3.1.28 \(\int x^2 \sin ^3(a+b x^2) \, dx\) [28]

Optimal. Leaf size=188 \[ -\frac {3 x \cos \left (a+b x^2\right )}{8 b}+\frac {x \cos \left (3 a+3 b x^2\right )}{24 b}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) C\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{24 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{8 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{24 b^{3/2}} \]

[Out]

-3/8*x*cos(b*x^2+a)/b+1/24*x*cos(3*b*x^2+3*a)/b-1/144*cos(3*a)*FresnelC(x*b^(1/2)*6^(1/2)/Pi^(1/2))*6^(1/2)*Pi
^(1/2)/b^(3/2)+1/144*FresnelS(x*b^(1/2)*6^(1/2)/Pi^(1/2))*sin(3*a)*6^(1/2)*Pi^(1/2)/b^(3/2)+3/16*cos(a)*Fresne
lC(x*b^(1/2)*2^(1/2)/Pi^(1/2))*2^(1/2)*Pi^(1/2)/b^(3/2)-3/16*FresnelS(x*b^(1/2)*2^(1/2)/Pi^(1/2))*sin(a)*2^(1/
2)*Pi^(1/2)/b^(3/2)

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Rubi [A]
time = 0.16, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3484, 3466, 3435, 3433, 3432} \begin {gather*} \frac {3 \sqrt {\frac {\pi }{2}} \cos (a) \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {b} x\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {b} x\right )}{24 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} \sin (a) S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{8 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} \sin (3 a) S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{24 b^{3/2}}-\frac {3 x \cos \left (a+b x^2\right )}{8 b}+\frac {x \cos \left (3 a+3 b x^2\right )}{24 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sin[a + b*x^2]^3,x]

[Out]

(-3*x*Cos[a + b*x^2])/(8*b) + (x*Cos[3*a + 3*b*x^2])/(24*b) + (3*Sqrt[Pi/2]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/Pi]
*x])/(8*b^(3/2)) - (Sqrt[Pi/6]*Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x])/(24*b^(3/2)) - (3*Sqrt[Pi/2]*FresnelS[
Sqrt[b]*Sqrt[2/Pi]*x]*Sin[a])/(8*b^(3/2)) + (Sqrt[Pi/6]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(24*b^(3/2))

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3435

Int[Cos[(c_) + (d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Dist[Cos[c], Int[Cos[d*(e + f*x)^2], x], x] - Dist[
Sin[c], Int[Sin[d*(e + f*x)^2], x], x] /; FreeQ[{c, d, e, f}, x]

Rule 3466

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

Rule 3484

Int[((e_.)*(x_))^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_), x_Symbol] :> Int[ExpandTrigReduce[(e
*x)^m, (a + b*Sin[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*} \int x^2 \sin ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac {3}{4} x^2 \sin \left (a+b x^2\right )-\frac {1}{4} x^2 \sin \left (3 a+3 b x^2\right )\right ) \, dx\\ &=-\left (\frac {1}{4} \int x^2 \sin \left (3 a+3 b x^2\right ) \, dx\right )+\frac {3}{4} \int x^2 \sin \left (a+b x^2\right ) \, dx\\ &=-\frac {3 x \cos \left (a+b x^2\right )}{8 b}+\frac {x \cos \left (3 a+3 b x^2\right )}{24 b}-\frac {\int \cos \left (3 a+3 b x^2\right ) \, dx}{24 b}+\frac {3 \int \cos \left (a+b x^2\right ) \, dx}{8 b}\\ &=-\frac {3 x \cos \left (a+b x^2\right )}{8 b}+\frac {x \cos \left (3 a+3 b x^2\right )}{24 b}+\frac {(3 \cos (a)) \int \cos \left (b x^2\right ) \, dx}{8 b}-\frac {\cos (3 a) \int \cos \left (3 b x^2\right ) \, dx}{24 b}-\frac {(3 \sin (a)) \int \sin \left (b x^2\right ) \, dx}{8 b}+\frac {\sin (3 a) \int \sin \left (3 b x^2\right ) \, dx}{24 b}\\ &=-\frac {3 x \cos \left (a+b x^2\right )}{8 b}+\frac {x \cos \left (3 a+3 b x^2\right )}{24 b}+\frac {3 \sqrt {\frac {\pi }{2}} \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )}{8 b^{3/2}}-\frac {\sqrt {\frac {\pi }{6}} \cos (3 a) C\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )}{24 b^{3/2}}-\frac {3 \sqrt {\frac {\pi }{2}} S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)}{8 b^{3/2}}+\frac {\sqrt {\frac {\pi }{6}} S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{24 b^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.30, size = 159, normalized size = 0.85 \begin {gather*} \frac {-54 \sqrt {b} x \cos \left (a+b x^2\right )+6 \sqrt {b} x \cos \left (3 \left (a+b x^2\right )\right )+27 \sqrt {2 \pi } \cos (a) C\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right )-\sqrt {6 \pi } \cos (3 a) C\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right )-27 \sqrt {2 \pi } S\left (\sqrt {b} \sqrt {\frac {2}{\pi }} x\right ) \sin (a)+\sqrt {6 \pi } S\left (\sqrt {b} \sqrt {\frac {6}{\pi }} x\right ) \sin (3 a)}{144 b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sin[a + b*x^2]^3,x]

