Optimal. Leaf size=200 \[ \frac {2}{15 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {1}{20 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c}}-\frac {4 \sqrt {a^2 x^2+1} \log \left (a^2 x^2+1\right )}{15 a c^3 \sqrt {a^2 c x^2+c}}+\frac {8 x \sinh ^{-1}(a x)}{15 c^3 \sqrt {a^2 c x^2+c}}+\frac {4 x \sinh ^{-1}(a x)}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \sinh ^{-1}(a x)}{5 c \left (a^2 c x^2+c\right )^{5/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {5690, 5687, 260, 261} \[ \frac {2}{15 a c^3 \sqrt {a^2 x^2+1} \sqrt {a^2 c x^2+c}}+\frac {1}{20 a c^3 \left (a^2 x^2+1\right )^{3/2} \sqrt {a^2 c x^2+c}}-\frac {4 \sqrt {a^2 x^2+1} \log \left (a^2 x^2+1\right )}{15 a c^3 \sqrt {a^2 c x^2+c}}+\frac {8 x \sinh ^{-1}(a x)}{15 c^3 \sqrt {a^2 c x^2+c}}+\frac {4 x \sinh ^{-1}(a x)}{15 c^2 \left (a^2 c x^2+c\right )^{3/2}}+\frac {x \sinh ^{-1}(a x)}{5 c \left (a^2 c x^2+c\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
Rule 5687
Rule 5690
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{7/2}} \, dx &=\frac {x \sinh ^{-1}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\sinh ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\sinh ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\left (4 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {\left (8 a \sqrt {1+a^2 x^2}\right ) \int \frac {x}{1+a^2 x^2} \, dx}{15 c^3 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{20 a c^3 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2}}+\frac {2}{15 a c^3 \sqrt {1+a^2 x^2} \sqrt {c+a^2 c x^2}}+\frac {x \sinh ^{-1}(a x)}{5 c \left (c+a^2 c x^2\right )^{5/2}}+\frac {4 x \sinh ^{-1}(a x)}{15 c^2 \left (c+a^2 c x^2\right )^{3/2}}+\frac {8 x \sinh ^{-1}(a x)}{15 c^3 \sqrt {c+a^2 c x^2}}-\frac {4 \sqrt {1+a^2 x^2} \log \left (1+a^2 x^2\right )}{15 a c^3 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 121, normalized size = 0.60 \[ \frac {\sqrt {a^2 c x^2+c} \left (4 a x \sqrt {a^2 x^2+1} \left (8 a^4 x^4+20 a^2 x^2+15\right ) \sinh ^{-1}(a x)-\left (a^2 x^2+1\right ) \left (-8 a^2 x^2+16 \left (a^2 x^2+1\right )^2 \log \left (a^2 x^2+1\right )-11\right )\right )}{60 a c^4 \left (a^2 x^2+1\right )^{7/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )}{a^{8} c^{4} x^{8} + 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} + 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.68, size = 124, normalized size = 0.62 \[ -\frac {1}{60} \, \sqrt {c} {\left (\frac {16 \, \log \left (a^{2} x^{2} + 1\right )}{a c^{4}} - \frac {24 \, a^{4} x^{4} + 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} + 1\right )}^{2} a c^{4}}\right )} + \frac {{\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} + \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}{15 \, {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 363, normalized size = 1.82 \[ \frac {16 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \arcsinh \left (a x \right )}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 x^{5} a^{5}-8 \sqrt {a^{2} x^{2}+1}\, x^{4} a^{4}+20 x^{3} a^{3}-16 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+15 a x -8 \sqrt {a^{2} x^{2}+1}\right ) \left (-64 x^{8} a^{8}-64 \sqrt {a^{2} x^{2}+1}\, x^{7} a^{7}-280 x^{6} a^{6}-248 \sqrt {a^{2} x^{2}+1}\, x^{5} a^{5}+160 \arcsinh \left (a x \right ) x^{4} a^{4}-456 x^{4} a^{4}-340 \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+380 \arcsinh \left (a x \right ) x^{2} a^{2}-328 a^{2} x^{2}-165 \sqrt {a^{2} x^{2}+1}\, x a +256 \arcsinh \left (a x \right )-88\right )}{60 \left (40 a^{10} x^{10}+215 x^{8} a^{8}+469 x^{6} a^{6}+517 x^{4} a^{4}+287 a^{2} x^{2}+64\right ) a \,c^{4}}-\frac {8 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \ln \left (1+\left (a x +\sqrt {a^{2} x^{2}+1}\right )^{2}\right )}{15 \sqrt {a^{2} x^{2}+1}\, a \,c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 143, normalized size = 0.72 \[ \frac {1}{60} \, a {\left (\frac {3}{{\left (a^{6} c^{\frac {5}{2}} x^{4} + 2 \, a^{4} c^{\frac {5}{2}} x^{2} + a^{2} c^{\frac {5}{2}}\right )} c} + \frac {8}{{\left (a^{4} c^{\frac {3}{2}} x^{2} + a^{2} c^{\frac {3}{2}}\right )} c^{2}} - \frac {16 \, \log \left (x^{2} + \frac {1}{a^{2}}\right )}{a^{2} c^{\frac {7}{2}}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \operatorname {arsinh}\left (a x\right ) \]
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 \[ \int \frac {\mathrm {asinh}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\operatorname {asinh}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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