3.167 \(\int \frac {C(d (a+b \log (c x^n)))}{x^2} \, dx\)

Optimal. Leaf size=217 \[ \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (c x^n\right )^{\frac {1}{n}} e^{\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{x}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (c x^n\right )^{\frac {1}{n}} e^{\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{x}-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

[Out]

(1/4+1/4*I)*exp(1/2*(2*a*b*n+I/d^2/Pi)/b^2/n^2)*(c*x^n)^(1/n)*erf((1/2+1/2*I)*(1/n-I*a*b*d^2*Pi-I*b^2*d^2*Pi*l
n(c*x^n))/b/d/Pi^(1/2))/x-(1/4+1/4*I)*exp(1/2*(2*a*b*n-I/d^2/Pi)/b^2/n^2)*(c*x^n)^(1/n)*erfi((1/2+1/2*I)*(1/n+
I*a*b*d^2*Pi+I*b^2*d^2*Pi*ln(c*x^n))/b/d/Pi^(1/2))/x-FresnelC(d*(a+b*ln(c*x^n)))/x

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Rubi [A]  time = 0.49, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {6472, 4618, 2278, 2274, 15, 2276, 2234, 2204, 2205} \[ \frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (c x^n\right )^{\frac {1}{n}} e^{\frac {2 a b n+\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {Erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (-i \pi a b d^2-i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{x}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (c x^n\right )^{\frac {1}{n}} e^{\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \text {Erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (i \pi a b d^2+i \pi b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}\right )}{\sqrt {\pi } b d}\right )}{x}-\frac {\text {FresnelC}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

Antiderivative was successfully verified.

[In]

Int[FresnelC[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

((1/4 + I/4)*E^((2*a*b*n + I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)*Erf[((1/2 + I/2)*(n^(-1) - I*a*b*d^2*Pi - I
*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x - ((1/4 + I/4)*E^((2*a*b*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-
1)*Erfi[((1/2 + I/2)*(n^(-1) + I*a*b*d^2*Pi + I*b^2*d^2*Pi*Log[c*x^n]))/(b*d*Sqrt[Pi])])/x - FresnelC[d*(a + b
*Log[c*x^n])]/x

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 2276

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]^2*(b_.))*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)/(
e*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(a*d*Log[F] + ((m + 1)*x)/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /;
FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 2278

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Int[(e*x)^m*F^(a^2*d
+ 2*a*b*d*Log[c*x^n] + b^2*d*Log[c*x^n]^2), x] /; FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 4618

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2, Int[(e*x)^m/
E^(I*d*(a + b*Log[c*x^n])^2), x], x] + Dist[1/2, Int[(e*x)^m*E^(I*d*(a + b*Log[c*x^n])^2), x], x] /; FreeQ[{a,
b, c, d, e, m, n}, x]

