Optimal. Leaf size=49 \[ -\frac {a+b \sinh ^{-1}(c+d x)}{d e^2 (c+d x)}-\frac {b \tanh ^{-1}\left (\sqrt {1+(c+d x)^2}\right )}{d e^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, 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 = {5859, 12, 5776,
272, 65, 213} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 65
Rule 213
Rule 272
Rule 5776
Rule 5859
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c+d x)}{(c e+d e x)^2} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \sinh ^{-1}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {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 \text {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 \text {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 \text {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 = 0.03, size = 43, normalized size = 0.88 \begin {gather*} -\frac {\frac {a+b \sinh ^{-1}(c+d x)}{c+d x}+b \tanh ^{-1}\left (\sqrt {1+(c+d x)^2}\right )}{d e^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.69, size = 54, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {-\frac {a}{e^{2} \left (d x +c \right )}+\frac {b \left (-\frac {\arcsinh \left (d x +c \right )}{d x +c}-\arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(54\) |
default | \(\frac {-\frac {a}{e^{2} \left (d x +c \right )}+\frac {b \left (-\frac {\arcsinh \left (d x +c \right )}{d x +c}-\arctanh \left (\frac {1}{\sqrt {1+\left (d x +c \right )^{2}}}\right )\right )}{e^{2}}}{d}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.28, size = 72, normalized size = 1.47 \begin {gather*} -b {\left (\frac {\operatorname {arsinh}\left (\frac {d e^{2}}{{\left | d^{2} x e^{2} + c d e^{2} \right |}}\right ) e^{\left (-2\right )}}{d} + \frac {\operatorname {arsinh}\left (d x + c\right )}{d^{2} x e^{2} + c d e^{2}}\right )} - \frac {a}{d^{2} x e^{2} + c d e^{2}} \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. 210 vs.
\(2 (45) = 90\).
time = 0.47, size = 210, normalized size = 4.29 \begin {gather*} \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 )}{{\left (c d^{2} x + c^{2} d\right )} \cosh \left (1\right )^{2} + 2 \, {\left (c d^{2} x + c^{2} d\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (c d^{2} x + c^{2} d\right )} \sinh \left (1\right )^{2}} \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*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 134 vs.
\(2 (47) = 94\).
time = 0.44, size = 134, normalized size = 2.73 \begin {gather*} -b {\left (\frac {\log \left (d x + c + \sqrt {{\left (d x + c\right )}^{2} + 1}\right )}{{\left (d e x + c e\right )} d e} + \frac {d \log \left (\sqrt {\frac {e^{2}}{{\left (d e x + c e\right )}^{2}} + 1} + \frac {\sqrt {d^{2} e^{4}}}{{\left (d e x + c e\right )} d e}\right )}{e^{2} {\left | d \right |}^{2} \mathrm {sgn}\left (\frac {1}{d e x + c e}\right ) \mathrm {sgn}\left (d\right ) \mathrm {sgn}\left (e\right )}\right )} - \frac {a}{{\left (d e x + c e\right )} d e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {asinh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________