Optimal. Leaf size=283 \[ \frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.90, antiderivative size = 283, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {198, 197, 6124,
6820, 12, 6847, 1633, 65, 214} \begin {gather*} \frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 65
Rule 197
Rule 198
Rule 214
Rule 1633
Rule 6124
Rule 6820
Rule 6847
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-a \int \frac {\frac {x}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {c+d x^2}}}{1-a^2 x^2} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{35 c^4 \left (1-a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right )}{\left (1-a^2 x^2\right ) \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4}\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \frac {35 c^3+70 c^2 d x+56 c d^2 x^2+16 d^3 x^3}{\left (1-a^2 x\right ) (c+d x)^{7/2}} \, dx,x,x^2\right )}{70 c^4}\\ &=\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {a \text {Subst}\left (\int \left (\frac {5 c^3 d}{\left (a^2 c+d\right ) (c+d x)^{7/2}}+\frac {c^2 d \left (11 a^2 c+6 d\right )}{\left (a^2 c+d\right )^2 (c+d x)^{5/2}}+\frac {c d \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{\left (a^2 c+d\right )^3 (c+d x)^{3/2}}+\frac {-35 a^6 c^3-70 a^4 c^2 d-56 a^2 c d^2-16 d^3}{\left (a^2 c+d\right )^3 \left (-1+a^2 x\right ) \sqrt {c+d x}}\right ) \, dx,x,x^2\right )}{70 c^4}\\ &=\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {\left (a \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{\left (-1+a^2 x\right ) \sqrt {c+d x}} \, dx,x,x^2\right )}{70 c^4 \left (a^2 c+d\right )^3}\\ &=\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}+\frac {\left (a \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right )\right ) \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 )}{35 c^4 d \left (a^2 c+d\right )^3}\\ &=\frac {a}{35 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac {a \left (11 a^2 c+6 d\right )}{105 c^2 \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}+\frac {a \left (19 a^4 c^2+22 a^2 c d+8 d^2\right )}{35 c^3 \left (a^2 c+d\right )^3 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac {6 x \coth ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac {16 x \coth ^{-1}(a x)}{35 c^4 \sqrt {c+d x^2}}-\frac {\left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{35 c^4 \left (a^2 c+d\right )^{7/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.87, size = 431, normalized size = 1.52 \begin {gather*} \frac {2 a c \sqrt {a^2 c+d} \left (c+d x^2\right ) \left (3 c^2 \left (a^2 c+d\right )^2+c \left (a^2 c+d\right ) \left (11 a^2 c+6 d\right ) \left (c+d x^2\right )+3 \left (19 a^4 c^2+22 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^2\right )+6 \left (a^2 c+d\right )^{7/2} x \left (35 c^3+70 c^2 d x^2+56 c d^2 x^4+16 d^3 x^6\right ) \coth ^{-1}(a x)+3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log (1-a x)+3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log (1+a x)-3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log \left (a c-d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )-3 \left (35 a^6 c^3+70 a^4 c^2 d+56 a^2 c d^2+16 d^3\right ) \left (c+d x^2\right )^{7/2} \log \left (a c+d x+\sqrt {a^2 c+d} \sqrt {c+d x^2}\right )}{210 c^4 \left (a^2 c+d\right )^{7/2} \left (c+d x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.36, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {9}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 639 vs.
\(2 (247) = 494\).
time = 0.48, size = 639, normalized size = 2.26 \begin {gather*} \frac {1}{210} \, a {\left (\frac {\frac {15 \, a^{5} d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{6} c^{4} + 3 \, a^{4} c^{3} d + 3 \, a^{2} c^{2} d^{2} + c d^{3}\right )} \sqrt {a^{2} c + d}} + \frac {2 \, {\left (15 \, {\left (d x^{2} + c\right )}^{2} a^{4} d + 3 \, a^{4} c^{2} d + 6 \, a^{2} c d^{2} + 3 \, d^{3} + 5 \, {\left (a^{4} c d + a^{2} d^{2}\right )} {\left (d x^{2} + c\right )}\right )}}{{\left (a^{6} c^{4} + 3 \, a^{4} c^{3} d + 3 \, a^{2} c^{2} d^{2} + c d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}}{d} + \frac {6 \, {\left (\frac {3 \, a^{3} d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{4} c^{4} + 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} \sqrt {a^{2} c + d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}\right )}}{{\left (a^{4} c^{4} + 2 \, a^{2} c^{3} d + c^{2} d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}\right )}}{d} + \frac {24 \, {\left (\frac {a d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{2} c^{4} + c^{3} d\right )} \sqrt {a^{2} c + d}} + \frac {2 \, d}{{\left (a^{2} c^{4} + c^{3} d\right )} \sqrt {d x^{2} + c}}\right )}}{d} + \frac {48 \, \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{\sqrt {a^{2} c + d} a c^{4}}\right )} + \frac {1}{35} \, {\left (\frac {16 \, x}{\sqrt {d x^{2} + c} c^{4}} + \frac {8 \, x}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{3}} + \frac {6 \, x}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} c^{2}} + \frac {5 \, x}{{\left (d x^{2} + c\right )}^{\frac {7}{2}} c}\right )} \operatorname {arcoth}\left (a x\right ) \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. 983 vs.
\(2 (247) = 494\).
time = 0.52, size = 2004, normalized size = 7.08 \begin {gather*} \text {Too large to display} \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 {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {9}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.43, size = 357, normalized size = 1.26 \begin {gather*} \frac {1}{105} \, a {\left (\frac {3 \, {\left (35 \, a^{6} c^{3} + 70 \, a^{4} c^{2} d + 56 \, a^{2} c d^{2} + 16 \, d^{3}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} a}{\sqrt {-a^{2} c - d}}\right )}{{\left (a^{6} c^{7} + 3 \, a^{4} c^{6} d + 3 \, a^{2} c^{5} d^{2} + c^{4} d^{3}\right )} \sqrt {-a^{2} c - d} a} + \frac {57 \, {\left (d x^{2} + c\right )}^{2} a^{4} c^{2} + 11 \, {\left (d x^{2} + c\right )} a^{4} c^{3} + 3 \, a^{4} c^{4} + 66 \, {\left (d x^{2} + c\right )}^{2} a^{2} c d + 17 \, {\left (d x^{2} + c\right )} a^{2} c^{2} d + 6 \, a^{2} c^{3} d + 24 \, {\left (d x^{2} + c\right )}^{2} d^{2} + 6 \, {\left (d x^{2} + c\right )} c d^{2} + 3 \, c^{2} d^{2}}{{\left (a^{6} c^{6} + 3 \, a^{4} c^{5} d + 3 \, a^{2} c^{4} d^{2} + c^{3} d^{3}\right )} {\left (d x^{2} + c\right )}^{\frac {5}{2}}}\right )} + \frac {{\left (2 \, {\left (4 \, x^{2} {\left (\frac {2 \, d^{3} x^{2}}{c^{4}} + \frac {7 \, d^{2}}{c^{3}}\right )} + \frac {35 \, d}{c^{2}}\right )} x^{2} + \frac {35}{c}\right )} x \log \left (-\frac {\frac {1}{a x} + 1}{\frac {1}{a x} - 1}\right )}{70 \, {\left (d x^{2} + c\right )}^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________