Optimal. Leaf size=124 \[ x \text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.16, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6659, 12, 4584,
2347, 2209} \begin {gather*} x \text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} x e^{-\frac {a}{b n}} \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(i b d n+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2209
Rule 2347
Rule 4584
Rule 6659
Rubi steps
\begin {align*} \int \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-(b d n) \int \frac {\cos \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\\ &=x \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-(b n) \int \frac {\cos \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{a+b \log \left (c x^n\right )} \, dx\\ &=x \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} \left (b e^{-i a d} n x^{i b d n} \left (c x^n\right )^{-i b d}\right ) \int \frac {x^{-i b d n}}{a+b \log \left (c x^n\right )} \, dx-\frac {1}{2} \left (b e^{i a d} n x^{-i b d n} \left (c x^n\right )^{i b d}\right ) \int \frac {x^{i b d n}}{a+b \log \left (c x^n\right )} \, dx\\ &=x \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} \left (b e^{-i a d} x \left (c x^n\right )^{-i b d-\frac {1-i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(1-i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )-\frac {1}{2} \left (b e^{i a d} x \left (c x^n\right )^{i b d-\frac {1+i b d n}{n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {(1+i b d n) x}{n}}}{a+b x} \, dx,x,\log \left (c x^n\right )\right )\\ &=x \text {Ci}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \text {Ei}\left (\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.96, size = 98, normalized size = 0.79 \begin {gather*} x \text {CosIntegral}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {a}{b n}} x \left (c x^n\right )^{-1/n} \left (\text {Ei}\left (\frac {(1-i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )+\text {Ei}\left (\frac {(1+i b d n) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \cosineIntegral \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
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 [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 445 vs. \(2 (114) = 228\).
time = 0.38, size = 445, normalized size = 3.59 \begin {gather*} -\frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {C}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + \frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} - \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n + i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) - \frac {1}{2} i \, \pi \sqrt {b^{2} d^{2} n^{2}} e^{\left (-\frac {\log \left (c\right )}{n} - \frac {a}{b n} + \frac {i}{2 \, \pi b^{2} d^{2} n^{2}}\right )} \operatorname {S}\left (\frac {{\left (\pi b^{2} d^{2} n^{2} \log \left (x\right ) + \pi b^{2} d^{2} n \log \left (c\right ) + \pi a b d^{2} n - i\right )} \sqrt {b^{2} d^{2} n^{2}}}{\pi b^{2} d^{2} n^{2}}\right ) + x \operatorname {C}\left (b d \log \left (c x^{n}\right ) + a d\right ) \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 \operatorname {Ci}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \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 = -0.01 \begin {gather*} \int \mathrm {cosint}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________