Optimal. Leaf size=236 \[ \frac {\sqrt {\pi } d e^{\frac {4 a}{b}} \text {erf}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {\sqrt {\pi } d e^{-\frac {4 a}{b}} \text {erfi}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}} \]
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Rubi [A] time = 0.77, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5777, 5699, 3312, 3307, 2180, 2204, 2205, 5779, 5448} \[ \frac {\sqrt {\pi } d e^{\frac {4 a}{b}} \text {Erf}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {\sqrt {\pi } d e^{-\frac {4 a}{b}} \text {Erfi}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {\sqrt {\frac {\pi }{2}} d e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}-\frac {2 d x \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}} \]
Antiderivative was successfully verified.
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Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 3312
Rule 5448
Rule 5699
Rule 5777
Rule 5779
Rubi steps
\begin {align*} \int \frac {x \left (d+c^2 d x^2\right )}{\left (a+b \sinh ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(2 d) \int \frac {\sqrt {1+c^2 x^2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b c}+\frac {(8 c d) \int \frac {x^2 \sqrt {1+c^2 x^2}}{\sqrt {a+b \sinh ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(2 d) \operatorname {Subst}\left (\int \frac {\cosh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {(8 d) \operatorname {Subst}\left (\int \frac {\cosh ^2(x) \sinh ^2(x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {(2 d) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cosh (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {(8 d) \operatorname {Subst}\left (\int \left (-\frac {1}{8 \sqrt {a+b x}}+\frac {\cosh (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {d \operatorname {Subst}\left (\int \frac {\cosh (4 x)}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d \operatorname {Subst}\left (\int \frac {e^{-4 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}+\frac {d \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}+\frac {d \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}+\frac {d \operatorname {Subst}\left (\int \frac {e^{4 x}}{\sqrt {a+b x}} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^2}\\ &=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d \operatorname {Subst}\left (\int e^{\frac {4 a}{b}-\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {d \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {d \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}+\frac {d \operatorname {Subst}\left (\int e^{-\frac {4 a}{b}+\frac {4 x^2}{b}} \, dx,x,\sqrt {a+b \sinh ^{-1}(c x)}\right )}{b^2 c^2}\\ &=-\frac {2 d x \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \sinh ^{-1}(c x)}}+\frac {d e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}+\frac {d e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c^2}+\frac {d e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \sinh ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c^2}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 227, normalized size = 0.96 \[ \frac {d e^{-\frac {4 a}{b}} \left (\sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+\sqrt {2} e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \sinh ^{-1}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )-e^{\frac {4 a}{b}} \left (\sqrt {2} e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {2 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\sinh ^{-1}(c x)} \Gamma \left (\frac {1}{2},\frac {4 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )+2 \sinh \left (2 \sinh ^{-1}(c x)\right )+\sinh \left (4 \sinh ^{-1}(c x)\right )\right )\right )}{4 b c^2 \sqrt {a+b \sinh ^{-1}(c x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {{\left (c^{2} d x^{2} + d\right )} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x \left (c^{2} d \,x^{2}+d \right )}{\left (a +b \arcsinh \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (c^{2} d x^{2} + d\right )} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x\,\left (d\,c^2\,x^2+d\right )}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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 \[ d \left (\int \frac {x}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{2} x^{3}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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