Optimal. Leaf size=335 \[ \frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b p q \text {Li}_2\left (-\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {h e^2+2 f^2}}\right )}{\sqrt {h}}-\frac {b p q \text {Li}_2\left (-\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{\sqrt {h} e+\sqrt {h e^2+2 f^2}}\right )}{\sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+2 f^2}}+1\right )}{\sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{\sqrt {e^2 h+2 f^2}+e \sqrt {h}}+1\right )}{\sqrt {h}}+\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}} \]
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Rubi [A] time = 0.83, antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {215, 2404, 12, 5799, 5561, 2190, 2279, 2391, 2445} \[ -\frac {b p q \text {PolyLog}\left (2,-\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+2 f^2}}\right )}{\sqrt {h}}-\frac {b p q \text {PolyLog}\left (2,-\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{\sqrt {e^2 h+2 f^2}+e \sqrt {h}}\right )}{\sqrt {h}}+\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+2 f^2}}+1\right )}{\sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{\sqrt {e^2 h+2 f^2}+e \sqrt {h}}+1\right )}{\sqrt {h}}+\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 215
Rule 2190
Rule 2279
Rule 2391
Rule 2404
Rule 2445
Rule 5561
Rule 5799
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {a+b \log \left (c d^q (e+f x)^{p q}\right )}{\sqrt {2+h x^2}} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\operatorname {Subst}\left ((b f p q) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}{\sqrt {h} (e+f x)} \, dx,c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\operatorname {Subst}\left (\frac {(b f p q) \int \frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}{e+f x} \, dx}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\operatorname {Subst}\left (\frac {(b f p q) \operatorname {Subst}\left (\int \frac {x \cosh (x)}{\frac {e \sqrt {h}}{\sqrt {2}}+f \sinh (x)} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )\right )}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}}+\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\operatorname {Subst}\left (\frac {(b f p q) \operatorname {Subst}\left (\int \frac {e^x x}{e^x f+\frac {e \sqrt {h}}{\sqrt {2}}-\frac {\sqrt {2 f^2+e^2 h}}{\sqrt {2}}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )\right )}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )-\operatorname {Subst}\left (\frac {(b f p q) \operatorname {Subst}\left (\int \frac {e^x x}{e^x f+\frac {e \sqrt {h}}{\sqrt {2}}+\frac {\sqrt {2 f^2+e^2 h}}{\sqrt {2}}} \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )\right )}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}+\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}+\operatorname {Subst}\left (\frac {(b p q) \operatorname {Subst}\left (\int \log \left (1+\frac {e^x f}{\frac {e \sqrt {h}}{\sqrt {2}}-\frac {\sqrt {2 f^2+e^2 h}}{\sqrt {2}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )\right )}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(b p q) \operatorname {Subst}\left (\int \log \left (1+\frac {e^x f}{\frac {e \sqrt {h}}{\sqrt {2}}+\frac {\sqrt {2 f^2+e^2 h}}{\sqrt {2}}}\right ) \, dx,x,\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )\right )}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}+\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}+\operatorname {Subst}\left (\frac {(b p q) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {f x}{\frac {e \sqrt {h}}{\sqrt {2}}-\frac {\sqrt {2 f^2+e^2 h}}{\sqrt {2}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}\right )}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )+\operatorname {Subst}\left (\frac {(b p q) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {f x}{\frac {e \sqrt {h}}{\sqrt {2}}+\frac {\sqrt {2 f^2+e^2 h}}{\sqrt {2}}}\right )}{x} \, dx,x,e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}\right )}{\sqrt {h}},c d^q (e+f x)^{p q},c \left (d (e+f x)^p\right )^q\right )\\ &=\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}-\frac {b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}+\frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b p q \text {Li}_2\left (-\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}-\frac {b p q \text {Li}_2\left (-\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 284, normalized size = 0.85 \[ \frac {\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (2 a+2 b \log \left (c \left (d (e+f x)^p\right )^q\right )-2 b p q \log \left (\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+2 f^2}}+1\right )-2 b p q \log \left (\frac {\sqrt {2} f e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{\sqrt {e^2 h+2 f^2}+e \sqrt {h}}+1\right )+b p q \sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )\right )-2 b p q \text {Li}_2\left (\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{\sqrt {h e^2+2 f^2}-e \sqrt {h}}\right )-2 b p q \text {Li}_2\left (-\frac {\sqrt {2} e^{\sinh ^{-1}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{\sqrt {h} e+\sqrt {h e^2+2 f^2}}\right )}{2 \sqrt {h}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {h x^{2} + 2} b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + \sqrt {h x^{2} + 2} a}{h x^{2} + 2}, x\right ) \]
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 \[ \int \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )+a}{\sqrt {h \,x^{2}+2}}\, dx \]
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 \[ b \int \frac {q \log \relax (d) + \log \left ({\left ({\left (f x + e\right )}^{p}\right )}^{q}\right ) + \log \relax (c)}{\sqrt {h x^{2} + 2}}\,{d x} + \frac {a \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {2} \sqrt {h} x\right )}{\sqrt {h}} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {h\,x^2+2}} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {h x^{2} + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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