Optimal. Leaf size=360 \[ -\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {8 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d} \]
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Rubi [A]
time = 0.57, antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5996, 12,
5886, 5951, 5885, 3384, 3379, 3382} \begin {gather*} \frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{3 b^4 d}-\frac {8 e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5885
Rule 5886
Rule 5951
Rule 5996
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^3}{\left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx &=\frac {\text {Subst}\left (\int \frac {e^3 x^3}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}-\frac {e^3 \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{b d}+\frac {\left (4 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^3 \text {Subst}\left (\int \frac {x}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (8 e^3\right ) \text {Subst}\left (\int \frac {x^3}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{3 b^2 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {8 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^3 d}-\frac {\left (8 e^3\right ) \text {Subst}\left (\int \left (-\frac {\cosh (2 x)}{2 (a+b x)}-\frac {\cosh (4 x)}{2 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {8 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (4 e^3\right ) \text {Subst}\left (\int \frac {\cosh (2 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (4 e^3\right ) \text {Subst}\left (\int \frac {\cosh (4 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^3 d}+\frac {\left (e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^3 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {8 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{b^4 d}+\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{b^4 d}+\frac {\left (4 e^3 \cosh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}+\frac {\left (4 e^3 \cosh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (4 e^3 \sinh \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}-\frac {\left (4 e^3 \sinh \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{3 b^3 d}\\ &=-\frac {e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {e^3 (c+d x)^2}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 e^3 (c+d x)^4}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {8 e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{3 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^3 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}+\frac {4 e^3 \cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac {e^3 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 a}{b}+2 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}-\frac {4 e^3 \sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (\frac {4 a}{b}+4 \cosh ^{-1}(c+d x)\right )}{3 b^4 d}\\ \end {align*}
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Mathematica [A]
time = 1.28, size = 330, normalized size = 0.92 \begin {gather*} \frac {e^3 \left (-\frac {2 b^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {b^2 \left (3 (c+d x)^2-4 (c+d x)^4\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {2 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-3 (c+d x)+8 (c+d x)^3\right )}{a+b \cosh ^{-1}(c+d x)}+6 \log \left (a+b \cosh ^{-1}(c+d x)\right )-30 \left (\cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\log \left (a+b \cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )+8 \left (4 \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac {4 a}{b}\right ) \text {Chi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+3 \log \left (a+b \cosh ^{-1}(c+d x)\right )-4 \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac {4 a}{b}\right ) \text {Shi}\left (4 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )\right )}{6 b^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(859\) vs.
\(2(332)=664\).
time = 0.14, size = 860, normalized size = 2.39 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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*} e^{3} \left (\int \frac {c^{3}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {3 c^{2} d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\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 {{\left (c\,e+d\,e\,x\right )}^3}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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