Optimal. Leaf size=155 \[ \frac {\sqrt {\frac {\pi }{2}} e e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\sqrt {\frac {\pi }{2}} e e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b d \sqrt {a+b \cosh ^{-1}(c+d x)}} \]
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Rubi [A] time = 0.23, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5866, 12, 5666, 3307, 2180, 2204, 2205} \[ \frac {\sqrt {\frac {\pi }{2}} e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {\sqrt {\frac {\pi }{2}} e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b d \sqrt {a+b \cosh ^{-1}(c+d x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2180
Rule 2204
Rule 2205
Rule 3307
Rule 5666
Rule 5866
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \operatorname {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(2 e) \operatorname {Subst}\left (\int \frac {\cosh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {e \operatorname {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}+\frac {e \operatorname {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(2 e) \operatorname {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^2 d}+\frac {(2 e) \operatorname {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{b^2 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}\\ \end {align*}
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Mathematica [B] time = 6.71, size = 314, normalized size = 2.03 \[ \frac {e \left (-2 \sqrt {\pi } c \left (\sinh \left (\frac {a}{b}\right )+\cosh \left (\frac {a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } \left (\sinh \left (\frac {2 a}{b}\right )+\cosh \left (\frac {2 a}{b}\right )\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-2 \sqrt {\pi } c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+\sqrt {2 \pi } e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-\frac {2 \sqrt {b} e^{-\frac {a}{b}} \left (e^{a/b} \sinh \left (2 \cosh ^{-1}(c+d x)\right )+c e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-c \sqrt {-\frac {a+b \cosh ^{-1}(c+d x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )\right )}{\sqrt {a+b \cosh ^{-1}(c+d x)}}\right )}{2 b^{3/2} d} \]
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 \[ \int \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {d e x +c e}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {d e x + c e}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ e \left (\int \frac {c}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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