Optimal. Leaf size=66 \[ \frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{c \sqrt {a^2 c-d}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {197, 5033, 12,
455, 65, 214} \begin {gather*} \frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{c \sqrt {a^2 c-d}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 65
Rule 197
Rule 214
Rule 455
Rule 5033
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)}{\left (c+d x^2\right )^{3/2}} \, dx &=\frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+a \int \frac {x}{c \left (1+a^2 x^2\right ) \sqrt {c+d x^2}} \, dx\\ &=\frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \int \frac {x}{\left (1+a^2 x^2\right ) \sqrt {c+d x^2}} \, dx}{c}\\ &=\frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \text {Subst}\left (\int \frac {1}{\left (1+a^2 x\right ) \sqrt {c+d x}} \, dx,x,x^2\right )}{2 c}\\ &=\frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}+\frac {a \text {Subst}\left (\int \frac {1}{1-\frac {a^2 c}{d}+\frac {a^2 x^2}{d}} \, dx,x,\sqrt {c+d x^2}\right )}{c d}\\ &=\frac {x \cot ^{-1}(a x)}{c \sqrt {c+d x^2}}-\frac {\tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c-d}}\right )}{c \sqrt {a^2 c-d}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 169, normalized size = 2.56 \begin {gather*} \frac {\frac {2 x \cot ^{-1}(a x)}{\sqrt {c+d x^2}}+\frac {-\log \left (\frac {4 a c \left (a c-i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\sqrt {a^2 c-d} (i+a x)}\right )-\log \left (\frac {4 a c \left (a c+i d x+\sqrt {a^2 c-d} \sqrt {c+d x^2}\right )}{\sqrt {a^2 c-d} (-i+a x)}\right )}{\sqrt {a^2 c-d}}}{2 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccot}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 154 vs.
\(2 (58) = 116\).
time = 2.45, size = 349, normalized size = 5.29 \begin {gather*} \left [\frac {4 \, {\left (a^{2} c - d\right )} \sqrt {d x^{2} + c} x \operatorname {arccot}\left (a x\right ) + \sqrt {a^{2} c - d} {\left (d x^{2} + c\right )} \log \left (\frac {a^{4} d^{2} x^{4} + 8 \, a^{4} c^{2} - 8 \, a^{2} c d + 2 \, {\left (4 \, a^{4} c d - 3 \, a^{2} d^{2}\right )} x^{2} - 4 \, {\left (a^{3} d x^{2} + 2 \, a^{3} c - a d\right )} \sqrt {a^{2} c - d} \sqrt {d x^{2} + c} + d^{2}}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right )}{4 \, {\left (a^{2} c^{3} - c^{2} d + {\left (a^{2} c^{2} d - c d^{2}\right )} x^{2}\right )}}, \frac {2 \, {\left (a^{2} c - d\right )} \sqrt {d x^{2} + c} x \operatorname {arccot}\left (a x\right ) - \sqrt {-a^{2} c + d} {\left (d x^{2} + c\right )} \arctan \left (-\frac {{\left (a^{2} d x^{2} + 2 \, a^{2} c - d\right )} \sqrt {-a^{2} c + d} \sqrt {d x^{2} + c}}{2 \, {\left (a^{3} c^{2} - a c d + {\left (a^{3} c d - a d^{2}\right )} x^{2}\right )}}\right )}{2 \, {\left (a^{2} c^{3} - c^{2} d + {\left (a^{2} c^{2} d - c d^{2}\right )} x^{2}\right )}}\right ] \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*} \int \frac {\operatorname {acot}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.42, size = 59, normalized size = 0.89 \begin {gather*} \frac {x \arctan \left (\frac {1}{a x}\right )}{\sqrt {d x^{2} + c} c} + \frac {\arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c + d}}\right )}{\sqrt {-a^{2} c + d} c} \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 {\mathrm {acot}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________