Optimal. Leaf size=49 \[ -\frac {a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}-\frac {b \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{d e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5865, 12, 5661, 266, 63, 207} \[ -\frac {a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}-\frac {b \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{d e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 63
Rule 207
Rule 266
Rule 5661
Rule 5865
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x^2}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,(c+d x)^2\right )}{2 d e^2}\\ &=-\frac {a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+(c+d x)^2}\right )}{d e^2}\\ &=-\frac {a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}-\frac {b \tanh ^{-1}\left (\sqrt {1+(c+d x)^2}\right )}{d e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 43, normalized size = 0.88 \[ -\frac {\frac {a+b \sinh ^{-1}(c+d x)}{c+d x}+b \tanh ^{-1}\left (\sqrt {(c+d x)^2+1}\right )}{d e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.58, size = 175, normalized size = 3.57 \[ \frac {b d x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - a c - {\left (b c d x + b c^{2}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} + 1\right ) + {\left (b d x + b c\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) + {\left (b c d x + b c^{2}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} - 1\right )}{c d^{2} e^{2} x + c^{2} d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.49, size = 131, normalized size = 2.67 \[ -b {\left (\frac {e^{\left (-1\right )} \log \left (d x + c + \sqrt {{\left (d x + c\right )}^{2} + 1}\right )}{{\left (d x e + c e\right )} d} + \frac {d e^{\left (-2\right )} \log \left (\sqrt {\frac {e^{2}}{{\left (d x e + c e\right )}^{2}} + 1} + \frac {\sqrt {d^{2}} e}{{\left (d x e + c e\right )} d}\right )}{{\left | d \right |}^{2} \mathrm {sgn}\left (\frac {1}{d x e + c e}\right ) \mathrm {sgn}\relax (d)}\right )} - \frac {a e^{\left (-1\right )}}{{\left (d x e + c e\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.00, size = 54, normalized size = 1.10 \[ \frac {-\frac {a}{e^{2} \left (d x +c \right )}+\frac {b \left (-\frac {\arcsinh \left (d x +c \right )}{d x +c}-\arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 80, normalized size = 1.63 \[ -b {\left (\frac {\operatorname {arsinh}\left (d x + c\right )}{d^{2} e^{2} x + c d e^{2}} + \frac {\operatorname {arsinh}\left (\frac {d e^{2}}{{\left | d^{2} e^{2} x + c d e^{2} \right |}}\right )}{d e^{2}}\right )} - \frac {a}{d^{2} e^{2} x + c d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________