Optimal. Leaf size=277 \[ -\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{128 b^2 c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b^2 c^4}+\frac {21 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}+\frac {9 \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{128 b^2 c^4}+\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{32 b^2 c^4}-\frac {21 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}-\frac {9 \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{256 b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
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Rubi [A] time = 1.22, antiderivative size = 273, normalized size of antiderivative = 0.99, number of steps used = 34, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5777, 5779, 5448, 3303, 3298, 3301} \[ -\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}+\frac {21 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac {9 \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}+\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}-\frac {21 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}-\frac {9 \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}-\frac {x^3 \left (c^2 x^2+1\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 5448
Rule 5777
Rule 5779
Rubi steps
\begin {align*} \int \frac {x^3 \left (1+c^2 x^2\right )^{5/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \int \frac {x^2 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac {(9 c) \int \frac {x^4 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh ^5(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh ^5(x) \sinh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {3 \operatorname {Subst}\left (\int \left (-\frac {5 \cosh (x)}{64 (a+b x)}+\frac {\cosh (3 x)}{64 (a+b x)}+\frac {3 \cosh (5 x)}{64 (a+b x)}+\frac {\cosh (7 x)}{64 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}+\frac {9 \operatorname {Subst}\left (\int \left (\frac {3 \cosh (x)}{128 (a+b x)}-\frac {\cosh (3 x)}{64 (a+b x)}-\frac {\cosh (5 x)}{64 (a+b x)}+\frac {\cosh (7 x)}{256 (a+b x)}+\frac {\cosh (9 x)}{256 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (7 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}+\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (9 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac {3 \operatorname {Subst}\left (\int \frac {\cosh (7 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac {9 \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac {27 \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 b c^4}-\frac {15 \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}\\ &=-\frac {x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\left (27 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 b c^4}-\frac {\left (15 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (3 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (9 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (9 \cosh \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}+\frac {\left (3 \cosh \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (9 \cosh \left (\frac {9 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}-\frac {\left (27 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{128 b c^4}+\frac {\left (15 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (3 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (9 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (9 \sinh \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}-\frac {\left (3 \sinh \left (\frac {7 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (9 \sinh \left (\frac {9 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{256 b c^4}\\ &=-\frac {x^3 \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}-\frac {3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}+\frac {21 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac {9 \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}+\frac {3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{128 b^2 c^4}+\frac {3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{32 b^2 c^4}-\frac {21 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}-\frac {9 \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 a}{b}+9 \sinh ^{-1}(c x)\right )}{256 b^2 c^4}\\ \end {align*}
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Mathematica [A] time = 1.38, size = 408, normalized size = 1.47 \[ -\frac {6 \cosh \left (\frac {a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+24 \cosh \left (\frac {3 a}{b}\right ) \left (a+b \sinh ^{-1}(c x)\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-21 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-21 b \cosh \left (\frac {7 a}{b}\right ) \sinh ^{-1}(c x) \text {Chi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-9 a \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-9 b \cosh \left (\frac {9 a}{b}\right ) \sinh ^{-1}(c x) \text {Chi}\left (9 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-6 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-6 b \sinh \left (\frac {a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-24 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-24 b \sinh \left (\frac {3 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+21 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+21 b \sinh \left (\frac {7 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+9 a \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+9 b \sinh \left (\frac {9 a}{b}\right ) \sinh ^{-1}(c x) \text {Shi}\left (9 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+256 b c^9 x^9+768 b c^7 x^7+768 b c^5 x^5+256 b c^3 x^3}{256 b^2 c^4 \left (a+b \sinh ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{7} + 2 \, c^{2} x^{5} + x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.74, size = 1070, normalized size = 3.86 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {{\left (c^{6} x^{9} + 3 \, c^{4} x^{7} + 3 \, c^{2} x^{5} + x^{3}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (c^{7} x^{10} + 3 \, c^{5} x^{8} + 3 \, c^{3} x^{6} + c x^{4}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} a b c^{2} x + a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c^{2} x^{2} + 1} b^{2} c^{2} x + b^{2} c\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )} + \int \frac {{\left (9 \, c^{7} x^{9} + 20 \, c^{5} x^{7} + 13 \, c^{3} x^{5} + 2 \, c x^{3}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} + 3 \, {\left (6 \, c^{8} x^{10} + 17 \, c^{6} x^{8} + 17 \, c^{4} x^{6} + 7 \, c^{2} x^{4} + x^{2}\right )} {\left (c^{2} x^{2} + 1\right )} + {\left (9 \, c^{9} x^{11} + 31 \, c^{7} x^{9} + 39 \, c^{5} x^{7} + 21 \, c^{3} x^{5} + 4 \, c x^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{a b c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} a b c^{3} x^{2} + 2 \, a b c^{3} x^{2} + a b c + {\left (b^{2} c^{5} x^{4} + {\left (c^{2} x^{2} + 1\right )} b^{2} c^{3} x^{2} + 2 \, b^{2} c^{3} x^{2} + b^{2} c + 2 \, {\left (b^{2} c^{4} x^{3} + b^{2} c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + 2 \, {\left (a b c^{4} x^{3} + a b c^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}\,{d x} \]
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 \[ \int \frac {x^3\,{\left (c^2\,x^2+1\right )}^{5/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x^{3} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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