Integrand size = 28, antiderivative size = 527 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 f^3 \sinh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2} \]
[Out]
Time = 0.64 (sec) , antiderivative size = 527, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5684, 32, 3377, 2717, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {6 f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 f^3 \sqrt {a^2+b^2} \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 f^2 \sqrt {a^2+b^2} (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 f \sqrt {a^2+b^2} (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^2 d}-\frac {a (e+f x)^4}{4 b^2 f}-\frac {6 f^3 \sinh (c+d x)}{b d^4}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac {(e+f x)^3 \cosh (c+d x)}{b d} \]
[In]
[Out]
Rule 32
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 2717
Rule 3377
Rule 3403
Rule 5684
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = -\frac {a \int (e+f x)^3 \, dx}{b^2}+\frac {\int (e+f x)^3 \sinh (c+d x) \, dx}{b}+\frac {\left (a^2+b^2\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^2} \\ & = -\frac {a (e+f x)^4}{4 b^2 f}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {\left (2 \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^2}-\frac {(3 f) \int (e+f x)^2 \cosh (c+d x) \, dx}{b d} \\ & = -\frac {a (e+f x)^4}{4 b^2 f}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac {\left (2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b}-\frac {\left (2 \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b}+\frac {\left (6 f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b d^2} \\ & = -\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac {\left (3 \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d}+\frac {\left (3 \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d}-\frac {\left (6 f^3\right ) \int \cosh (c+d x) \, dx}{b d^3} \\ & = -\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {6 f^3 \sinh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2}-\frac {\left (6 \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^2}+\frac {\left (6 \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \operatorname {PolyLog}\left (2,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^2} \\ & = -\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}-\frac {6 f^3 \sinh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^3}-\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \int \operatorname {PolyLog}\left (3,-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^2 d^3} \\ & = -\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}-\frac {6 f^3 \sinh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2}+\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4}-\frac {\left (6 \sqrt {a^2+b^2} f^3\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}\left (3,-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 d^4} \\ & = -\frac {a (e+f x)^4}{4 b^2 f}+\frac {6 f^2 (e+f x) \cosh (c+d x)}{b d^3}+\frac {(e+f x)^3 \cosh (c+d x)}{b d}+\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d}-\frac {\sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d}+\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {3 \sqrt {a^2+b^2} f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^2}-\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^2 (e+f x) \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^3}+\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 d^4}-\frac {6 f^3 \sinh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \sinh (c+d x)}{b d^2} \\ \end{align*}
Time = 1.55 (sec) , antiderivative size = 933, normalized size of antiderivative = 1.77 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {4 a d^4 e^3 x+6 a d^4 e^2 f x^2+4 a d^4 e f^2 x^3+a d^4 f^3 x^4+8 \sqrt {a^2+b^2} d^3 e^3 \text {arctanh}\left (\frac {a+b e^{c+d x}}{\sqrt {a^2+b^2}}\right )-4 b d^3 e^3 \cosh (c+d x)-24 b d e f^2 \cosh (c+d x)-12 b d^3 e^2 f x \cosh (c+d x)-24 b d f^3 x \cosh (c+d x)-12 b d^3 e f^2 x^2 \cosh (c+d x)-4 b d^3 f^3 x^3 \cosh (c+d x)-12 \sqrt {a^2+b^2} d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-12 \sqrt {a^2+b^2} d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )-4 \sqrt {a^2+b^2} d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )+12 \sqrt {a^2+b^2} d^3 e^2 f x \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+12 \sqrt {a^2+b^2} d^3 e f^2 x^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+4 \sqrt {a^2+b^2} d^3 f^3 x^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-12 \sqrt {a^2+b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+12 \sqrt {a^2+b^2} d^2 f (e+f x)^2 \operatorname {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+24 \sqrt {a^2+b^2} d e f^2 \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+24 \sqrt {a^2+b^2} d f^3 x \operatorname {PolyLog}\left (3,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )-24 \sqrt {a^2+b^2} d e f^2 \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-24 \sqrt {a^2+b^2} d f^3 x \operatorname {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )-24 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,\frac {b e^{c+d x}}{-a+\sqrt {a^2+b^2}}\right )+24 \sqrt {a^2+b^2} f^3 \operatorname {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )+12 b d^2 e^2 f \sinh (c+d x)+24 b f^3 \sinh (c+d x)+24 b d^2 e f^2 x \sinh (c+d x)+12 b d^2 f^3 x^2 \sinh (c+d x)}{4 b^2 d^4} \]
[In]
[Out]
\[\int \frac {\left (f x +e \right )^{3} \cosh \left (d x +c \right )^{2}}{a +b \sinh \left (d x +c \right )}d x\]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 2020 vs. \(2 (483) = 966\).
Time = 0.29 (sec) , antiderivative size = 2020, normalized size of antiderivative = 3.83 \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]
[In]
[Out]
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
\[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{3} \cosh \left (d x + c\right )^{2}}{b \sinh \left (d x + c\right ) + a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]
[In]
[Out]