\(\int \frac {\cos ^2(a+b x^2)}{x^{3/2}} \, dx\) [33]

Optimal result

Integrand size = 16, antiderivative size = 117 \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=-\frac {1}{\sqrt {x}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-\frac {i b e^{2 i a} x^{3/2} \Gamma \left (\frac {3}{4},-2 i b x^2\right )}{2^{3/4} \left (-i b x^2\right )^{3/4}}+\frac {i b e^{-2 i a} x^{3/2} \Gamma \left (\frac {3}{4},2 i b x^2\right )}{2^{3/4} \left (i b x^2\right )^{3/4}} \]

[Out]

-1/2*I*b*exp(2*I*a)*x^(3/2)*GAMMA(3/4,-2*I*b*x^2)*2^(1/4)/(-I*b*x^2)^(3/4)+1/2*I*b*x^(3/2)*GAMMA(3/4,2*I*b*x^2
)*2^(1/4)/exp(2*I*a)/(I*b*x^2)^(3/4)-1/x^(1/2)-cos(2*b*x^2+2*a)/x^(1/2)

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3483, 3485, 3469, 3470, 2250} \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-\frac {i e^{2 i a} b x^{3/2} \Gamma \left (\frac {3}{4},-2 i b x^2\right )}{2^{3/4} \left (-i b x^2\right )^{3/4}}+\frac {i e^{-2 i a} b x^{3/2} \Gamma \left (\frac {3}{4},2 i b x^2\right )}{2^{3/4} \left (i b x^2\right )^{3/4}}-\frac {1}{\sqrt {x}} \]

[In]

Int[Cos[a + b*x^2]^2/x^(3/2),x]

[Out]

-(1/Sqrt[x]) - Cos[2*(a + b*x^2)]/Sqrt[x] - (I*b*E^((2*I)*a)*x^(3/2)*Gamma[3/4, (-2*I)*b*x^2])/(2^(3/4)*((-I)*
b*x^2)^(3/4)) + (I*b*x^(3/2)*Gamma[3/4, (2*I)*b*x^2])/(2^(3/4)*E^((2*I)*a)*(I*b*x^2)^(3/4))

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 3469

Int[Cos[(c_.) + (d_.)*(x_)^(n_)]*((e_.)*(x_))^(m_), x_Symbol] :> Simp[(e*x)^(m + 1)*(Cos[c + d*x^n]/(e*(m + 1)
)), x] + Dist[d*(n/(e^n*(m + 1))), Int[(e*x)^(m + n)*Sin[c + d*x^n], x], x] /; FreeQ[{c, d, e}, x] && IGtQ[n,
0] && LtQ[m, -1]

Rule 3470

Int[((e_.)*(x_))^(m_.)*Sin[(c_.) + (d_.)*(x_)^(n_)], x_Symbol] :> Dist[I/2, Int[(e*x)^m*E^((-c)*I - d*I*x^n),
x], x] - Dist[I/2, Int[(e*x)^m*E^(c*I + d*I*x^n), x], x] /; FreeQ[{c, d, e, m}, x] && IGtQ[n, 0]

Rule 3483

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_.)*(x_))^(m_), x_Symbol] :> With[{k = Denominator[m
]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Cos[c + d*(x^(k*n)/e^n)])^p, x], x, (e*x)^(1/k)], x]] /; Free
Q[{a, b, c, d, e}, x] && IntegerQ[p] && IGtQ[n, 0] && FractionQ[m]

