Optimal. Leaf size=56 \[ \frac {b \tan ^{-1}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{d e^2}-\frac {a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5866, 12, 5662, 92, 203} \[ \frac {b \tan ^{-1}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{d e^2}-\frac {a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 92
Rule 203
Rule 5662
Rule 5866
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e^2 x^2} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{x^2} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{d e^2}\\ &=-\frac {a+b \cosh ^{-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} \sqrt {1+c+d x}\right )}{d e^2}\\ &=-\frac {a+b \cosh ^{-1}(c+d x)}{d e^2 (c+d x)}+\frac {b \tan ^{-1}\left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{d e^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.14, size = 78, normalized size = 1.39 \[ \frac {\frac {-a-b \cosh ^{-1}(c+d x)}{c+d x}+\frac {b \sqrt {(c+d x)^2-1} \tan ^{-1}\left (\sqrt {(c+d x)^2-1}\right )}{\sqrt {c+d x-1} \sqrt {c+d x+1}}}{d e^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.82, size = 133, normalized size = 2.38 \[ \frac {b d x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - a c + 2 \, {\left (b c d x + b c^{2}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + {\left (b d x + b c\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )}{c d^{2} e^{2} x + c^{2} d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 88, normalized size = 1.57 \[ -\frac {a}{d \,e^{2} \left (d x +c \right )}-\frac {b \,\mathrm {arccosh}\left (d x +c \right )}{d \,e^{2} \left (d x +c \right )}-\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{d \,e^{2} \sqrt {\left (d x +c \right )^{2}-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.55, size = 80, normalized size = 1.43 \[ -b {\left (\frac {\operatorname {arcosh}\left (d x + c\right )}{d^{2} e^{2} x + c d e^{2}} + \frac {\arcsin \left (\frac {d e^{2}}{{\left | d^{2} e^{2} x + c d e^{2} \right |}}\right )}{d e^{2}}\right )} - \frac {a}{d^{2} e^{2} x + c d e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________