Optimal. Leaf size=431 \[ -\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d} \]
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Rubi [A]
time = 0.98, antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5996, 12,
5886, 5951, 5885, 3388, 2211, 2236, 2235, 5880, 5953} \begin {gather*} \frac {\sqrt {\pi } e^2 e^{a/b} \text {Erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 \sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {Erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {Erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 \sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {Erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {24 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {16 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {2 e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 5880
Rule 5885
Rule 5886
Rule 5951
Rule 5953
Rule 5996
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^{7/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^{7/2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}-\frac {\left (4 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^{5/2}} \, dx,x,c+d x\right )}{5 b d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {\left (8 e^2\right ) \text {Subst}\left (\int \frac {1}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}+\frac {\left (12 e^2\right ) \text {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \sqrt {a+b \cosh ^{-1}(x)}} \, dx,x,c+d x\right )}{15 b^3 d}-\frac {\left (24 e^2\right ) \text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 \sqrt {a+b x}}-\frac {3 \cosh (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int \frac {\cosh (x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (18 e^2\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (8 e^2\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}-\frac {\left (8 e^2\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{15 b^3 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {e^{-x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {e^x}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {e^{-3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {e^{3 x}}{\sqrt {a+b x}} \, dx,x,\cosh ^{-1}(c+d x)\right )}{5 b^3 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{15 b^4 d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac {\left (18 e^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}+\frac {\left (18 e^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \cosh ^{-1}(c+d x)}\right )}{5 b^4 d}\\ &=-\frac {2 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}}+\frac {8 e^2 (c+d x)}{15 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )^{3/2}}+\frac {16 e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{15 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}-\frac {24 e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{5 b^3 d \sqrt {a+b \cosh ^{-1}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}+\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \cosh ^{-1}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}\\ \end {align*}
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Mathematica [A]
time = 2.27, size = 452, normalized size = 1.05 \begin {gather*} \frac {e^2 \left (-6 b^2 \sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)-2 e^{-\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (-2 a+b-2 b \cosh ^{-1}(c+d x)+2 e^{\frac {a}{b}+\cosh ^{-1}(c+d x)} \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 )\right )-2 e^{-\frac {a}{b}} \left (a+b \cosh ^{-1}(c+d x)\right ) \left (e^{\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (2 a+b+2 b \cosh ^{-1}(c+d x)\right )+2 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 )\right )-3 \left (a+b \cosh ^{-1}(c+d x)\right ) \left (12 \sqrt {3} b e^{-\frac {3 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )+2 e^{-3 \cosh ^{-1}(c+d x)} \left (b+6 a \left (-1+e^{6 \cosh ^{-1}(c+d x)}\right )-6 b \cosh ^{-1}(c+d x)+b e^{6 \cosh ^{-1}(c+d x)} \left (1+6 \cosh ^{-1}(c+d x)\right )+6 \sqrt {3} e^{3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )} \sqrt {\frac {a}{b}+\cosh ^{-1}(c+d x)} \left (a+b \cosh ^{-1}(c+d x)\right ) \Gamma \left (\frac {1}{2},\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )\right )\right )-6 b^2 \sinh \left (3 \cosh ^{-1}(c+d x)\right )\right )}{60 b^3 d \left (a+b \cosh ^{-1}(c+d x)\right )^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{\frac {7}{2}}}\, dx\]
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^{2} \left (\int \frac {c^{2}}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {acosh}{\left (c + d x \right )}} \operatorname {acosh}^{3}{\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 )}^2}{{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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