Rule 3485

Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 1] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\cos ^2\left (a+b x^4\right )}{x^2} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {1}{2 x^2}+\frac {\cos \left (2 a+2 b x^4\right )}{2 x^2}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {1}{\sqrt {x}}+\text {Subst}\left (\int \frac {\cos \left (2 a+2 b x^4\right )}{x^2} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {1}{\sqrt {x}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-(8 b) \text {Subst}\left (\int x^2 \sin \left (2 a+2 b x^4\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {1}{\sqrt {x}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-(4 i b) \text {Subst}\left (\int e^{-2 i a-2 i b x^4} x^2 \, dx,x,\sqrt {x}\right )+(4 i b) \text {Subst}\left (\int e^{2 i a+2 i b x^4} x^2 \, dx,x,\sqrt {x}\right ) \\ & = -\frac {1}{\sqrt {x}}-\frac {\cos \left (2 \left (a+b x^2\right )\right )}{\sqrt {x}}-\frac {i b e^{2 i a} x^{3/2} \Gamma \left (\frac {3}{4},-2 i b x^2\right )}{2^{3/4} \left (-i b x^2\right )^{3/4}}+\frac {i b e^{-2 i a} x^{3/2} \Gamma \left (\frac {3}{4},2 i b x^2\right )}{2^{3/4} \left (i b x^2\right )^{3/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.17 \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=\frac {-4 \left (b^2 x^4\right )^{3/4} \cos ^2\left (a+b x^2\right )+\sqrt [4]{2} b x^2 \left (i b x^2\right )^{3/4} \Gamma \left (\frac {3}{4},-2 i b x^2\right ) (-i \cos (2 a)+\sin (2 a))+i \sqrt [4]{2} \left (-i b x^2\right )^{7/4} \Gamma \left (\frac {3}{4},2 i b x^2\right ) (i \cos (2 a)+\sin (2 a))}{2 \sqrt {x} \left (b^2 x^4\right )^{3/4}} \]

[In]

Integrate[Cos[a + b*x^2]^2/x^(3/2),x]

[Out]

(-4*(b^2*x^4)^(3/4)*Cos[a + b*x^2]^2 + 2^(1/4)*b*x^2*(I*b*x^2)^(3/4)*Gamma[3/4, (-2*I)*b*x^2]*((-I)*Cos[2*a] +
Sin[2*a]) + I*2^(1/4)*((-I)*b*x^2)^(7/4)*Gamma[3/4, (2*I)*b*x^2]*(I*Cos[2*a] + Sin[2*a]))/(2*Sqrt[x]*(b^2*x^4
)^(3/4))

Maple [F]

\[\int \frac {\cos ^{2}\left (b \,x^{2}+a \right )}{x^{\frac {3}{2}}}d x\]

[In]

int(cos(b*x^2+a)^2/x^(3/2),x)

[Out]

int(cos(b*x^2+a)^2/x^(3/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=-\frac {4 \, \sqrt {x} \cos \left (b x^{2} + a\right )^{2} - {\left (x \cos \left (2 \, a\right ) - i \, x \sin \left (2 \, a\right )\right )} \left (2 i \, b\right )^{\frac {1}{4}} \Gamma \left (\frac {3}{4}, 2 i \, b x^{2}\right ) - {\left (x \cos \left (2 \, a\right ) + i \, x \sin \left (2 \, a\right )\right )} \left (-2 i \, b\right )^{\frac {1}{4}} \Gamma \left (\frac {3}{4}, -2 i \, b x^{2}\right )}{2 \, x} \]

[In]

integrate(cos(b*x^2+a)^2/x^(3/2),x, algorithm="fricas")

[Out]

-1/2*(4*sqrt(x)*cos(b*x^2 + a)^2 - (x*cos(2*a) - I*x*sin(2*a))*(2*I*b)^(1/4)*gamma(3/4, 2*I*b*x^2) - (x*cos(2*
a) + I*x*sin(2*a))*(-2*I*b)^(1/4)*gamma(3/4, -2*I*b*x^2))/x

Sympy [F]

\[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=\int \frac {\cos ^{2}{\left (a + b x^{2} \right )}}{x^{\frac {3}{2}}}\, dx \]

[In]

integrate(cos(b*x**2+a)**2/x**(3/2),x)

[Out]

Integral(cos(a + b*x**2)**2/x**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(cos(b*x^2+a)^2/x^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> Encountered operator mismatch in maxima-to-sr translation

Giac [F]

\[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=\int { \frac {\cos \left (b x^{2} + a\right )^{2}}{x^{\frac {3}{2}}} \,d x } \]

[In]

integrate(cos(b*x^2+a)^2/x^(3/2),x, algorithm="giac")

[Out]

integrate(cos(b*x^2 + a)^2/x^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^2\left (a+b x^2\right )}{x^{3/2}} \, dx=\int \frac {{\cos \left (b\,x^2+a\right )}^2}{x^{3/2}} \,d x \]

[In]

int(cos(a + b*x^2)^2/x^(3/2),x)

[Out]

int(cos(a + b*x^2)^2/x^(3/2), x)