Optimal. Leaf size=120 \[ \frac {f \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {c^2 d x^2+d}}+\frac {g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {c^2 d x^2+d}}-\frac {b g x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5835, 5821, 5675, 5717, 8} \[ \frac {f \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {c^2 d x^2+d}}+\frac {g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {c^2 d x^2+d}}-\frac {b g x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 5675
Rule 5717
Rule 5821
Rule 5835
Rubi steps
\begin {align*} \int \frac {(f+g x) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {\sqrt {1+c^2 x^2} \int \frac {(f+g x) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\sqrt {1+c^2 x^2} \int \left (\frac {f \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {g x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\left (f \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}+\frac {\left (g \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {d+c^2 d x^2}}+\frac {f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {\left (b g \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{c \sqrt {d+c^2 d x^2}}\\ &=-\frac {b g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{c^2 \sqrt {d+c^2 d x^2}}+\frac {f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{2 b c \sqrt {d+c^2 d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 158, normalized size = 1.32 \[ \frac {2 \sqrt {d} g \left (a c^2 x^2+a-b c x \sqrt {c^2 x^2+1}\right )+2 a c f \sqrt {c^2 d x^2+d} \log \left (\sqrt {d} \sqrt {c^2 d x^2+d}+c d x\right )+b c \sqrt {d} f \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2+2 b \sqrt {d} g \left (c^2 x^2+1\right ) \sinh ^{-1}(c x)}{2 c^2 \sqrt {d} \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a g x + a f + {\left (b g x + b f\right )} \operatorname {arsinh}\left (c x\right )}{\sqrt {c^{2} d x^{2} + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {c^{2} d x^{2} + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.41, size = 209, normalized size = 1.74 \[ \frac {a g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}+\frac {a f \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, f \arcsinh \left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c d}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \arcsinh \left (c x \right ) x^{2}}{d \left (c^{2} x^{2}+1\right )}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g x}{c d \sqrt {c^{2} x^{2}+1}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, g \arcsinh \left (c x \right )}{c^{2} d \left (c^{2} x^{2}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.42, size = 87, normalized size = 0.72 \[ \frac {b f \operatorname {arsinh}\left (c x\right )^{2}}{2 \, c \sqrt {d}} - \frac {b g x}{c \sqrt {d}} + \frac {a f \operatorname {arsinh}\left (c x\right )}{c \sqrt {d}} + \frac {\sqrt {c^{2} d x^{2} + d} b g \operatorname {arsinh}\left (c x\right )}{c^{2} d} + \frac {\sqrt {c^{2} d x^{2} + d} a g}{c^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {d\,c^2\,x^2+d}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________