Optimal. Leaf size=58 \[ \frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {2 c \tan ^{-1}\left (\sqrt {\frac {c \left (1-\frac {a}{c}\right )-b x}{a+b x+c}}\right )}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5893, 6313, 1961, 12, 203} \[ \frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {2 c \tan ^{-1}\left (\sqrt {\frac {c \left (1-\frac {a}{c}\right )-b x}{a+b x+c}}\right )}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 203
Rule 1961
Rule 5893
Rule 6313
Rubi steps
\begin {align*} \int \cosh ^{-1}\left (\frac {c}{a+b x}\right ) \, dx &=\int \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right ) \, dx\\ &=\frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}+\int \frac {\sqrt {\frac {1-\frac {a}{c}-\frac {b x}{c}}{1+\frac {a}{c}+\frac {b x}{c}}}}{1-\frac {a}{c}-\frac {b x}{c}} \, dx\\ &=\frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {c^2}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt {\frac {1-\frac {a}{c}-\frac {b x}{c}}{1+\frac {a}{c}+\frac {b x}{c}}}\right )}{c}\\ &=\frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-\frac {a}{c}-\frac {b x}{c}}{1+\frac {a}{c}+\frac {b x}{c}}}\right )}{b}\\ &=\frac {(a+b x) \text {sech}^{-1}\left (\frac {a}{c}+\frac {b x}{c}\right )}{b}-\frac {2 c \tan ^{-1}\left (\sqrt {\frac {\left (1-\frac {a}{c}\right ) c-b x}{a+c+b x}}\right )}{b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.56, size = 180, normalized size = 3.10 \[ \frac {2 c \sqrt {-\frac {a+b x-c}{a+b x+c}} \left (a \sqrt {b} \sqrt {a+b x+c} \tanh ^{-1}\left (\frac {\sqrt {-b c} \sqrt {a+b x-c}}{\sqrt {b c} \sqrt {a+b x+c}}\right )-c \sqrt {-b c} \sqrt {\frac {a+b x+c}{c}} \sinh ^{-1}\left (\frac {\sqrt {b} \sqrt {a+b x-c}}{\sqrt {2} \sqrt {b c}}\right )\right )}{\sqrt {b} \sqrt {-b^2 c^2} \sqrt {a+b x-c}}+x \cosh ^{-1}\left (\frac {c}{a+b x}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.62, size = 276, normalized size = 4.76 \[ \frac {2 \, b x \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{b x + a}\right ) - 2 \, c \arctan \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}\right ) + a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + c}{x}\right ) - a \log \left (\frac {{\left (b x + a\right )} \sqrt {-\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - c}{x}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 13.04, size = 119, normalized size = 2.05 \[ \frac {c \arcsin \left (-\frac {b x + a}{c}\right ) \mathrm {sgn}\relax (b) \mathrm {sgn}\relax (c)}{{\left | b \right |}} + x \log \left (\sqrt {\frac {c}{b x + a} + 1} \sqrt {\frac {c}{b x + a} - 1} + \frac {c}{b x + a}\right ) - \frac {a \log \left (\frac {{\left | -2 \, b c - 2 \, \sqrt {-b^{2} x^{2} - 2 \, a b x - a^{2} + c^{2}} {\left | b \right |} \right |}}{{\left | -2 \, b^{2} x - 2 \, a b \right |}}\right )}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 91, normalized size = 1.57 \[ \mathrm {arccosh}\left (\frac {c}{b x +a}\right ) x +\frac {\mathrm {arccosh}\left (\frac {c}{b x +a}\right ) a}{b}+\frac {c \sqrt {\frac {c}{b x +a}-1}\, \sqrt {\frac {c}{b x +a}+1}\, \arctan \left (\frac {1}{\sqrt {\frac {c^{2}}{\left (b x +a \right )^{2}}-1}}\right )}{b \sqrt {\frac {c^{2}}{\left (b x +a \right )^{2}}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, b x \log \left (\sqrt {b x + a + c} \sqrt {-b x - a + c} b x + \sqrt {b x + a + c} \sqrt {-b x - a + c} a + {\left (b x + a\right )} c\right ) - 2 \, b x \log \left (b x + a\right ) + {\left (a + c\right )} \log \left (b x + a + c\right ) - 2 \, {\left (b x + a\right )} \log \left (b x + a\right ) + {\left (a - c\right )} \log \left (-b x - a + c\right )}{2 \, b} + \int \frac {b^{2} c x^{2} + a b c x}{b^{2} c x^{2} + 2 \, a b c x + a^{2} c - c^{3} + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} - c^{2}\right )} e^{\left (\frac {1}{2} \, \log \left (b x + a + c\right ) + \frac {1}{2} \, \log \left (-b x - a + c\right )\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.51, size = 53, normalized size = 0.91 \[ \frac {\mathrm {acosh}\left (\frac {c}{a+b\,x}\right )\,\left (a+b\,x\right )}{b}+\frac {c\,\mathrm {atan}\left (\frac {1}{\sqrt {\frac {c}{a+b\,x}-1}\,\sqrt {\frac {c}{a+b\,x}+1}}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acosh}{\left (\frac {c}{a + b x} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________