Integrand size = 20, antiderivative size = 20 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{(-b)^{3/2} f^3}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{b^{3/2} f^3}-\frac {4 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{b^{3/2} f^3}+\frac {4 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{b^{3/2} f^3}-\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{b^{3/2} f^3}-\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{b^{3/2} f^3}-\frac {4 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{(-b)^{3/2} f^3}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{(-b)^{3/2} f^3}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (-\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{(-b)^{3/2} f^3}+\frac {4 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{(-b)^{3/2} f^3}-\frac {2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{b^{3/2} f^3}-\frac {2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{b^{3/2} f^3}+\frac {d^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{b^{3/2} f^3}+\frac {d^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b} \left (\sqrt {-b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}\right )}{b^{3/2} f^3}-\frac {2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{(-b)^{3/2} f^3}+\frac {d^2 \operatorname {PolyLog}\left (2,1-\frac {2 \left (\sqrt {b}-\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}+\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{(-b)^{3/2} f^3}+\frac {d^2 \operatorname {PolyLog}\left (2,1+\frac {2 \left (\sqrt {b}+\sqrt {b \tanh (e+f x)}\right )}{\left (\sqrt {-b}-\sqrt {b}\right ) \left (1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}\right )}{(-b)^{3/2} f^3}-\frac {2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{(-b)^{3/2} f^3}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\text {Int}\left ((c+d x)^2 \sqrt {b \tanh (e+f x)},x\right )}{b^2} \]
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Not integrable
Time = 1.15 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {(4 d) \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx}{b f} \\ & = \frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}-\frac {\left (4 d^2\right ) \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \, dx}{(-b)^{3/2} f^2}-\frac {\left (4 d^2\right ) \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \, dx}{b^{3/2} f^2} \\ & = \frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b \sqrt {b x}}{(-b)^{3/2}}\right )}{-1+x^2} \, dx,x,\tanh (e+f x)\right )}{(-b)^{3/2} f^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {\sqrt {b x}}{\sqrt {b}}\right )}{1-x^2} \, dx,x,\tanh (e+f x)\right )}{b^{3/2} f^3} \\ & = \frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {\left (8 d^2\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{-1+\frac {x^4}{b^2}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{(-b)^{5/2} f^3}-\frac {\left (8 d^2\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{1-\frac {x^4}{b^2}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^{5/2} f^3} \\ & = \frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {\left (8 d^2\right ) \text {Subst}\left (\int \left (-\frac {b x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (b-x^2\right )}-\frac {b x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (b+x^2\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{(-b)^{5/2} f^3}-\frac {\left (8 d^2\right ) \text {Subst}\left (\int \left (\frac {b x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (b-x^2\right )}+\frac {b x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (b+x^2\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^{5/2} f^3} \\ & = \frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{b-x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{(-b)^{3/2} f^3}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{b+x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{(-b)^{3/2} f^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{b-x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^{3/2} f^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {x \text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{b+x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^{3/2} f^3} \\ & = \frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{(-b)^{3/2} f^3}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{b^{3/2} f^3}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int \left (\frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (\sqrt {b}-x\right )}-\frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{2 \left (\sqrt {b}+x\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{(-b)^{3/2} f^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{1-\frac {x}{\sqrt {b}}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^2 f^3}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{1-\frac {b x}{(-b)^{3/2}}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^2 f^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \left (-\frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (\sqrt {-b}-x\right )}+\frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{2 \left (\sqrt {-b}+x\right )}\right ) \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^{3/2} f^3} \\ & = \frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{(-b)^{3/2} f^2}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{(-b)^{3/2} f^3}+\frac {4 d (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{b^{3/2} f^2}+\frac {2 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{b^{3/2} f^3}-\frac {4 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \log \left (\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{b^{3/2} f^3}+\frac {4 d^2 \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \log \left (\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{(-b)^{3/2} f^3}-\frac {2 (c+d x)^2}{b f \sqrt {b \tanh (e+f x)}}+\frac {\int (c+d x)^2 \sqrt {b \tanh (e+f x)} \, dx}{b^2}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{\sqrt {b}-x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{(-b)^{3/2} f^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {b x}{(-b)^{3/2}}\right )}{\sqrt {b}+x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{(-b)^{3/2} f^3}+\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {b}}}\right )}{1-\frac {x^2}{b}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^2 f^3}-\frac {\left (4 d^2\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {b x}{(-b)^{3/2}}}\right )}{1+\frac {x^2}{b}} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^2 f^3}+\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{\sqrt {-b}-x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^{3/2} f^3}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {\text {arctanh}\left (\frac {x}{\sqrt {b}}\right )}{\sqrt {-b}+x} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{b^{3/2} f^3} \\ & = \text {Too large to display} \\ \end{align*}
Not integrable
Time = 31.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {\left (d x +c \right )^{2}}{\left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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Not integrable
Time = 1.40 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int \frac {\left (c + d x\right )^{2}}{\left (b \tanh {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Not integrable
Time = 0.58 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{\left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int { \frac {{\left (d x + c\right )}^{2}}{\left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 2.11 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {(c+d x)^2}{(b \tanh (e+f x))^{3/2}} \, dx=\int \frac {{\left (c+d\,x\right )}^2}{{\left (b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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