Optimal. Leaf size=176 \[ -\frac {i x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m-i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}+\frac {i x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}+\frac {(e x)^{m+1} \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]
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Rubi [A] time = 0.31, antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6526, 12, 4497, 2310, 2178} \[ -\frac {i x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m-i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}+\frac {i x (e x)^m e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \text {Ei}\left (\frac {(m+i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (m+1)}+\frac {(e x)^{m+1} \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (m+1)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2178
Rule 2310
Rule 4497
Rule 6526
Rubi steps
\begin {align*} \int (e x)^m \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=\frac {(e x)^{1+m} \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b d n) \int \frac {(e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{d \left (a+b \log \left (c x^n\right )\right )} \, dx}{1+m}\\ &=\frac {(e x)^{1+m} \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {(b n) \int \frac {(e x)^m \sin \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx}{1+m}\\ &=\frac {(e x)^{1+m} \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i b e^{-i a d} n x^{-m+i b d n} (e x)^m \left (c x^n\right )^{-i b d}\right ) \int \frac {x^{m-i b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)}+\frac {\left (i b e^{i a d} n x^{-m-i b d n} (e x)^m \left (c x^n\right )^{i b d}\right ) \int \frac {x^{m+i b d n}}{a+b \log \left (c x^n\right )} \, dx}{2 (1+m)}\\ &=\frac {(e x)^{1+m} \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}-\frac {\left (i b e^{-i a d} x (e x)^m \left (c x^n\right )^{-i b d-\frac {1+m-i b d n}{n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(1+m-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}+\frac {\left (i b e^{i a d} x (e x)^m \left (c x^n\right )^{i b d-\frac {1+m+i b d n}{n}}\right ) \operatorname {Subst}\left (\int \frac {e^{\frac {(1+m+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )}{2 (1+m)}\\ &=-\frac {i e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {Ei}\left (\frac {(1+m-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}+\frac {i e^{-\frac {a (1+m)}{b n}} x (e x)^m \left (c x^n\right )^{-\frac {1+m}{n}} \text {Ei}\left (\frac {(1+m+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{2 (1+m)}+\frac {(e x)^{1+m} \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{e (1+m)}\\ \end {align*}
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Mathematica [A] time = 3.45, size = 128, normalized size = 0.73 \[ \frac {(e x)^m \left (2 x \text {Si}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-i x^{-m} \exp \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{b n}\right ) \left (\text {Ei}\left (\frac {(m-i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\text {Ei}\left (\frac {(m+i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right )\right )}{2 (m+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (e x\right )^{m} \operatorname {Si}\left (b d \log \left (c x^{n}\right ) + a d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (e x \right )^{m} \Si \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} {\rm Si}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {sinint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )\,{\left (e\,x\right )}^m \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e x\right )^{m} \operatorname {Si}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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