Optimal. Leaf size=275 \[ -\frac {x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2} \]
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Rubi [A]
time = 0.57, antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps
used = 28, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5814, 5791,
3393, 3384, 3379, 3382, 5819, 5556} \begin {gather*} \frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{64 b^2 c^2}-\frac {x \left (c^2 x^2+1\right )^3}{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 3393
Rule 5556
Rule 5791
Rule 5814
Rule 5819
Rubi steps
\begin {align*} \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{\left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\int \frac {\left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b c}+\frac {(7 c) \int \frac {x^2 \left (1+c^2 x^2\right )^2}{a+b \sinh ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\text {Subst}\left (\int \frac {\cosh ^5(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {7 \text {Subst}\left (\int \frac {\cosh ^5(x) \sinh ^2(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\text {Subst}\left (\int \left (\frac {5 \cosh (x)}{8 (a+b x)}+\frac {5 \cosh (3 x)}{16 (a+b x)}+\frac {\cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}+\frac {7 \text {Subst}\left (\int \left (-\frac {5 \cosh (x)}{64 (a+b x)}+\frac {\cosh (3 x)}{64 (a+b x)}+\frac {3 \cosh (5 x)}{64 (a+b x)}+\frac {\cosh (7 x)}{64 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^2}\\ &=-\frac {x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\text {Subst}\left (\int \frac {\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}+\frac {7 \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}+\frac {7 \text {Subst}\left (\int \frac {\cosh (7 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}+\frac {5 \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^2}+\frac {21 \text {Subst}\left (\int \frac {\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}-\frac {35 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}+\frac {5 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^2}\\ &=-\frac {x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}-\frac {\left (35 \cosh \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 )}{64 b c^2}+\frac {\left (5 \cosh \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^2}+\frac {\left (7 \cosh \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 )}{64 b c^2}+\frac {\left (5 \cosh \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^2}+\frac {\cosh \left (\frac {5 a}{b}\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^2}+\frac {\left (21 \cosh \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 )}{64 b c^2}+\frac {\left (7 \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}+\frac {\left (35 \sinh \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 )}{64 b c^2}-\frac {\left (5 \sinh \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^2}-\frac {\left (7 \sinh \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 )}{64 b c^2}-\frac {\left (5 \sinh \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^2}-\frac {\sinh \left (\frac {5 a}{b}\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^2}-\frac {\left (21 \sinh \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 )}{64 b c^2}-\frac {\left (7 \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{64 b c^2}\\ &=-\frac {x \left (1+c^2 x^2\right )^3}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \sinh ^{-1}(c x)\right )}{64 b^2 c^2}\\ \end {align*}
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Mathematica [A]
time = 0.73, size = 404, normalized size = 1.47 \begin {gather*} -\frac {64 b c x+192 b c^3 x^3+192 b c^5 x^5+64 b c^7 x^7-5 \left (a+b \sinh ^{-1}(c x)\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-27 \left (a+b \sinh ^{-1}(c x)\right ) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-25 a \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-25 b \sinh ^{-1}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-7 b \sinh ^{-1}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+5 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+5 b \sinh ^{-1}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+27 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+27 b \sinh ^{-1}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+25 a \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+25 b \sinh ^{-1}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+7 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+7 b \sinh ^{-1}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )}{64 b^2 c^2 \left (a+b \sinh ^{-1}(c x)\right )} \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. \(957\) vs.
\(2(259)=518\).
time = 7.58, size = 958, normalized size = 3.48
method | result | size |
default | \(\text {Expression too large to display}\) | \(958\) |
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 {x \left (c^{2} x^{2} + 1\right )^{\frac {5}{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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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 {x\,{\left (c^2\,x^2+1\right )}^{5/2}}{{\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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