Optimal. Leaf size=200 \[ -\frac{\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \tanh ^{-1}\left (\frac{a \sqrt{c+d x^2}}{\sqrt{a^2 c+d}}\right )}{15 c^3 \left (a^2 c+d\right )^{5/2}}+\frac{a \left (7 a^2 c+4 d\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt{c+d x^2}}+\frac{a}{15 c \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac{8 x \coth ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \coth ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{x \coth ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.04748, antiderivative size = 200, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 192
Rule 191
Rule 5977
Rule 6688
Rule 12
Rule 6715
Rule 897
Rule 1261
Rule 208
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*}
Mathematica [A] time = 0.636018, size = 329, normalized size = 1.64 \[ \frac{\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \log (1-a x) \left (c+d x^2\right )^{5/2}+\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \log (a x+1) \left (c+d x^2\right )^{5/2}-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c-d x\right )-\left (15 a^4 c^2+20 a^2 c d+8 d^2\right ) \left (c+d x^2\right )^{5/2} \log \left (\sqrt{a^2 c+d} \sqrt{c+d x^2}+a c+d x\right )+2 x \left (a^2 c+d\right )^{5/2} \coth ^{-1}(a x) \left (15 c^2+20 c d x^2+8 d^2 x^4\right )+2 a c \sqrt{a^2 c+d} \left (c+d x^2\right ) \left (a^2 c \left (8 c+7 d x^2\right )+d \left (5 c+4 d x^2\right )\right )}{30 c^3 \left (a^2 c+d\right )^{5/2} \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.455, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccoth} \left (ax\right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.24513, size = 2610, normalized size = 13.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x\right )}{{\left (d x^{2} + c\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]