[Out]

(-54*Sqrt[b]*x*Cos[a + b*x^2] + 6*Sqrt[b]*x*Cos[3*(a + b*x^2)] + 27*Sqrt[2*Pi]*Cos[a]*FresnelC[Sqrt[b]*Sqrt[2/
Pi]*x] - Sqrt[6*Pi]*Cos[3*a]*FresnelC[Sqrt[b]*Sqrt[6/Pi]*x] - 27*Sqrt[2*Pi]*FresnelS[Sqrt[b]*Sqrt[2/Pi]*x]*Sin
[a] + Sqrt[6*Pi]*FresnelS[Sqrt[b]*Sqrt[6/Pi]*x]*Sin[3*a])/(144*b^(3/2))

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Maple [A]
time = 0.10, size = 132, normalized size = 0.70

method	result	size
default	\(-\frac {3 x \cos \left (b \,x^{2}+a \right )}{8 b}+\frac {3 \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (a \right ) \FresnelC \left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )-\sin \left (a \right ) \mathrm {S}\left (\frac {x \sqrt {b}\, \sqrt {2}}{\sqrt {\pi }}\right )\right )}{16 b^{\frac {3}{2}}}+\frac {x \cos \left (3 b \,x^{2}+3 a \right )}{24 b}-\frac {\sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (3 a \right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )-\sin \left (3 a \right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {b}\, x}{\sqrt {\pi }}\right )\right )}{144 b^{\frac {3}{2}}}\)	\(132\)
risch	\(-\frac {{\mathrm e}^{-3 i a} \sqrt {\pi }\, \sqrt {3}\, \erf \left (\sqrt {3}\, \sqrt {i b}\, x \right )}{288 b \sqrt {i b}}-\frac {{\mathrm e}^{3 i a} \sqrt {\pi }\, \erf \left (\sqrt {-3 i b}\, x \right )}{96 b \sqrt {-3 i b}}+\frac {3 \,{\mathrm e}^{i a} \sqrt {\pi }\, \erf \left (\sqrt {-i b}\, x \right )}{32 b \sqrt {-i b}}+\frac {3 \,{\mathrm e}^{-i a} \sqrt {\pi }\, \erf \left (\sqrt {i b}\, x \right )}{32 b \sqrt {i b}}-\frac {3 x \cos \left (b \,x^{2}+a \right )}{8 b}+\frac {x \cos \left (3 b \,x^{2}+3 a \right )}{24 b}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(b*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

-3/8*x*cos(b*x^2+a)/b+3/16/b^(3/2)*2^(1/2)*Pi^(1/2)*(cos(a)*FresnelC(x*b^(1/2)*2^(1/2)/Pi^(1/2))-sin(a)*Fresne
lS(x*b^(1/2)*2^(1/2)/Pi^(1/2)))+1/24*x*cos(3*b*x^2+3*a)/b-1/144/b^(3/2)*2^(1/2)*Pi^(1/2)*3^(1/2)*(cos(3*a)*Fre
snelC(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x)-sin(3*a)*FresnelS(2^(1/2)/Pi^(1/2)*3^(1/2)*b^(1/2)*x))

