Optimal. Leaf size=380 \[ -\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {b^2 c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {PolyLog}\left (2,-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \]
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Rubi [A]
time = 0.54, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {5963, 5980,
3405, 3401, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} -\frac {b c \sqrt {-\frac {1-c x}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {e e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}+1\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3401
Rule 3405
Rule 5963
Rule 5980
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(d+e x)^3} \, dx &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {(b c) \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x} (d+e x)^2} \, dx}{e}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {a+b x}{(c d+e \cosh (x))^2} \, dx,x,\cosh ^{-1}(c x)\right )}{e}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 d^2-e^2}+\frac {\left (b c^3 d\right ) \text {Subst}\left (\int \frac {a+b x}{c d+e \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {1}{c d+x} \, dx,x,c e x\right )}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 b c^3 d\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{e+2 c d e^x+e e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {\left (2 b c^3 d\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d-2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}-\frac {\left (2 b c^3 d\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c d+2 \sqrt {c^2 d^2-e^2}+2 e e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e e^x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}-\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d-2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {\left (b^2 c^3 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 e x}{2 c d+2 \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ &=-\frac {b c \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \left (a+b \cosh ^{-1}(c x)\right )}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{2 e (d+e x)^2}+\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b c^3 d \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 c^2 \log (d+e x)}{e \left (c^2 d^2-e^2\right )}+\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 c^3 d \text {Li}_2\left (-\frac {e e^{\cosh ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e \left (c^2 d^2-e^2\right )^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.64, size = 1099, normalized size = 2.89 \begin {gather*} -\frac {a^2}{2 e (d+e x)^2}+a b c^2 \left (-\frac {\sqrt {-1+c x} \sqrt {1+c x}}{(c d-e) (c d+e) (c d+c e x)}-\frac {\cosh ^{-1}(c x)}{e (c d+c e x)^2}+\frac {2 c d \text {ArcTan}\left (\frac {\sqrt {c d-e} \sqrt {\frac {-1+c x}{1+c x}}}{\sqrt {-c d-e}}\right )}{(-c d-e)^{3/2} (c d-e)^{3/2} e}\right )+b^2 c^2 \left (-\frac {\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \cosh ^{-1}(c x)}{(c d-e) (c d+e) (c d+c e x)}-\frac {\cosh ^{-1}(c x)^2}{2 e (c d+c e x)^2}+\frac {\log \left (1+\frac {e x}{d}\right )}{c^2 d^2 e-e^3}+\frac {c d \left (2 \cosh ^{-1}(c x) \text {ArcTan}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )-2 i \text {ArcCos}\left (-\frac {c d}{e}\right ) \text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\left (\text {ArcCos}\left (-\frac {c d}{e}\right )+2 \left (\text {ArcTan}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{-\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c (d+e x)}}\right )+\left (\text {ArcCos}\left (-\frac {c d}{e}\right )-2 \left (\text {ArcTan}\left (\frac {(c d+e) \coth \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )+\text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right )\right ) \log \left (\frac {\sqrt {-c^2 d^2+e^2} e^{\frac {1}{2} \cosh ^{-1}(c x)}}{\sqrt {2} \sqrt {e} \sqrt {c (d+e x)}}\right )-\left (\text {ArcCos}\left (-\frac {c d}{e}\right )+2 \text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (c d-e+i \sqrt {-c^2 d^2+e^2}\right ) \left (-1+\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\left (\text {ArcCos}\left (-\frac {c d}{e}\right )-2 \text {ArcTan}\left (\frac {(-c d+e) \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )}{\sqrt {-c^2 d^2+e^2}}\right )\right ) \log \left (\frac {(c d+e) \left (-c d+e+i \sqrt {-c^2 d^2+e^2}\right ) \left (1+\tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )+i \left (\text {PolyLog}\left (2,\frac {\left (c d-i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )-\text {PolyLog}\left (2,\frac {\left (c d+i \sqrt {-c^2 d^2+e^2}\right ) \left (c d+e-i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}{e \left (c d+e+i \sqrt {-c^2 d^2+e^2} \tanh \left (\frac {1}{2} \cosh ^{-1}(c x)\right )\right )}\right )\right )\right )}{e \left (-c^2 d^2+e^2\right )^{3/2}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1173\) vs.
\(2(410)=820\).
time = 9.19, size = 1174, normalized size = 3.09 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{\left (d + e x\right )^{3}}\, 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 (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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