3.198 \(\int \sqrt{c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=299 \[ -\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \cos \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{384 b^{3/2}}+\frac{3 \sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{128 b^{3/2}}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \sin \left (6 a-\frac{6 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{384 b^{3/2}}-\frac{3 \sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{128 b^{3/2}}-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b} \]

[Out]

(-3*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b) + (Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(192*b) - (Sqrt[d]*Sqrt[Pi/3]*Co
s[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(384*b^(3/2)) + (3*Sqrt[d]*Sqrt[Pi]
*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(3/2)) + (Sqrt[d]*Sqrt[Pi
/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(384*b^(3/2)) - (3*Sqrt[d]*Sq
rt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(3/2))

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Rubi [A]  time = 0.457678, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {4406, 3296, 3306, 3305, 3351, 3304, 3352} \[ -\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \cos \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{384 b^{3/2}}+\frac{3 \sqrt{\pi } \sqrt{d} \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{128 b^{3/2}}+\frac{\sqrt{\frac{\pi }{3}} \sqrt{d} \sin \left (6 a-\frac{6 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{384 b^{3/2}}-\frac{3 \sqrt{\pi } \sqrt{d} \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{128 b^{3/2}}-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(-3*Sqrt[c + d*x]*Cos[2*a + 2*b*x])/(64*b) + (Sqrt[c + d*x]*Cos[6*a + 6*b*x])/(192*b) - (Sqrt[d]*Sqrt[Pi/3]*Co
s[6*a - (6*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(384*b^(3/2)) + (3*Sqrt[d]*Sqrt[Pi]
*Cos[2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(128*b^(3/2)) + (Sqrt[d]*Sqrt[Pi
/3]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(384*b^(3/2)) - (3*Sqrt[d]*Sq
rt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(128*b^(3/2))

Rule 4406

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

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3306

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[(c*f)/d +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[(c*f)/d + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3305

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[(f*x^2)/d], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3351

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

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

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

Rubi steps

\begin{align*} \int \sqrt{c+d x} \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{3}{32} \sqrt{c+d x} \sin (2 a+2 b x)-\frac{1}{32} \sqrt{c+d x} \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac{1}{32} \int \sqrt{c+d x} \sin (6 a+6 b x) \, dx\right )+\frac{3}{32} \int \sqrt{c+d x} \sin (2 a+2 b x) \, dx\\ &=-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b}-\frac{d \int \frac{\cos (6 a+6 b x)}{\sqrt{c+d x}} \, dx}{384 b}+\frac{(3 d) \int \frac{\cos (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{128 b}\\ &=-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b}-\frac{\left (d \cos \left (6 a-\frac{6 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{6 b c}{d}+6 b x\right )}{\sqrt{c+d x}} \, dx}{384 b}+\frac{\left (3 d \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\cos \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx}{128 b}+\frac{\left (d \sin \left (6 a-\frac{6 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{6 b c}{d}+6 b x\right )}{\sqrt{c+d x}} \, dx}{384 b}-\frac{\left (3 d \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \int \frac{\sin \left (\frac{2 b c}{d}+2 b x\right )}{\sqrt{c+d x}} \, dx}{128 b}\\ &=-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b}-\frac{\cos \left (6 a-\frac{6 b c}{d}\right ) \operatorname{Subst}\left (\int \cos \left (\frac{6 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{192 b}+\frac{\left (3 \cos \left (2 a-\frac{2 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{64 b}+\frac{\sin \left (6 a-\frac{6 b c}{d}\right ) \operatorname{Subst}\left (\int \sin \left (\frac{6 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{192 b}-\frac{\left (3 \sin \left (2 a-\frac{2 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{2 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{64 b}\\ &=-\frac{3 \sqrt{c+d x} \cos (2 a+2 b x)}{64 b}+\frac{\sqrt{c+d x} \cos (6 a+6 b x)}{192 b}-\frac{\sqrt{d} \sqrt{\frac{\pi }{3}} \cos \left (6 a-\frac{6 b c}{d}\right ) C\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{384 b^{3/2}}+\frac{3 \sqrt{d} \sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) C\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{128 b^{3/2}}+\frac{\sqrt{d} \sqrt{\frac{\pi }{3}} S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right ) \sin \left (6 a-\frac{6 b c}{d}\right )}{384 b^{3/2}}-\frac{3 \sqrt{d} \sqrt{\pi } S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{128 b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.38502, size = 264, normalized size = 0.88 \[ \frac{-\sqrt{3 \pi } \cos \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )+27 \sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+\sqrt{3 \pi } \sin \left (6 a-\frac{6 b c}{d}\right ) S\left (2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right )-27 \sqrt{\pi } \sin \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )-54 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+6 \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))}{1152 b \sqrt{\frac{b}{d}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(-54*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 6*Sqrt[b/d]*Sqrt[c + d*x]*Cos[6*(a + b*x)] - Sqrt[3*Pi]*Cos[6*
a - (6*b*c)/d]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] + 27*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*FresnelC[(2*S
qrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + Sqrt[3*Pi]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6*a - (6*b*c
)/d] - 27*Sqrt[Pi]*FresnelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d])/(1152*b*Sqrt[b/d])