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Maxima [C] Result contains complex when optimal does not.
time = 0.52, size = 143, normalized size = 0.76 \begin {gather*} \frac {24 \, b^{2} x \cos \left (3 \, b x^{2} + 3 \, a\right ) - 216 \, b^{2} x \cos \left (b x^{2} + a\right ) + 9^{\frac {1}{4}} \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (3 \, a\right ) + \left (i + 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3 i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \left (3 \, a\right ) - \left (i - 1\right ) \, \sin \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {-3 i \, b} x\right )\right )} b^{\frac {3}{2}} - 27 \, \sqrt {2} \sqrt {\pi } {\left ({\left (\left (i - 1\right ) \, \cos \left (a\right ) + \left (i + 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {i \, b} x\right ) + {\left (-\left (i + 1\right ) \, \cos \left (a\right ) - \left (i - 1\right ) \, \sin \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-i \, b} x\right )\right )} b^{\frac {3}{2}}}{576 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(24*b^2*x*cos(3*b*x^2 + 3*a) - 216*b^2*x*cos(b*x^2 + a) + 9^(1/4)*sqrt(2)*sqrt(pi)*(((I - 1)*cos(3*a) +
(I + 1)*sin(3*a))*erf(sqrt(3*I*b)*x) + (-(I + 1)*cos(3*a) - (I - 1)*sin(3*a))*erf(sqrt(-3*I*b)*x))*b^(3/2) - 2
7*sqrt(2)*sqrt(pi)*(((I - 1)*cos(a) + (I + 1)*sin(a))*erf(sqrt(I*b)*x) + (-(I + 1)*cos(a) - (I - 1)*sin(a))*er
f(sqrt(-I*b)*x))*b^(3/2))/b^3

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Fricas [A]
time = 0.37, size = 147, normalized size = 0.78 \begin {gather*} \frac {24 \, b x \cos \left (b x^{2} + a\right )^{3} - \sqrt {6} \pi \sqrt {\frac {b}{\pi }} \cos \left (3 \, a\right ) \operatorname {C}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) + 27 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \cos \left (a\right ) \operatorname {C}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) + \sqrt {6} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {6} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (3 \, a\right ) - 27 \, \sqrt {2} \pi \sqrt {\frac {b}{\pi }} \operatorname {S}\left (\sqrt {2} x \sqrt {\frac {b}{\pi }}\right ) \sin \left (a\right ) - 72 \, b x \cos \left (b x^{2} + a\right )}{144 \, b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/144*(24*b*x*cos(b*x^2 + a)^3 - sqrt(6)*pi*sqrt(b/pi)*cos(3*a)*fresnel_cos(sqrt(6)*x*sqrt(b/pi)) + 27*sqrt(2)
*pi*sqrt(b/pi)*cos(a)*fresnel_cos(sqrt(2)*x*sqrt(b/pi)) + sqrt(6)*pi*sqrt(b/pi)*fresnel_sin(sqrt(6)*x*sqrt(b/p
i))*sin(3*a) - 27*sqrt(2)*pi*sqrt(b/pi)*fresnel_sin(sqrt(2)*x*sqrt(b/pi))*sin(a) - 72*b*x*cos(b*x^2 + a))/b^2

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 439 vs. \(2 (194) = 388\).
time = 2.40, size = 439, normalized size = 2.34 \begin {gather*} - \frac {3 b^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}} \cos {\left (a \right )} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {9}{4} \end {matrix}\middle | {- \frac {b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right )} + \frac {3 b^{\frac {3}{2}} x^{5} \sqrt {\frac {1}{b}} \cos {\left (3 a \right )} \Gamma \left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4} \\ \frac {3}{2}, \frac {7}{4}, \frac {9}{4} \end {matrix}\middle | {- \frac {9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {7}{4}\right ) \Gamma \left (\frac {9}{4}\right )} - \frac {3 \sqrt {b} x^{3} \sqrt {\frac {1}{b}} \sin {\left (a \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {7}{4} \end {matrix}\middle | {- \frac {b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right )} + \frac {\sqrt {b} x^{3} \sqrt {\frac {1}{b}} \sin {\left (3 a \right )} \Gamma \left (\frac {1}{4}\right ) \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{3}\left (\begin {matrix} \frac {1}{4}, \frac {3}{4} \\ \frac {1}{2}, \frac {5}{4}, \frac {7}{4} \end {matrix}\middle | {- \frac {9 b^{2} x^{4}}{4}} \right )}}{32 \Gamma \left (\frac {5}{4}\right ) \Gamma \left (\frac {7}{4}\right )} + \frac {3 \sqrt {2} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \sin {\left (a \right )} C\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )}{8} - \frac {\sqrt {6} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \sin {\left (3 a \right )} C\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )}{24} + \frac {3 \sqrt {2} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \cos {\left (a \right )} S\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\pi }}\right )}{8} - \frac {\sqrt {6} \sqrt {\pi } x^{2} \sqrt {\frac {1}{b}} \cos {\left (3 a \right )} S\left (\frac {\sqrt {6} \sqrt {b} x}{\sqrt {\pi }}\right )}{24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*sin(b*x**2+a)**3,x)

