Optimal. Leaf size=404 \[ \frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {3 d (c+d x)^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {6 d^2 (c+d x) \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^2 (c+d x) \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^3 \text {PolyLog}\left (4,-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^4}-\frac {6 d^3 \text {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.56, antiderivative size = 404, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3403, 2296,
2221, 2611, 6744, 2320, 6724} \begin {gather*} -\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^3 \sqrt {a^2+b^2}}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^3 \sqrt {a^2+b^2}}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^2 \sqrt {a^2+b^2}}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^2 \sqrt {a^2+b^2}}+\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}+1\right )}{f \sqrt {a^2+b^2}}-\frac {(c+d x)^3 \log \left (\frac {b e^{e+f x}}{\sqrt {a^2+b^2}+a}+1\right )}{f \sqrt {a^2+b^2}}+\frac {6 d^3 \text {Li}_4\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{f^4 \sqrt {a^2+b^2}}-\frac {6 d^3 \text {Li}_4\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{f^4 \sqrt {a^2+b^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3403
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{a+b \sinh (e+f x)} \, dx &=2 \int \frac {e^{e+f x} (c+d x)^3}{-b+2 a e^{e+f x}+b e^{2 (e+f x)}} \, dx\\ &=\frac {(2 b) \int \frac {e^{e+f x} (c+d x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\sqrt {a^2+b^2}}-\frac {(2 b) \int \frac {e^{e+f x} (c+d x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{e+f x}} \, dx}{\sqrt {a^2+b^2}}\\ &=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(3 d) \int (c+d x)^2 \log \left (1+\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f}+\frac {(3 d) \int (c+d x)^2 \log \left (1+\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f}\\ &=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {\left (6 d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f^2}+\frac {\left (6 d^2\right ) \int (c+d x) \text {Li}_2\left (-\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f^2}\\ &=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {\left (6 d^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{e+f x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f^3}-\frac {\left (6 d^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{e+f x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{\sqrt {a^2+b^2} f^3}\\ &=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\sqrt {a^2+b^2} f^4}-\frac {\left (6 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{e+f x}\right )}{\sqrt {a^2+b^2} f^4}\\ &=\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}-\frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f}+\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {3 d (c+d x)^2 \text {Li}_2\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^2}-\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^2 (c+d x) \text {Li}_3\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^3}+\frac {6 d^3 \text {Li}_4\left (-\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^4}-\frac {6 d^3 \text {Li}_4\left (-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )}{\sqrt {a^2+b^2} f^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.18, size = 318, normalized size = 0.79 \begin {gather*} \frac {(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a-\sqrt {a^2+b^2}}\right )-(c+d x)^3 \log \left (1+\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+\frac {3 d \left (f^2 (c+d x)^2 \text {PolyLog}\left (2,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )-2 d f (c+d x) \text {PolyLog}\left (3,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )+2 d^2 \text {PolyLog}\left (4,\frac {b e^{e+f x}}{-a+\sqrt {a^2+b^2}}\right )\right )}{f^3}-\frac {3 d \left (f^2 (c+d x)^2 \text {PolyLog}\left (2,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )-2 d f (c+d x) \text {PolyLog}\left (3,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )+2 d^2 \text {PolyLog}\left (4,-\frac {b e^{e+f x}}{a+\sqrt {a^2+b^2}}\right )\right )}{f^3}}{\sqrt {a^2+b^2} f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{3}}{a +b \sinh \left (f x +e \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1382 vs.
\(2 (370) = 740\).
time = 0.36, size = 1382, normalized size = 3.42 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (c + d x\right )^{3}}{a + b \sinh {\left (e + f x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (c+d\,x\right )}^3}{a+b\,\mathrm {sinh}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________