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Maple [A]  time = 0.041, size = 293, normalized size = 1. \begin{align*} 2\,{\frac{1}{d} \left ( -{\frac{3\,d\sqrt{dx+c}}{128\,b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }+{\frac{3\,d\sqrt{\pi }}{256\,b} \left ( \cos \left ( 2\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ( 2\,{\frac{\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 2\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ( 2\,{\frac{\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}}+{\frac{d\sqrt{dx+c}}{384\,b}\cos \left ( 6\,{\frac{ \left ( dx+c \right ) b}{d}}+6\,{\frac{ad-bc}{d}} \right ) }-{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{6}}{4608\,b} \left ( \cos \left ( 6\,{\frac{ad-bc}{d}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) -\sin \left ( 6\,{\frac{ad-bc}{d}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{6}\sqrt{dx+c}b}{d\sqrt{\pi }}{\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x)

[Out]

2/d*(-3/128/b*d*(d*x+c)^(1/2)*cos(2/d*(d*x+c)*b+2*(a*d-b*c)/d)+3/256/b*d*Pi^(1/2)/(b/d)^(1/2)*(cos(2*(a*d-b*c)
/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)-sin(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)/(b/d)^(1/2)*(d*x
+c)^(1/2)*b/d))+1/384/b*d*(d*x+c)^(1/2)*cos(6/d*(d*x+c)*b+6*(a*d-b*c)/d)-1/4608/b*d*2^(1/2)*Pi^(1/2)*6^(1/2)/(
b/d)^(1/2)*(cos(6*(a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)-sin(6*(a*d-b*c
)/d)*FresnelS(2^(1/2)/Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)))

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Maxima [C]  time = 2.19434, size = 1681, normalized size = 5.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/18432*sqrt(6)*sqrt(2)*(8*sqrt(6)*sqrt(2)*sqrt(d*x + c)*d*abs(b)*cos(6*((d*x + c)*b - b*c + a*d)/d)/abs(d) -
72*sqrt(6)*sqrt(2)*sqrt(d*x + c)*d*abs(b)*cos(2*((d*x + c)*b - b*c + a*d)/d)/abs(d) - (sqrt(2)*(sqrt(pi)*cos(1
/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arct
an2(0, d/sqrt(d^2))) - I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + I*sqrt(pi)*s
in(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d*sqrt(abs(b)/abs(d))*cos(-6*(b*c - a*d)/d) - s
qrt(2)*(I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + I*sqrt(pi)*cos(-1/4*pi + 1/
2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sq
rt(d^2))) - sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d*sqrt(abs(b)/abs(d))*sin
(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(6*I*b/d)) + (sqrt(6)*(9*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1
/2*arctan2(0, d/sqrt(d^2))) + 9*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 9*I*
sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan
2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d*sqrt(abs(b)/abs(d))*cos(-2*(b*c - a*d)/d) + sqrt(6)*(-9*I*sqrt(pi)*c
os(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 9*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) +
1/2*arctan2(0, d/sqrt(d^2))) - 9*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*sq
rt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d*sqrt(abs(b)/abs(d))*sin(-2*(b*c - a*d
)/d))*erf(sqrt(d*x + c)*sqrt(2*I*b/d)) + (sqrt(6)*(9*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0,
d/sqrt(d^2))) + 9*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*I*sqrt(pi)*sin(1
/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 9*I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*
arctan2(0, d/sqrt(d^2))))*d*sqrt(abs(b)/abs(d))*cos(-2*(b*c - a*d)/d) + sqrt(6)*(9*I*sqrt(pi)*cos(1/4*pi + 1/2
*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0,
d/sqrt(d^2))) - 9*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 9*sqrt(pi)*sin(-1/4
*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d*sqrt(abs(b)/abs(d))*sin(-2*(b*c - a*d)/d))*erf(sqrt(
d*x + c)*sqrt(-2*I*b/d)) - (sqrt(2)*(sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) +
sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(
0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - I*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2)
)))*d*sqrt(abs(b)/abs(d))*cos(-6*(b*c - a*d)/d) - sqrt(2)*(-I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*ar
ctan2(0, d/sqrt(d^2))) - I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + sqrt(pi)*
sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2
*arctan2(0, d/sqrt(d^2))))*d*sqrt(abs(b)/abs(d))*sin(-6*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-6*I*b/d)))*abs
(d)/(b*d*abs(b))

