Optimal. Leaf size=220 \[ -\frac {3^{-\frac {1+m}{n}} e^{3 a} (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )}{8 e n}+\frac {3 e^a (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{8 e n}-\frac {3 e^{-a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{8 e n}+\frac {3^{-\frac {1+m}{n}} e^{-3 a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 b x^n\right )}{8 e n} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {5470, 5468,
2250} \begin {gather*} -\frac {e^{3 a} 3^{-\frac {m+1}{n}} (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-3 b x^n\right )}{8 e n}+\frac {3 e^a (e x)^{m+1} \left (-b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},-b x^n\right )}{8 e n}-\frac {3 e^{-a} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},b x^n\right )}{8 e n}+\frac {e^{-3 a} 3^{-\frac {m+1}{n}} (e x)^{m+1} \left (b x^n\right )^{-\frac {m+1}{n}} \text {Gamma}\left (\frac {m+1}{n},3 b x^n\right )}{8 e n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2250
Rule 5468
Rule 5470
Rubi steps
\begin {align*} \int (e x)^m \sinh ^3\left (a+b x^n\right ) \, dx &=\int \left (-\frac {3}{4} (e x)^m \sinh \left (a+b x^n\right )+\frac {1}{4} (e x)^m \sinh \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac {1}{4} \int (e x)^m \sinh \left (3 a+3 b x^n\right ) \, dx-\frac {3}{4} \int (e x)^m \sinh \left (a+b x^n\right ) \, dx\\ &=-\left (\frac {1}{8} \int e^{-3 a-3 b x^n} (e x)^m \, dx\right )+\frac {1}{8} \int e^{3 a+3 b x^n} (e x)^m \, dx+\frac {3}{8} \int e^{-a-b x^n} (e x)^m \, dx-\frac {3}{8} \int e^{a+b x^n} (e x)^m \, dx\\ &=-\frac {3^{-\frac {1+m}{n}} e^{3 a} (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )}{8 e n}+\frac {3 e^a (e x)^{1+m} \left (-b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )}{8 e n}-\frac {3 e^{-a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},b x^n\right )}{8 e n}+\frac {3^{-\frac {1+m}{n}} e^{-3 a} (e x)^{1+m} \left (b x^n\right )^{-\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},3 b x^n\right )}{8 e n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 1.51, size = 185, normalized size = 0.84 \begin {gather*} \frac {3^{-\frac {1+m}{n}} e^{-3 a} x (e x)^m \left (-b^2 x^{2 n}\right )^{-\frac {1+m}{n}} \left (-e^{6 a} \left (b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-3 b x^n\right )+3^{\frac {1+m+n}{n}} e^{4 a} \left (b x^n\right )^{\frac {1+m}{n}} \Gamma \left (\frac {1+m}{n},-b x^n\right )+\left (-b x^n\right )^{\frac {1+m}{n}} \left (-3^{\frac {1+m+n}{n}} e^{2 a} \Gamma \left (\frac {1+m}{n},b x^n\right )+\Gamma \left (\frac {1+m}{n},3 b x^n\right )\right )\right )}{8 n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.93, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (\sinh ^{3}\left (a +b \,x^{n}\right )\right )\, dx\]
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 [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{m} \sinh ^{3}{\left (a + b x^{n} \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 {\mathrm {sinh}\left (a+b\,x^n\right )}^3\,{\left (e\,x\right )}^m \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________