Optimal. Leaf size=269 \[ \frac{2 a \left (1-a^2 x^2\right ) \left (3 a^2 c+2 d\right )}{15 c^2 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right )^2 \sqrt{c+d x^2}}-\frac{8 \sqrt{a^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a^2 x^2-1}}{a \sqrt{c+d x^2}}\right )}{15 c^3 \sqrt{d} \sqrt{a x-1} \sqrt{a x+1}}+\frac{a \left (1-a^2 x^2\right )}{15 c \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
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Rubi [A] time = 0.805397, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.688, Rules used = {192, 191, 5705, 12, 519, 6715, 949, 78, 63, 217, 206} \[ \frac{2 a \left (1-a^2 x^2\right ) \left (3 a^2 c+2 d\right )}{15 c^2 \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right )^2 \sqrt{c+d x^2}}-\frac{8 \sqrt{a^2 x^2-1} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a^2 x^2-1}}{a \sqrt{c+d x^2}}\right )}{15 c^3 \sqrt{d} \sqrt{a x-1} \sqrt{a x+1}}+\frac{a \left (1-a^2 x^2\right )}{15 c \sqrt{a x-1} \sqrt{a x+1} \left (a^2 c+d\right ) \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 192
Rule 191
Rule 5705
Rule 12
Rule 519
Rule 6715
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 )^{7/2}} \, dx &=\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-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 \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}} \, dx\\ &=\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-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 )}{\sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3}\\ &=\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \int \frac{x \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{\sqrt{-1+a^2 x^2} \left (c+d x^2\right )^{5/2}} \, dx}{15 c^3 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{15 c^2+20 c d x+8 d^2 x^2}{\sqrt{-1+a^2 x} (c+d x)^{5/2}} \, dx,x,x^2\right )}{30 c^3 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}-\frac{\left (a \sqrt{-1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{3 c \left (7 a^2 c+6 d\right )+12 d \left (a^2 c+d\right ) x}{\sqrt{-1+a^2 x} (c+d x)^{3/2}} \, dx,x,x^2\right )}{45 c^3 \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}-\frac{\left (4 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 )}{15 c^3 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}-\frac{\left (8 \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 )}{15 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}-\frac{\left (8 \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 )}{15 a c^3 \sqrt{-1+a x} \sqrt{1+a x}}\\ &=\frac{a \left (1-a^2 x^2\right )}{15 c \left (a^2 c+d\right ) \sqrt{-1+a x} \sqrt{1+a x} \left (c+d x^2\right )^{3/2}}+\frac{2 a \left (3 a^2 c+2 d\right ) \left (1-a^2 x^2\right )}{15 c^2 \left (a^2 c+d\right )^2 \sqrt{-1+a x} \sqrt{1+a x} \sqrt{c+d x^2}}+\frac{x \cosh ^{-1}(a x)}{5 c \left (c+d x^2\right )^{5/2}}+\frac{4 x \cosh ^{-1}(a x)}{15 c^2 \left (c+d x^2\right )^{3/2}}+\frac{8 x \cosh ^{-1}(a x)}{15 c^3 \sqrt{c+d x^2}}-\frac{8 \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 )}{15 c^3 \sqrt{d} \sqrt{-1+a x} \sqrt{1+a x}}\\ \end{align*}
Mathematica [C] time = 3.21406, size = 655, normalized size = 2.43 \[ \frac{\frac{16 (a x-1)^{3/2} \left (c+d x^2\right )^2 \sqrt{\frac{(a x+1) \left (a \sqrt{c}-i \sqrt{d}\right )}{(a x-1) \left (a \sqrt{c}+i \sqrt{d}\right )}} \left (\frac{a \left (\sqrt{d}-i a \sqrt{c}\right ) \left (\sqrt{d} x+i \sqrt{c}\right ) \sqrt{\frac{\frac{i a \sqrt{c}}{\sqrt{d}}+a (-x)+\frac{i \sqrt{d} x}{\sqrt{c}}+1}{1-a x}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{2-2 a x}}\right ),\frac{4 i a \sqrt{c} \sqrt{d}}{\left (a \sqrt{c}+i \sqrt{d}\right )^2}\right )}{a x-1}+a \sqrt{c} \left (-a \sqrt{c}+i \sqrt{d}\right ) \sqrt{\frac{\left (a^2 c+d\right ) \left (c+d x^2\right )}{c d (a x-1)^2}} \sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{1-a x}} \Pi \left (\frac{2 a \sqrt{c}}{\sqrt{c} a+i \sqrt{d}};\sin ^{-1}\left (\sqrt{-\frac{\frac{i \sqrt{d} x}{\sqrt{c}}+a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )-1}{2-2 a x}}\right )|\frac{4 i a \sqrt{c} \sqrt{d}}{\left (\sqrt{c} a+i \sqrt{d}\right )^2}\right )\right )}{a c^3 \sqrt{a x+1} \left (a^2 c+d\right ) \sqrt{-\frac{a \left (x+\frac{i \sqrt{c}}{\sqrt{d}}\right )+\frac{i \sqrt{d} x}{\sqrt{c}}-1}{1-a x}}}-\frac{a \sqrt{a x-1} \sqrt{a x+1} \left (c+d x^2\right ) \left (a^2 c \left (7 c+6 d x^2\right )+d \left (5 c+4 d x^2\right )\right )}{c^2 \left (a^2 c+d\right )^2}+\frac{x \cosh ^{-1}(a x) \left (15 c^2+20 c d x^2+8 d^2 x^4\right )}{c^3}}{15 \left (c+d x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.153, size = 0, normalized size = 0. \begin{align*} \int{{\rm arccosh} \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.
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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.
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Fricas [B] time = 3.53707, size = 2221, normalized size = 8.26 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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