Rule 6472

Int[FresnelC[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((e*x)^(m + 1)
*FresnelC[d*(a + b*Log[c*x^n])])/(e*(m + 1)), x] - Dist[(b*d*n)/(m + 1), Int[(e*x)^m*Cos[(Pi*(d*(a + b*Log[c*x
^n]))^2)/2], x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx &=-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+(b d n) \int \frac {\cos \left (\frac {1}{2} d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2\right )}{x^2} \, dx\\ &=-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {1}{2} (b d n) \int \frac {e^{-\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx+\frac {1}{2} (b d n) \int \frac {e^{\frac {1}{2} i d^2 \pi \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx\\ &=-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {1}{2} (b d n) \int \frac {\exp \left (-\frac {1}{2} i a^2 d^2 \pi -i a b d^2 \pi \log \left (c x^n\right )-\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right )}{x^2} \, dx+\frac {1}{2} (b d n) \int \frac {\exp \left (\frac {1}{2} i a^2 d^2 \pi +i a b d^2 \pi \log \left (c x^n\right )+\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right )}{x^2} \, dx\\ &=-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {1}{2} (b d n) \int \frac {\exp \left (-\frac {1}{2} i a^2 d^2 \pi -\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) \left (c x^n\right )^{-i a b d^2 \pi }}{x^2} \, dx+\frac {1}{2} (b d n) \int \frac {\exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) \left (c x^n\right )^{i a b d^2 \pi }}{x^2} \, dx\\ &=-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {1}{2} \left (b d n x^{i a b d^2 n \pi } \left (c x^n\right )^{-i a b d^2 \pi }\right ) \int \exp \left (-\frac {1}{2} i a^2 d^2 \pi -\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{-2-i a b d^2 n \pi } \, dx+\frac {1}{2} \left (b d n x^{-i a b d^2 n \pi } \left (c x^n\right )^{i a b d^2 \pi }\right ) \int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {1}{2} i b^2 d^2 \pi \log ^2\left (c x^n\right )\right ) x^{-2+i a b d^2 n \pi } \, dx\\ &=-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (b d \left (c x^n\right )^{-i a b d^2 \pi -\frac {-1-i a b d^2 n \pi }{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-\frac {1}{2} i a^2 d^2 \pi +\frac {\left (-1-i a b d^2 n \pi \right ) x}{n}-\frac {1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 x}+\frac {\left (b d \left (c x^n\right )^{i a b d^2 \pi -\frac {-1+i a b d^2 n \pi }{n}}\right ) \operatorname {Subst}\left (\int \exp \left (\frac {1}{2} i a^2 d^2 \pi +\frac {\left (-1+i a b d^2 n \pi \right ) x}{n}+\frac {1}{2} i b^2 d^2 \pi x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 x}\\ &=-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (b d e^{\frac {2 a b n-\frac {i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{-i a b d^2 \pi -\frac {-1-i a b d^2 n \pi }{n}}\right ) \operatorname {Subst}\left (\int \exp \left (\frac {i \left (\frac {-1-i a b d^2 n \pi }{n}-i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 x}+\frac {\left (b d e^{\frac {2 a b n+\frac {i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{i a b d^2 \pi -\frac {-1+i a b d^2 n \pi }{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-\frac {i \left (\frac {-1+i a b d^2 n \pi }{n}+i b^2 d^2 \pi x\right )^2}{2 b^2 d^2 \pi }\right ) \, dx,x,\log \left (c x^n\right )\right )}{2 x}\\ &=\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{\frac {2 a b n+\frac {i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}-i a b d^2 \pi -i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) e^{\frac {2 a b n-\frac {i}{d^2 \pi }}{2 b^2 n^2}} \left (c x^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\frac {1}{n}+i a b d^2 \pi +i b^2 d^2 \pi \log \left (c x^n\right )\right )}{b d \sqrt {\pi }}\right )}{x}-\frac {C\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}\\ \end {align*}

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Mathematica [A]  time = 4.62, size = 194, normalized size = 0.89 \[ -\frac {4 C\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\sqrt [4]{-1} \sqrt {2} \left (c x^n\right )^{\frac {1}{n}} e^{\frac {2 a b n-\frac {i}{\pi d^2}}{2 b^2 n^2}} \left (i e^{\frac {i}{\pi b^2 d^2 n^2}} \text {erfi}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )+i\right )}{\sqrt {\pi } b d n}\right )+\text {erfi}\left (\frac {(-1)^{3/4} \left (\pi a b d^2 n+\pi b^2 d^2 n \log \left (c x^n\right )-i\right )}{\sqrt {2 \pi } b d n}\right )\right )}{4 x} \]

Antiderivative was successfully verified.

[In]

Integrate[FresnelC[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

-1/4*((-1)^(1/4)*Sqrt[2]*E^((2*a*b*n - I/(d^2*Pi))/(2*b^2*n^2))*(c*x^n)^n^(-1)*(Erfi[((-1)^(3/4)*(-I + a*b*d^2
*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[2*Pi])] + I*E^(I/(b^2*d^2*n^2*Pi))*Erfi[((1/2 + I/2)*(I + a*b*d^
2*n*Pi + b^2*d^2*n*Pi*Log[c*x^n]))/(b*d*n*Sqrt[Pi])]) + 4*FresnelC[d*(a + b*Log[c*x^n])])/x

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fricas [F]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\rm fresnelc}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(d*(a+b*log(c*x^n)))/x^2,x, algorithm="fricas")

[Out]

integral(fresnelc(b*d*log(c*x^n) + a*d)/x^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(d*(a+b*log(c*x^n)))/x^2,x, algorithm="giac")

[Out]

integrate(fresnelc((b*log(c*x^n) + a)*d)/x^2, x)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \frac {\FresnelC \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(d*(a+b*ln(c*x^n)))/x^2,x)

[Out]

int(FresnelC(d*(a+b*ln(c*x^n)))/x^2,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\rm fresnelc}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(d*(a+b*log(c*x^n)))/x^2,x, algorithm="maxima")

[Out]

integrate(fresnelc((b*log(c*x^n) + a)*d)/x^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {FresnelC}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(FresnelC(d*(a + b*log(c*x^n)))/x^2,x)

[Out]

int(FresnelC(d*(a + b*log(c*x^n)))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {C\left (a d + b d \log {\left (c x^{n} \right )}\right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(fresnelc(d*(a+b*ln(c*x**n)))/x**2,x)

[Out]

Integral(fresnelc(a*d + b*d*log(c*x**n))/x**2, x)

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