Integrand size = 30, antiderivative size = 335 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\frac {b p q \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}}-\frac {b p q \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}-\frac {b p q \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (1+\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}+\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )}{\sqrt {h}} \]
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Time = 0.54 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {221, 2451, 12, 5827, 5680, 2221, 2317, 2438, 2495} \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {h}}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {h e^2+2 f^2}}\right )}{\sqrt {h}}-\frac {b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{\sqrt {h} e+\sqrt {h e^2+2 f^2}}\right )}{\sqrt {h}}-\frac {b p q \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} f e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{e \sqrt {h}-\sqrt {e^2 h+2 f^2}}+1\right )}{\sqrt {h}}-\frac {b p q \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \log \left (\frac {\sqrt {2} f e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )}}{\sqrt {e^2 h+2 f^2}+e \sqrt {h}}+1\right )}{\sqrt {h}}+\frac {b p q \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )^2}{2 \sqrt {h}} \]
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Rule 12
Rule 221
Rule 2221
Rule 2317
Rule 2438
Rule 2451
Rule 2495
Rule 5680
Rule 5827
Rubi steps \begin{align*} \text {integral}& = \text {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}}-\text {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}}-\text {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}}-\text {Subst}\left (\frac {(b f p q) \text {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}}-\text {Subst}\left (\frac {(b f p q) \text {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 )-\text {Subst}\left (\frac {(b f p q) \text {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}}+\text {Subst}\left (\frac {(b p q) \text {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 )+\text {Subst}\left (\frac {(b p q) \text {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}}+\text {Subst}\left (\frac {(b p q) \text {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 )+\text {Subst}\left (\frac {(b p q) \text {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*}
Time = 0.14 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\frac {\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right ) \left (2 a+b p q \text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )-2 b p q \log \left (1+\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}-\sqrt {2 f^2+e^2 h}}\right )-2 b p q \log \left (1+\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )+2 b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )-2 b p q \operatorname {PolyLog}\left (2,\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{-e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )-2 b p q \operatorname {PolyLog}\left (2,-\frac {\sqrt {2} e^{\text {arcsinh}\left (\frac {\sqrt {h} x}{\sqrt {2}}\right )} f}{e \sqrt {h}+\sqrt {2 f^2+e^2 h}}\right )}{2 \sqrt {h}} \]
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\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\sqrt {h \,x^{2}+2}}d x\]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + 2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\sqrt {h x^{2} + 2}}\, dx \]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + 2}} \,d x } \]
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\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\int { \frac {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}{\sqrt {h x^{2} + 2}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{\sqrt {2+h x^2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{\sqrt {h\,x^2+2}} \,d x \]
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