Optimal. Leaf size=252 \[ -\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^3 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3 d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.53, antiderivative size = 252, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5996, 12,
5886, 5951, 5887, 5556, 3384, 3379, 3382, 5881} \begin {gather*} -\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}-\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c+d x)\right )}{b}\right )}{8 b^3 d}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5881
Rule 5886
Rule 5887
Rule 5951
Rule 5996
Rubi steps
\begin {align*} \int \frac {(c e+d e x)^2}{\left (a+b \cosh ^{-1}(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {e^2 x^2}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int \frac {x^2}{\left (a+b \cosh ^{-1}(x)\right )^3} \, dx,x,c+d x\right )}{d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}-\frac {e^2 \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{b d}+\frac {\left (3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x} \left (a+b \cosh ^{-1}(x)\right )^2} \, dx,x,c+d x\right )}{2 b d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {e^2 \text {Subst}\left (\int \frac {1}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{b^2 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {x^2}{a+b \cosh ^{-1}(x)} \, dx,x,c+d x\right )}{2 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)}+\frac {\sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c+d x)\right )}{2 b^2 d}-\frac {\left (e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b^3 d}+\frac {\left (e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cosh ^{-1}(c+d x)\right )}{b^3 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^3 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}+\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^3 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}+\frac {\left (9 e^2 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}+\frac {\left (9 e^2 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^2 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}-\frac {\left (9 e^2 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c+d x)\right )}{8 b^2 d}\\ &=-\frac {e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{2 b d \left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {e^2 (c+d x)}{b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {3 e^2 (c+d x)^3}{2 b^2 d \left (a+b \cosh ^{-1}(c+d x)\right )}-\frac {9 e^2 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )}{8 b^3 d}+\frac {e^2 \text {Chi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^3 d}-\frac {9 e^2 \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {3 a}{b}\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )}{8 b^3 d}+\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c+d x)\right )}{8 b^3 d}-\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c+d x)}{b}\right )}{b^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.51, size = 223, normalized size = 0.88 \begin {gather*} \frac {e^2 \left (-\frac {4 b^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{\left (a+b \cosh ^{-1}(c+d x)\right )^2}+\frac {4 b \left (2 (c+d x)-3 (c+d x)^3\right )}{a+b \cosh ^{-1}(c+d x)}+8 \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-8 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+9 \left (-\text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right ) \sinh \left (\frac {a}{b}\right )-\text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+\cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )+\cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c+d x)\right )\right )\right )\right )}{8 b^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(556\) vs.
\(2(236)=472\).
time = 0.11, size = 557, normalized size = 2.21
method | result | size |
derivativedivides | \(\frac {-\frac {\left (-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (3 b \,\mathrm {arccosh}\left (d x +c \right )+3 a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b \,\mathrm {arccosh}\left (d x +c \right )+a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\) | \(557\) |
default | \(\frac {-\frac {\left (-4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}+\sqrt {d x +c -1}\, \sqrt {d x +c +1}+4 \left (d x +c \right )^{3}-3 d x -3 c \right ) e^{2} \left (3 b \,\mathrm {arccosh}\left (d x +c \right )+3 a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {9 e^{2} {\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (d x +c \right )+\frac {3 a}{b}\right )}{16 b^{3}}-\frac {\left (-\sqrt {d x +c -1}\, \sqrt {d x +c +1}+d x +c \right ) e^{2} \left (b \,\mathrm {arccosh}\left (d x +c \right )+a -b \right )}{16 b^{2} \left (b^{2} \mathrm {arccosh}\left (d x +c \right )^{2}+2 a b \,\mathrm {arccosh}\left (d x +c \right )+a^{2}\right )}+\frac {e^{2} {\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \mathrm {arccosh}\left (d x +c \right )+\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {e^{2} \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {e^{2} {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arccosh}\left (d x +c \right )-\frac {a}{b}\right )}{16 b^{3}}-\frac {e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )^{2}}-\frac {3 e^{2} \left (4 \left (d x +c \right )^{3}-3 d x -3 c +4 \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, \left (d x +c \right )^{2}-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{16 b^{2} \left (a +b \,\mathrm {arccosh}\left (d x +c \right )\right )}-\frac {9 e^{2} {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (d x +c \right )-\frac {3 a}{b}\right )}{16 b^{3}}}{d}\) | \(557\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{2} \left (\int \frac {c^{2}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} + 3 a^{2} b \operatorname {acosh}{\left (c + d x \right )} + 3 a b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + b^{3} \operatorname {acosh}^{3}{\left (c + d x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________