Optimal. Leaf size=155 \[ -\frac {2 e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {e e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d}+\frac {e e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {Erfi}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{b^{3/2} d} \]
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Rubi [A]
time = 0.16, 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 = {5996, 12, 5885,
3388, 2211, 2236, 2235} \begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5885
Rule 5996
Rubi steps
\begin {align*} \int \frac {c e+d e x}{\left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e x}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e \text {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) \text {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 \text {Subst}\left (\int \frac {e^{-2 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b d}+\frac {e \text {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) \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 )}{b^2 d}+\frac {(2 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 )}{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] Leaf count is larger than twice the leaf count of optimal. \(314\) vs. \(2(155)=310\).
time = 5.30, size = 314, normalized size = 2.03 \begin {gather*} \frac {e \left (-2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )+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 )-2 c \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {a}{b}\right )+\sinh \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \text {Erf}\left (\frac {\sqrt {2} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right ) \left (\cosh \left (\frac {2 a}{b}\right )+\sinh \left (\frac {2 a}{b}\right )\right )-\frac {2 \sqrt {b} e^{-\frac {a}{b}} \left (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 )+e^{a/b} \sinh \left (2 \cosh ^{-1}(c+d x)\right )\right )}{\sqrt {a+b \cosh ^{-1}(c+d x)}}\right )}{2 b^{3/2} d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {d e x +c e}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx\]
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 \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 \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 ) \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 = -0.01 \begin {gather*} \int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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