Optimal. Leaf size=369 \[ \frac{4 a \left (1-a^2 x^2\right ) \left (11 a^4 c^2+15 a^2 c d+6 d^2\right )}{105 c^3 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right )^3 \sqrt{c+d x^2}}+\frac{2 a \left (1-a^2 x^2\right ) \left (5 a^2 c+3 d\right )}{105 c^2 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}-\frac{16 \sqrt{a^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a^2 x^2-1}}{a \sqrt{c+d x^2}}\right )}{35 c^4 \sqrt{d} \sqrt{a x-1} \sqrt{a x+1}}+\frac{a \left (1-a^2 x^2\right )}{35 c \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.01382, antiderivative size = 369, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 12, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {192, 191, 5705, 12, 519, 6715, 1622, 949, 78, 63, 217, 206} \[ \frac{4 a \left (1-a^2 x^2\right ) \left (11 a^4 c^2+15 a^2 c d+6 d^2\right )}{105 c^3 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right )^3 \sqrt{c+d x^2}}+\frac{2 a \left (1-a^2 x^2\right ) \left (5 a^2 c+3 d\right )}{105 c^2 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right )^2 \left (c+d x^2\right )^{3/2}}-\frac{16 \sqrt{a^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a^2 x^2-1}}{a \sqrt{c+d x^2}}\right )}{35 c^4 \sqrt{d} \sqrt{a x-1} \sqrt{a x+1}}+\frac{a \left (1-a^2 x^2\right )}{35 c \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{5/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 192
Rule 191
Rule 5705
Rule 12
Rule 519
Rule 6715
Rule 1622
Rule 949
Rule 78
Rule 63
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}(a x)}{\left (c+d x^2\right )^{9/2}} \, dx &=\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-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 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{7/2}} \, dx\\ &=\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-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 )}{\sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4}\\ &=\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \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 )}{\sqrt{-1+a^2 x^2} \left (c+d x^2\right )^{7/2}} \, dx}{35 c^4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{35 c^3+70 c^2 d x+56 c d^2 x^2+16 d^3 x^3}{\sqrt{-1+a^2 x} (c+d x)^{7/2}} \, dx,x,x^2\right )}{70 c^4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{5 c^2 \left (17 a^2 c+15 d\right )+100 c d \left (a^2 c+d\right ) x+40 d^2 \left (a^2 c+d\right ) x^2}{\sqrt{-1+a^2 x} (c+d x)^{5/2}} \, dx,x,x^2\right )}{175 c^4 \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}}+\frac{2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (2 a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{5 c \left (23 a^4 c^2+39 a^2 c d+18 d^2\right )+60 d \left (a^2 c+d\right )^2 x}{\sqrt{-1+a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{525 c^4 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}}+\frac{2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (8 a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1+a^2 x} \sqrt{c+d x}} \, dx,x,x^2\right )}{35 c^4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}}+\frac{2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (16 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c+\frac{d}{a^2}+\frac{d x^2}{a^2}}} \, dx,x,\sqrt{-1+a^2 x^2}\right )}{35 a c^4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}}+\frac{2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{\left (16 \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\frac{d x^2}{a^2}} \, dx,x,\frac{\sqrt{-1+a^2 x^2}}{\sqrt{c+d x^2}}\right )}{35 a c^4 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{35 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}}+\frac{2 a \left (5 a^2 c+3 d\right ) \left (1-a^2 x^2\right )}{105 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{4 a \left (11 a^4 c^2+15 a^2 c d+6 d^2\right ) \left (1-a^2 x^2\right )}{105 c^3 \left (a^2 c+d\right )^3 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{7 c \left (c+d x^2\right )^{7/2}}+\frac{6 x \cosh ^{-1}(a x)}{35 c^2 \left (c+d x^2\right )^{5/2}}+\frac{8 x \cosh ^{-1}(a x)}{35 c^3 \left (c+d x^2\right )^{3/2}}+\frac{16 x \cosh ^{-1}(a x)}{35 c^4 \sqrt{c+d x^2}}-\frac{16 \sqrt{-1+a^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{-1+a^2 x^2}}{a \sqrt{c+d x^2}}\right )}{35 c^4 \sqrt{d} \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [C] time = 6.17026, size = 723, normalized size = 1.96 \[ \frac{\frac{32 (a x-1)^{3/2} \left (c+d x^2\right )^3 \sqrt{\frac{(a x+1) \left (a \sqrt{c}-i \sqrt{d}\right )}{(a x-1) \left (a \sqrt{c}+i \sqrt{d}\right )}} \left (\frac{a \left (\sqrt{d}-i a \sqrt{c}\right ) \left (\sqrt{d} x+i \sqrt{c}\right ) \sqrt{\frac{\frac{i a \sqrt{c}}{\sqrt{d}}+a (-x)+\frac{i \sqrt{d} x}{\sqrt{c}}+1}{1-a x}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{2-2 a x}}\right ),\frac{4 i a \sqrt{c} \sqrt{d}}{\left (a \sqrt{c}+i \sqrt{d}\right )^2}\right )}{a x-1}+a \sqrt{c} \left (-a \sqrt{c}+i \sqrt{d}\right ) \sqrt{\frac{\left (a^2 c+d\right ) \left (c+d x^2\right )}{c d (a x-1)^2}} \sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{1-a x}} \Pi \left (\frac{2 a \sqrt{c}}{\sqrt{c} a+i \sqrt{d}};\sin ^{-1}\left (\sqrt{-\frac{\frac{i \sqrt{d} x}{\sqrt{c}}+a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )-1}{2-2 a x}}\right )|\frac{4 i a \sqrt{c} \sqrt{d}}{\left (\sqrt{c} a+i \sqrt{d}\right )^2}\right )\right )}{a c^4 \sqrt{a x+1} \left (a^2 c+d\right ) \sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{1-a x}}}-\frac{a \sqrt{a x-1} \sqrt{a x+1} \left (c+d x^2\right ) \left (a^4 c^2 \left (57 c^2+98 c d x^2+44 d^2 x^4\right )+2 a^2 c d \left (41 c^2+68 c d x^2+30 d^2 x^4\right )+3 d^2 \left (11 c^2+18 c d x^2+8 d^2 x^4\right )\right )}{3 c^3 \left (a^2 c+d\right )^3}+\frac{x \cosh ^{-1}(a x) \left (70 c^2 d x^2+35 c^3+56 c d^2 x^4+16 d^3 x^6\right )}{c^4}}{35 \left (c+d x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.145, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccosh} \left (ax\right ) \left ( d{x}^{2}+c \right ) ^{-{\frac{9}{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 = 5.56558, size = 3546, normalized size = 9.61 \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(-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]