Optimal. Leaf size=495 \[ -\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {d^2 \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {d e \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{2 b^2 c^3}-\frac {e^2 \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^5}-\frac {3 d e \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{2 b^2 c^3}+\frac {9 e^2 \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \text {Chi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^5}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{2 b^2 c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^5}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}-\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5} \]
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Rubi [A]
time = 0.55, antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5793, 5773,
5819, 3384, 3379, 3382, 5778} \begin {gather*} -\frac {e^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^5}+\frac {9 e^2 \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \sinh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^5}-\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^5}+\frac {d e \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{2 b^2 c^3}-\frac {3 d e \sinh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}-\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{2 b^2 c^3}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{2 b^2 c^3}-\frac {d^2 \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{b^2 c}-\frac {d^2 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {c^2 x^2+1}}{b c \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5773
Rule 5778
Rule 5793
Rule 5819
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=\int \left (\frac {d^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac {2 d e x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2}+\frac {e^2 x^4}{\left (a+b \sinh ^{-1}(c x)\right )^2}\right ) \, dx\\ &=d^2 \int \frac {1}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+(2 d e) \int \frac {x^2}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx+e^2 \int \frac {x^4}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\left (c d^2\right ) \int \frac {x}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )} \, dx}{b}+\frac {(2 d e) \text {Subst}\left (\int \left (-\frac {\sinh (x)}{4 (a+b x)}+\frac {3 \sinh (3 x)}{4 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^3}+\frac {e^2 \text {Subst}\left (\int \left (\frac {\sinh (x)}{8 (a+b x)}-\frac {9 \sinh (3 x)}{16 (a+b x)}+\frac {5 \sinh (5 x)}{16 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^5}\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {d^2 \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c}-\frac {(d e) \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {(3 d e) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {e^2 \text {Subst}\left (\int \frac {\sinh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^5}+\frac {\left (5 e^2\right ) \text {Subst}\left (\int \frac {\sinh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}-\frac {\left (9 e^2\right ) \text {Subst}\left (\int \frac {\sinh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\left (d^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,\sinh ^{-1}(c x)\right )}{b c}-\frac {\left (d e \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}+\frac {\left (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,\sinh ^{-1}(c x)\right )}{8 b c^5}+\frac {\left (3 d e \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,\sinh ^{-1}(c x)\right )}{2 b c^3}-\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,\sinh ^{-1}(c x)\right )}{16 b c^5}+\frac {\left (5 e^2 \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}-\frac {\left (d^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,\sinh ^{-1}(c x)\right )}{b c}+\frac {\left (d e \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 b c^3}-\frac {\left (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,\sinh ^{-1}(c x)\right )}{8 b c^5}-\frac {\left (3 d e \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,\sinh ^{-1}(c x)\right )}{2 b c^3}+\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,\sinh ^{-1}(c x)\right )}{16 b c^5}-\frac {\left (5 e^2 \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^5}\\ &=-\frac {d^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {2 d e x^2 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {e^2 x^4 \sqrt {1+c^2 x^2}}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {d^2 \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {d e \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{2 b^2 c^3}-\frac {e^2 \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )}{8 b^2 c^5}-\frac {3 d e \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{2 b^2 c^3}+\frac {9 e^2 \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {3 a}{b}\right )}{16 b^2 c^5}-\frac {5 e^2 \text {Chi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c x)\right ) \sinh \left (\frac {5 a}{b}\right )}{16 b^2 c^5}+\frac {d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{b^2 c}-\frac {d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{2 b^2 c^3}+\frac {e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{8 b^2 c^5}+\frac {3 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{2 b^2 c^3}-\frac {9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}+\frac {5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c x)\right )}{16 b^2 c^5}\\ \end {align*}
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Mathematica [A]
time = 1.35, size = 356, normalized size = 0.72 \begin {gather*} -\frac {\frac {16 b c^4 d^2 \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)}+\frac {32 b c^4 d e x^2 \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)}+\frac {16 b c^4 e^2 x^4 \sqrt {1+c^2 x^2}}{a+b \sinh ^{-1}(c x)}+2 \left (8 c^4 d^2-4 c^2 d e+e^2\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right ) \sinh \left (\frac {a}{b}\right )+3 \left (8 c^2 d-3 e\right ) e \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right ) \sinh \left (\frac {3 a}{b}\right )+5 e^2 \text {Chi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right ) \sinh \left (\frac {5 a}{b}\right )-16 c^4 d^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+8 c^2 d e \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-2 e^2 \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-24 c^2 d e \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+9 e^2 \cosh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-5 e^2 \cosh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{16 b^2 c^5} \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. \(1035\) vs.
\(2(469)=938\).
time = 9.45, size = 1036, normalized size = 2.09
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1036\) |
default | \(\text {Expression too large to display}\) | \(1036\) |
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]
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 (d + e x^{2}\right )^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, 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 (e\,x^2+d\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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