Optimal. Leaf size=31 \[ a x+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}+b x \csc ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5215, 266, 63, 208} \[ a x+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}+b x \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 208
Rule 266
Rule 5215
Rubi steps
\begin {align*} \int \left (a+b \csc ^{-1}(c x)\right ) \, dx &=a x+b \int \csc ^{-1}(c x) \, dx\\ &=a x+b x \csc ^{-1}(c x)+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x} \, dx}{c}\\ &=a x+b x \csc ^{-1}(c x)-\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{c^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 c}\\ &=a x+b x \csc ^{-1}(c x)+(b c) \operatorname {Subst}\left (\int \frac {1}{c^2-c^2 x^2} \, dx,x,\sqrt {1-\frac {1}{c^2 x^2}}\right )\\ &=a x+b x \csc ^{-1}(c x)+\frac {b \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 58, normalized size = 1.87 \[ a x+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}} \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )}{\sqrt {c^2 x^2-1}}+b x \csc ^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.47, size = 64, normalized size = 2.06 \[ \frac {a c x - 2 \, b c \arctan \left (-c x + \sqrt {c^{2} x^{2} - 1}\right ) + {\left (b c x - b c\right )} \operatorname {arccsc}\left (c x\right ) - b \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.13, size = 62, normalized size = 2.00 \[ \frac {1}{2} \, b c {\left (\frac {2 \, x \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}}\right )} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 37, normalized size = 1.19 \[ a x +b x \,\mathrm {arccsc}\left (c x \right )+\frac {b \ln \left (c x +c x \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.35, size = 53, normalized size = 1.71 \[ a x + \frac {{\left (2 \, c x \operatorname {arccsc}\left (c x\right ) + \log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (-\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )\right )} b}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.00, size = 33, normalized size = 1.06 \[ a\,x+b\,x\,\mathrm {asin}\left (\frac {1}{c\,x}\right )+\frac {b\,\mathrm {atanh}\left (\frac {1}{\sqrt {1-\frac {1}{c^2\,x^2}}}\right )}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.43, size = 32, normalized size = 1.03 \[ a x + b \left (x \operatorname {acsc}{\left (c x \right )} + \frac {\begin {cases} \operatorname {acosh}{\left (c x \right )} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\- i \operatorname {asin}{\left (c x \right )} & \text {otherwise} \end {cases}}{c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________