Optimal. Leaf size=444 \[ -\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {b c^2 f \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {b c^2 f \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}} \]
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Rubi [A]
time = 0.45, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5845, 5843,
3405, 3403, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} -\frac {g \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right ) (f+g x)}+\frac {c^2 f \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{c f-\sqrt {c^2 f^2+g^2}}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {c^2 f \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {g e^{\sinh ^{-1}(c x)}}{\sqrt {c^2 f^2+g^2}+c f}+1\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c^2 f \sqrt {c^2 x^2+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}-\frac {b c^2 f \sqrt {c^2 x^2+1} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )^{3/2}}+\frac {b c \sqrt {c^2 x^2+1} \log (f+g x)}{\sqrt {c^2 d x^2+d} \left (c^2 f^2+g^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3403
Rule 3405
Rule 5843
Rule 5845
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{(f+g x)^2 \sqrt {d+c^2 d x^2}} \, dx &=\frac {\sqrt {1+c^2 x^2} \int \frac {a+b \sinh ^{-1}(c x)}{(f+g x)^2 \sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\left (c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{(c f+g \sinh (x))^2} \, dx,x,\sinh ^{-1}(c x)\right )}{\sqrt {d+c^2 d x^2}}\\ &=-\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {\left (c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {a+b x}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {\left (b c g \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x)}{c f+g \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}\\ &=-\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {\left (b c \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {1}{c f+x} \, dx,x,c g x\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {\left (2 c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c e^x f-g+e^{2 x} g} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}\\ &=-\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {\left (2 c^2 f g \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g-2 \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {\left (2 c^2 f g \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {e^x (a+b x)}{2 c f+2 e^x g+2 \sqrt {c^2 f^2+g^2}} \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}\\ &=-\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}-\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+\frac {2 e^x g}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}\\ &=-\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}-\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f-2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {\left (b c^2 f \sqrt {1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 g x}{2 c f+2 \sqrt {c^2 f^2+g^2}}\right )}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}\\ &=-\frac {g \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )}{\left (c^2 f^2+g^2\right ) (f+g x) \sqrt {d+c^2 d x^2}}+\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {c^2 f \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}+\frac {b c \sqrt {1+c^2 x^2} \log (f+g x)}{\left (c^2 f^2+g^2\right ) \sqrt {d+c^2 d x^2}}+\frac {b c^2 f \sqrt {1+c^2 x^2} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}-\frac {b c^2 f \sqrt {1+c^2 x^2} \text {Li}_2\left (-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{\left (c^2 f^2+g^2\right )^{3/2} \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 1.50, size = 448, normalized size = 1.01 \begin {gather*} \frac {-a g \sqrt {c^2 f^2+g^2} \left (d+c^2 d x^2\right )+a c^2 \sqrt {d} f (f+g x) \sqrt {d+c^2 d x^2} \log (f+g x)-a c^2 \sqrt {d} f (f+g x) \sqrt {d+c^2 d x^2} \log \left (d \left (g-c^2 f x\right )+\sqrt {d} \sqrt {c^2 f^2+g^2} \sqrt {d+c^2 d x^2}\right )-b d \sqrt {1+c^2 x^2} \left (g \sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)-c^2 f (f+g x) \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )+c^2 f (f+g x) \sinh ^{-1}(c x) \log \left (1+\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )-c \sqrt {c^2 f^2+g^2} (f+g x) \log (c (f+g x))-c^2 f (f+g x) \text {PolyLog}\left (2,\frac {e^{\sinh ^{-1}(c x)} g}{-c f+\sqrt {c^2 f^2+g^2}}\right )+c^2 f (f+g x) \text {PolyLog}\left (2,-\frac {e^{\sinh ^{-1}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )\right )}{d \left (c^2 f^2+g^2\right )^{3/2} (f+g x) \sqrt {d+c^2 d x^2}} \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. \(1769\) vs.
\(2(442)=884\).
time = 8.37, size = 1770, normalized size = 3.99
method | result | size |
default | \(\text {Expression too large to display}\) | \(1770\) |
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 {a + b \operatorname {asinh}{\left (c x \right )}}{\sqrt {d \left (c^{2} x^{2} + 1\right )} \left (f + g x\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 {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\sqrt {d\,c^2\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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