[Out]

-3*b**(3/2)*x**5*sqrt(1/b)*cos(a)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4, 9/4), -b**2*x**4/4)/(32*g
amma(7/4)*gamma(9/4)) + 3*b**(3/2)*x**5*sqrt(1/b)*cos(3*a)*gamma(3/4)*gamma(5/4)*hyper((3/4, 5/4), (3/2, 7/4,
9/4), -9*b**2*x**4/4)/(32*gamma(7/4)*gamma(9/4)) - 3*sqrt(b)*x**3*sqrt(1/b)*sin(a)*gamma(1/4)*gamma(3/4)*hyper
((1/4, 3/4), (1/2, 5/4, 7/4), -b**2*x**4/4)/(32*gamma(5/4)*gamma(7/4)) + sqrt(b)*x**3*sqrt(1/b)*sin(3*a)*gamma
(1/4)*gamma(3/4)*hyper((1/4, 3/4), (1/2, 5/4, 7/4), -9*b**2*x**4/4)/(32*gamma(5/4)*gamma(7/4)) + 3*sqrt(2)*sqr
t(pi)*x**2*sqrt(1/b)*sin(a)*fresnelc(sqrt(2)*sqrt(b)*x/sqrt(pi))/8 - sqrt(6)*sqrt(pi)*x**2*sqrt(1/b)*sin(3*a)*
fresnelc(sqrt(6)*sqrt(b)*x/sqrt(pi))/24 + 3*sqrt(2)*sqrt(pi)*x**2*sqrt(1/b)*cos(a)*fresnels(sqrt(2)*sqrt(b)*x/
sqrt(pi))/8 - sqrt(6)*sqrt(pi)*x**2*sqrt(1/b)*cos(3*a)*fresnels(sqrt(6)*sqrt(b)*x/sqrt(pi))/24

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Giac [C] Result contains complex when optimal does not.
time = 5.88, size = 259, normalized size = 1.38 \begin {gather*} \frac {x e^{\left (3 i \, b x^{2} + 3 i \, a\right )}}{48 \, b} - \frac {3 \, x e^{\left (i \, b x^{2} + i \, a\right )}}{16 \, b} - \frac {3 \, x e^{\left (-i \, b x^{2} - i \, a\right )}}{16 \, b} + \frac {x e^{\left (-3 i \, b x^{2} - 3 i \, a\right )}}{48 \, b} + \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {6} \sqrt {b} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (3 i \, a\right )}}{288 \, b^{\frac {3}{2}} {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )}} - \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (i \, a\right )}}{32 \, b {\left (-\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} - \frac {3 \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}\right ) e^{\left (-i \, a\right )}}{32 \, b {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )} \sqrt {{\left | b \right |}}} + \frac {\sqrt {6} \sqrt {\pi } \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {6} \sqrt {b} x {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}\right ) e^{\left (-3 i \, a\right )}}{288 \, b^{\frac {3}{2}} {\left (\frac {i \, b}{{\left | b \right |}} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*x*e^(3*I*b*x^2 + 3*I*a)/b - 3/16*x*e^(I*b*x^2 + I*a)/b - 3/16*x*e^(-I*b*x^2 - I*a)/b + 1/48*x*e^(-3*I*b*x
^2 - 3*I*a)/b + 1/288*sqrt(6)*sqrt(pi)*erf(-1/2*sqrt(6)*sqrt(b)*x*(-I*b/abs(b) + 1))*e^(3*I*a)/(b^(3/2)*(-I*b/
abs(b) + 1)) - 3/32*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x*(-I*b/abs(b) + 1)*sqrt(abs(b)))*e^(I*a)/(b*(-I*b/abs(b
) + 1)*sqrt(abs(b))) - 3/32*sqrt(2)*sqrt(pi)*erf(-1/2*sqrt(2)*x*(I*b/abs(b) + 1)*sqrt(abs(b)))*e^(-I*a)/(b*(I*
b/abs(b) + 1)*sqrt(abs(b))) + 1/288*sqrt(6)*sqrt(pi)*erf(-1/2*sqrt(6)*sqrt(b)*x*(I*b/abs(b) + 1))*e^(-3*I*a)/(
b^(3/2)*(I*b/abs(b) + 1))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\sin \left (b\,x^2+a\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*sin(a + b*x^2)^3,x)

[Out]

int(x^2*sin(a + b*x^2)^3, x)

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