Integrand size = 18, antiderivative size = 1363 \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=-\frac {2 b^{3/2} d \arctan \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}-\frac {(-b)^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}+\frac {2 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {b^{3/2} d \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 )}{f^2}+\frac {b^{3/2} d \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 )}{f^2}-\frac {b^{3/2} d \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 )}{2 f^2}-\frac {b^{3/2} d \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 )}{2 f^2}+\frac {(-b)^{3/2} d \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 )}{f^2}-\frac {(-b)^{3/2} d \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 )}{2 f^2}-\frac {(-b)^{3/2} d \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 )}{2 f^2}-\frac {(-b)^{3/2} d \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 )}{f^2}-\frac {b^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}-\frac {b^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}+\frac {b^{3/2} d \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 )}{4 f^2}+\frac {b^{3/2} d \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 )}{4 f^2}+\frac {(-b)^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {(-b)^{3/2} d \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 )}{4 f^2}-\frac {(-b)^{3/2} d \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 )}{4 f^2}+\frac {(-b)^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2}{1+\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f} \]
[Out]
Time = 1.14 (sec) , antiderivative size = 1363, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.944, Rules used = {3801, 3557, 335, 304, 209, 212, 3819, 213, 281, 6857, 6139, 6057, 2449, 2352, 2497, 6131, 6055} \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=-\frac {(-b)^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}-\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {(-b)^{3/2} d \log \left (\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f^2}-\frac {(-b)^{3/2} d \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 ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 f^2}-\frac {(-b)^{3/2} d \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 ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{2 f^2}-\frac {(-b)^{3/2} d \log \left (\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right ) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f^2}+\frac {b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {2 b^{3/2} d \arctan \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {2 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {b^{3/2} d \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 )}{f^2}+\frac {b^{3/2} d \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 )}{f^2}-\frac {b^{3/2} d \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 )}{2 f^2}-\frac {b^{3/2} d \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 )}{2 f^2}-\frac {b^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}-\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}-\frac {b^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {b}}{\sqrt {b}+\sqrt {b \tanh (e+f x)}}\right )}{2 f^2}+\frac {b^{3/2} d \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 )}{4 f^2}+\frac {b^{3/2} d \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 )}{4 f^2}+\frac {(-b)^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}}\right )}{2 f^2}-\frac {(-b)^{3/2} d \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 )}{4 f^2}-\frac {(-b)^{3/2} d \operatorname {PolyLog}\left (2,\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 )}+1\right )}{4 f^2}+\frac {(-b)^{3/2} d \operatorname {PolyLog}\left (2,1-\frac {2}{\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}+1}\right )}{2 f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f} \]
[In]
[Out]
Rule 209
Rule 212
Rule 213
Rule 281
Rule 304
Rule 335
Rule 2352
Rule 2449
Rule 2497
Rule 3557
Rule 3801
Rule 3819
Rule 6055
Rule 6057
Rule 6131
Rule 6139
Rule 6857
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}+b^2 \int \frac {c+d x}{\sqrt {b \tanh (e+f x)}} \, dx+\frac {(2 b d) \int \sqrt {b \tanh (e+f x)} \, dx}{f} \\ & = -\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{-b^2+x^2} \, dx,x,b \tanh (e+f x)\right )}{f^2}+\frac {\left ((-b)^{3/2} d\right ) \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right ) \, dx}{f}-\frac {\left (b^{3/2} d\right ) \int \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right ) \, dx}{f} \\ & = -\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}+\frac {\left ((-b)^{3/2} d\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 )}{f^2}-\frac {\left (b^{3/2} d\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 )}{f^2}-\frac {\left (4 b^2 d\right ) \text {Subst}\left (\int \frac {x^2}{-b^2+x^4} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2} \\ & = -\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}-\frac {\left (2 \sqrt {-b} d\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 )}{f^2}-\frac {\left (2 \sqrt {b} d\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 )}{f^2}+\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2}-\frac {\left (2 b^2 d\right ) \text {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {b \tanh (e+f x)}\right )}{f^2} \\ & = -\frac {2 b^{3/2} d \arctan \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {2 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}-\frac {\left (2 \sqrt {-b} d\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 )}{f^2}-\frac {\left (2 \sqrt {b} d\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 )}{f^2} \\ & = -\frac {2 b^{3/2} d \arctan \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}+\frac {2 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}-\frac {\left ((-b)^{3/2} d\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 )}{f^2}-\frac {\left ((-b)^{3/2} d\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 )}{f^2}-\frac {\left (b^{3/2} d\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 )}{f^2}-\frac {\left (b^{3/2} d\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 )}{f^2} \\ & = -\frac {2 b^{3/2} d \arctan \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}-\frac {(-b)^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}+\frac {2 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}-\frac {\left ((-b)^{3/2} d\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 )}{f^2}-\frac {(b d) \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 )}{f^2}+\frac {(b d) \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 )}{f^2}-\frac {\left (b^{3/2} d\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 )}{f^2} \\ & = -\frac {2 b^{3/2} d \arctan \left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}-\frac {(-b)^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )}{f}-\frac {(-b)^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {-b}}\right )^2}{2 f^2}+\frac {2 b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f^2}+\frac {b^{3/2} (c+d x) \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )}{f}+\frac {b^{3/2} d \text {arctanh}\left (\frac {\sqrt {b \tanh (e+f x)}}{\sqrt {b}}\right )^2}{2 f^2}-\frac {b^{3/2} d \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 )}{f^2}-\frac {(-b)^{3/2} d \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 )}{f^2}-\frac {2 b (c+d x) \sqrt {b \tanh (e+f x)}}{f}-\frac {\left ((-b)^{3/2} d\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 )}{2 f^2}+\frac {\left ((-b)^{3/2} d\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 )}{2 f^2}+\frac {(b d) \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 )}{f^2}-\frac {(b d) \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 )}{f^2}+\frac {\left (b^{3/2} d\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 )}{2 f^2}-\frac {\left (b^{3/2} d\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 )}{2 f^2} \\ & = \text {Too large to display} \\ \end{align*}
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int (c+d x) (b \tanh (e+f x))^{3/2} \, dx \]
[In]
[Out]
\[\int \left (d x +c \right ) \left (b \tanh \left (f x +e \right )\right )^{\frac {3}{2}}d x\]
[In]
[Out]
Exception generated. \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int \left (b \tanh {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (c + d x\right )\, dx \]
[In]
[Out]
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int { {\left (d x + c\right )} \left (b \tanh \left (f x + e\right )\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int (c+d x) (b \tanh (e+f x))^{3/2} \, dx=\int {\left (b\,\mathrm {tanh}\left (e+f\,x\right )\right )}^{3/2}\,\left (c+d\,x\right ) \,d x \]
[In]
[Out]