Optimal. Leaf size=296 \[ -\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {2 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}} \]
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Rubi [A]
time = 0.27, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5790, 5814,
5791, 3393, 3388, 2211, 2235, 2236, 5819, 5556} \begin {gather*} \frac {2 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {a^2 c x^2+c} \text {Erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {\pi } c \sqrt {a^2 c x^2+c} \text {Erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}+\frac {2 \sqrt {2 \pi } c \sqrt {a^2 c x^2+c} \text {Erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {a^2 x^2+1}}-\frac {2 \sqrt {a^2 x^2+1} \left (a^2 c x^2+c\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c}}{3 \sqrt {\sinh ^{-1}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3393
Rule 5556
Rule 5790
Rule 5791
Rule 5814
Rule 5819
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{3/2}}{\sinh ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}+\frac {\left (8 a c \sqrt {c+a^2 c x^2}\right ) \int \frac {x \left (1+a^2 x^2\right )}{\sinh ^{-1}(a x)^{3/2}} \, dx}{3 \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (16 c \sqrt {c+a^2 c x^2}\right ) \int \frac {\sqrt {1+a^2 x^2}}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{3 \sqrt {1+a^2 x^2}}+\frac {\left (64 a^2 c \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2 \sqrt {1+a^2 x^2}}{\sqrt {\sinh ^{-1}(a x)}} \, dx}{3 \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (16 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (64 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (16 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}+\frac {\cosh (2 x)}{2 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (64 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{8 \sqrt {x}}+\frac {\cosh (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (4 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (4 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (4 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (4 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int \frac {e^{4 x}}{\sqrt {x}} \, dx,x,\sinh ^{-1}(a x)\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{-2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{2 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {\left (8 c \sqrt {c+a^2 c x^2}\right ) \text {Subst}\left (\int e^{4 x^2} \, dx,x,\sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}\\ &=-\frac {2 \sqrt {1+a^2 x^2} \left (c+a^2 c x^2\right )^{3/2}}{3 a \sinh ^{-1}(a x)^{3/2}}-\frac {16 c x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2}}{3 \sqrt {\sinh ^{-1}(a x)}}+\frac {2 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erf}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erf}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {\pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (2 \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}+\frac {2 c \sqrt {2 \pi } \sqrt {c+a^2 c x^2} \text {erfi}\left (\sqrt {2} \sqrt {\sinh ^{-1}(a x)}\right )}{3 a \sqrt {1+a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.28, size = 262, normalized size = 0.89 \begin {gather*} -\frac {c e^{-4 \sinh ^{-1}(a x)} \sqrt {c+a^2 c x^2} \left (1+14 e^{4 \sinh ^{-1}(a x)}+e^{8 \sinh ^{-1}(a x)}+16 a^2 e^{4 \sinh ^{-1}(a x)} x^2-8 \sinh ^{-1}(a x)+8 e^{8 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)+64 a e^{4 \sinh ^{-1}(a x)} x \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)+16 e^{4 \sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-4 \sinh ^{-1}(a x)\right )+16 \sqrt {2} e^{4 \sinh ^{-1}(a x)} \left (-\sinh ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-2 \sinh ^{-1}(a x)\right )+16 \sqrt {2} e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},2 \sinh ^{-1}(a x)\right )+16 e^{4 \sinh ^{-1}(a x)} \sinh ^{-1}(a x)^{3/2} \Gamma \left (\frac {1}{2},4 \sinh ^{-1}(a x)\right )\right )}{24 a \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}{\arcsinh \left (a x \right )^{\frac {5}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}{\operatorname {asinh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (c\,a^2\,x^2+c\right )}^{3/2}}{{\mathrm {asinh}\left (a\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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