^2 \sqrt {a^2+c^2+2 a b x+b^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.14, size = 46, normalized size = 0.94
method | result | size |
derivativedivides | \(-\frac {c \left (-\frac {\left (b x +a \right ) \arcsinh \left (\frac {c}{b x +a}\right )}{c}-\arctanh \left (\frac {1}{\sqrt {1+\frac {c^{2}}{\left (b x +a \right )^{2}}}}\right )\right )}{b}\) | \(46\) |
default | \(-\frac {c \left (-\frac {\left (b x +a \right ) \arcsinh \left (\frac {c}{b x +a}\right )}{c}-\arctanh \left (\frac {1}{\sqrt {1+\frac {c^{2}}{\left (b x +a \right )^{2}}}}\right )\right )}{b}\) | \(46\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 242 vs.
\(2 (47) = 94\).
time = 0.39, size = 242, normalized size = 4.94 \begin {gather*} \frac {b x \log \left (\frac {{\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{b x + a}\right ) + a \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a + c\right ) - a \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a - c\right ) - c \log \left (-b x + {\left (b x + a\right )} \sqrt {\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - a\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {asinh}{\left (\frac {c}{a + b x} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.09, size = 41, normalized size = 0.84 \begin {gather*} \frac {c\,\mathrm {atanh}\left (\sqrt {\frac {c^2}{{\left (a+b\,x\right )}^2}+1}\right )}{b}+\frac {\mathrm {asinh}\left (\frac {c}{a+b\,x}\right )\,\left (a+b\,x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________