Optimal. Leaf size=216 \[ -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 e e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d} \]
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Rubi [A]
time = 0.50, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5996, 12,
5886, 5951, 5887, 5556, 3389, 2211, 2236, 2235, 5893} \begin {gather*} -\frac {2 \sqrt {2 \pi } e e^{\frac {2 a}{b}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 \sqrt {2 \pi } e e^{-\frac {2 a}{b}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5886
Rule 5887
Rule 5893
Rule 5951
Rule 5996
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {(2 e) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}+\frac {(4 e) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(16 e) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(16 e) \text {Subst}\left (\int \frac {\cosh (x) \sinh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(16 e) \text {Subst}\left (\int \frac {\sinh (2 x)}{2 \sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {(8 e) \text {Subst}\left (\int \frac {\sinh (2 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {(4 e) \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}+\frac {(4 e) \text {Subst}\left (\int \frac {e^{2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^2 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {(8 e) \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{3 b^3 d}+\frac {(8 e) \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{3 b^3 d}\\ &=-\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {4 e}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {2 e e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}+\frac {2 e e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(687\) vs. \(2(216)=432\).
time = 4.74, size = 687, normalized size = 3.18 \begin {gather*} \frac {e \left (4 a \sqrt {b} c (c+d x)+4 b^{3/2} c (c+d x) \cosh ^{-1}(c+d x)-2 \sqrt {b} c e^{-\cosh ^{-1}(c+d x)} \left (1+e^{2 \cosh ^{-1}(c+d x)}\right ) \left (a+b \cosh ^{-1}(c+d x)\right )-4 a \sqrt {b} \cosh \left (2 \cosh ^{-1}(c+d x)\right )-4 b^{3/2} \cosh ^{-1}(c+d x) \cosh \left (2 \cosh ^{-1}(c+d x)\right )+2 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \cosh \left (\frac {a}{b}\right ) \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-2 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \cosh \left (\frac {2 a}{b}\right ) \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )-2 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \cosh \left (\frac {a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+2 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \cosh \left (\frac {2 a}{b}\right ) \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+2 \sqrt {b} c e^{a/b} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-2 b^{3/2} c e^{-\frac {a}{b}} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )+2 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+2 c \sqrt {\pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )-2 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-2 \sqrt {2 \pi } \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )-b^{3/2} \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )}{3 b^{5/2} d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {d e x +c e}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {5}{2}}}\, dx\]
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*} e \left (\int \frac {c}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 2 a b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )}}\, dx\right ) \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 {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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