Optimal. Leaf size=431 \[ \frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{48 b^4 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^4 d}+\frac {125 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{96 b^4 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{48 b^4 d}-\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{32 b^4 d}-\frac {125 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{96 b^4 d}-\frac {25 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {2 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]
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Rubi [A] time = 1.10, antiderivative size = 427, normalized size of antiderivative = 0.99, number of steps used = 24, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {5866, 12, 5668, 5775, 5666, 3303, 3298, 3301} \[ \frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}+\frac {125 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}-\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}-\frac {125 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}-\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {25 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^4 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3298
Rule 3301
Rule 3303
Rule 5666
Rule 5668
Rule 5775
Rule 5866
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^4} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^4 x^4}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^4 \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b \cosh ^{-1}(x)\right )^4} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}-\frac {\left (4 e^4\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{3 b d}+\frac {\left (5 e^4\right ) \operatorname {Subst}\left (\int \frac {x^5}{\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^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{6 b^2 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {25 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (2 e^4\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{4 (a+b x)}-\frac {3 \cosh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{b^3 d}-\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \left (-\frac {\cosh (x)}{8 (a+b x)}-\frac {9 \cosh (3 x)}{16 (a+b x)}-\frac {5 \cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{6 b^3 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {25 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^4 \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac {\left (25 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 b^3 d}+\frac {\left (125 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (5 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{96 b^3 d}-\frac {\left (3 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac {\left (75 e^4\right ) \operatorname {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^3 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {25 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {\left (e^4 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac {\left (25 e^4 \cosh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 b^3 d}-\frac {\left (3 e^4 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}+\frac {\left (75 e^4 \cosh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^3 d}+\frac {\left (125 e^4 \cosh \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{96 b^3 d}+\frac {\left (e^4 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}-\frac {\left (25 e^4 \sinh \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{48 b^3 d}+\frac {\left (3 e^4 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^3 d}-\frac {\left (75 e^4 \sinh \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{32 b^3 d}-\frac {\left (125 e^4 \sinh \left (\frac {5 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{96 b^3 d}\\ &=-\frac {e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{3 b d \left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {2 e^4 (c+d x)^3}{3 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {5 e^4 (c+d x)^5}{6 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {2 e^4 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {25 e^4 \sqrt {-1+c+d x} (c+d x)^4 \sqrt {1+c+d x}}{6 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^4 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}+\frac {27 e^4 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}+\frac {125 e^4 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}-\frac {e^4 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{48 b^4 d}-\frac {27 e^4 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{32 b^4 d}-\frac {125 e^4 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \cosh ^{-1}(c+d x)\right )}{96 b^4 d}\\ \end {align*}
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Mathematica [A] time = 2.01, size = 424, normalized size = 0.98 \[ \frac {e^4 \left (-\frac {32 b^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^4}{\left (a+b \cosh ^{-1}(c+d x)\right )^3}+\frac {16 b^2 \left (4 (c+d x)^3-5 (c+d x)^5\right )}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+384 \left (\cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )-544 \left (3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )-3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-\sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )+125 \left (10 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+5 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )-10 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )-5 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )-\frac {16 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (25 (c+d x)^4-12 (c+d x)^2\right )}{a+b \cosh ^{-1}(c+d x)}\right )}{96 b^4 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {d^{4} e^{4} x^{4} + 4 \, c d^{3} e^{4} x^{3} + 6 \, c^{2} d^{2} e^{4} x^{2} + 4 \, c^{3} d e^{4} x + c^{4} e^{4}}{b^{4} \operatorname {arcosh}\left (d x + c\right )^{4} + 4 \, a b^{3} \operatorname {arcosh}\left (d x + c\right )^{3} + 6 \, a^{2} b^{2} \operatorname {arcosh}\left (d x + c\right )^{2} + 4 \, a^{3} b \operatorname {arcosh}\left (d x + c\right ) + a^{4}}, x\right ) \]
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 \[ \int \frac {{\left (d e x + c e\right )}^{4}}{{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.47, size = 1375, normalized size = 3.19 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (c\,e+d\,e\,x\right )}^4}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4} \,d x \]
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 \[ e^{4} \left (\int \frac {c^{4}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {d^{4} x^{4}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {4 c d^{3} x^{3}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {6 c^{2} d^{2} x^{2}}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx + \int \frac {4 c^{3} d x}{a^{4} + 4 a^{3} b \operatorname {acosh}{\left (c + d x \right )} + 6 a^{2} b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + 4 a b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )} + b^{4} \operatorname {acosh}^{4}{\left (c + d x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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