Optimal. Leaf size=200 \[ \frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
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Rubi [A] time = 1.05, antiderivative size = 200, 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 = {192, 191, 5977, 6688, 12, 6715, 897, 1261, 208} \[ -\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 192
Rule 208
Rule 897
Rule 1261
Rule 5977
Rule 6688
Rule 6715
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(a x)}{\left (c+d x^2\right )^{7/2}} \, dx &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-a \int \frac {\frac {x}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x}{15 c^3 \sqrt {c+d x^2}}}{1-a^2 x^2} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{15 c^3 \left (1-a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \int \frac {x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\left (1-a^2 x^2\right ) \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3}\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \operatorname {Subst}\left (\int \frac {15 c^2+20 c d x+8 d^2 x^2}{\left (1-a^2 x\right ) (c+d x)^{5/2}} \, dx,x,x^2\right )}{30 c^3}\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \operatorname {Subst}\left (\int \frac {3 c^2+4 c x^2+8 x^4}{x^4 \left (\frac {a^2 c+d}{d}-\frac {a^2 x^2}{d}\right )} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d}\\ &=\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {a \operatorname {Subst}\left (\int \left (\frac {3 c^2 d}{\left (a^2 c+d\right ) x^4}+\frac {c d \left (7 a^2 c+4 d\right )}{\left (a^2 c+d\right )^2 x^2}+\frac {d \left (15 a^4 c^2+20 a^2 c d+8 d^2\right )}{\left (a^2 c+d\right )^2 \left (a^2 c+d-a^2 x^2\right )}\right ) \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 d}\\ &=\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (a \left (15 a^4 c^2+20 a^2 c d+8 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a^2 c+d-a^2 x^2} \, dx,x,\sqrt {c+d x^2}\right )}{15 c^3 \left (a^2 c+d\right )^2}\\ &=\frac {a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac {a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt {c+d x^2}}+\frac {x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac {4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac {8 x \coth ^{-1}(a x)}{15 c^3 \sqrt {c+d x^2}}-\frac {\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac {a \sqrt {c+d x^2}}{\sqrt {a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 329, normalized size = 1.64 \[ \frac {2 x \left (a^2 c+d\right )^{5/2} \coth ^{-1}(a x) \left (15 c^2+20 c d x^2+8 d^2 x^4\right )+2 a c \sqrt {a^2 c+d} \left (c+d x^2\right ) \left (a^2 c \left (8 c+7 d x^2\right )+d \left (5 c+4 d x^2\right )\right )+\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \log (1-a x) \left (c+d x^2\right )^{5/2}+\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \log (a x+1) \left (c+d x^2\right )^{5/2}-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (\sqrt {a^2 c+d} \sqrt {c+d x^2}+a c-d x\right )-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (\sqrt {a^2 c+d} \sqrt {c+d x^2}+a c+d x\right )}{30 c^3 \left (a^2 c+d\right )^{5/2} \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 1278, normalized size = 6.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.95, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccoth}\left (a x \right )}{\left (d \,x^{2}+c \right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 401, normalized size = 2.00 \[ \frac {1}{30} \, a {\left (\frac {\frac {3 \, a^{3} d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} \sqrt {a^{2} c + d}} + \frac {2 \, {\left (3 \, {\left (d x^{2} + c\right )} a^{2} d + a^{2} c d + d^{2}\right )}}{{\left (a^{4} c^{3} + 2 \, a^{2} c^{2} d + c d^{2}\right )} {\left (d x^{2} + c\right )}^{\frac {3}{2}}}}{d} + \frac {4 \, {\left (\frac {a d \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {a^{2} c + d}} + \frac {2 \, d}{{\left (a^{2} c^{3} + c^{2} d\right )} \sqrt {d x^{2} + c}}\right )}}{d} + \frac {8 \, \log \left (\frac {\sqrt {d x^{2} + c} a^{2} - \sqrt {a^{2} c + d} a}{\sqrt {d x^{2} + c} a^{2} + \sqrt {a^{2} c + d} a}\right )}{\sqrt {a^{2} c + d} a c^{3}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {d x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (d x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (d x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arcoth}\left (a x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {acoth}\left (a\,x\right )}{{\left (d\,x^2+c\right )}^{7/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acoth}{\left (a x \right )}}{\left (c + d x^{2}\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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