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Fricas [A]  time = 0.64598, size = 632, normalized size = 2.11 \begin{align*} -\frac{\sqrt{3} \pi d \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (2 \, \sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - \sqrt{3} \pi d \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (2 \, \sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) - 27 \, \pi d \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{C}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + 27 \, \pi d \sqrt{\frac{b}{\pi d}} \operatorname{S}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - 48 \,{\left (4 \, b \cos \left (b x + a\right )^{6} - 6 \, b \cos \left (b x + a\right )^{4} + b\right )} \sqrt{d x + c}}{1152 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/1152*(sqrt(3)*pi*d*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_cos(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))
- sqrt(3)*pi*d*sqrt(b/(pi*d))*fresnel_sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/d) - 27*
pi*d*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d))) + 27*pi*d*sqrt(b/(pi*d))
*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) - 48*(4*b*cos(b*x + a)^6 - 6*b*cos(b*x + a)
^4 + b)*sqrt(d*x + c))/b^2

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)*cos(b*x+a)**3*sin(b*x+a)**3,x)

[Out]

Timed out

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Giac [C]  time = 1.55546, size = 643, normalized size = 2.15 \begin{align*} \frac{\frac{\sqrt{3} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{3} \sqrt{b d} \sqrt{d x + c}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac{6 i \, b c - 6 i \, a d}{d}\right )}}{\sqrt{b d}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} + \frac{\sqrt{3} \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{3} \sqrt{b d} \sqrt{d x + c}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac{-6 i \, b c + 6 i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} - \frac{27 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{b d} \sqrt{d x + c}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac{2 i \, b c - 2 i \, a d}{d}\right )}}{\sqrt{b d}{\left (\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} - \frac{27 \, \sqrt{\pi } d^{2} \operatorname{erf}\left (-\frac{\sqrt{b d} \sqrt{d x + c}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )}}{d}\right ) e^{\left (\frac{-2 i \, b c + 2 i \, a d}{d}\right )}}{\sqrt{b d}{\left (-\frac{i \, b d}{\sqrt{b^{2} d^{2}}} + 1\right )} b} + \frac{6 \, \sqrt{d x + c} d e^{\left (\frac{6 i \,{\left (d x + c\right )} b - 6 i \, b c + 6 i \, a d}{d}\right )}}{b} - \frac{54 \, \sqrt{d x + c} d e^{\left (\frac{2 i \,{\left (d x + c\right )} b - 2 i \, b c + 2 i \, a d}{d}\right )}}{b} - \frac{54 \, \sqrt{d x + c} d e^{\left (\frac{-2 i \,{\left (d x + c\right )} b + 2 i \, b c - 2 i \, a d}{d}\right )}}{b} + \frac{6 \, \sqrt{d x + c} d e^{\left (\frac{-6 i \,{\left (d x + c\right )} b + 6 i \, b c - 6 i \, a d}{d}\right )}}{b}}{2304 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/2304*(sqrt(3)*sqrt(pi)*d^2*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((6*I*b*c - 6
*I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) + sqrt(3)*sqrt(pi)*d^2*erf(-sqrt(3)*sqrt(b*d)*sqrt(d*x + c)
*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-6*I*b*c + 6*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) - 27*sqrt(p
i)*d^2*erf(-sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((2*I*b*c - 2*I*a*d)/d)/(sqrt(b*d)*(I*b*d/s
qrt(b^2*d^2) + 1)*b) - 27*sqrt(pi)*d^2*erf(-sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-2*I*b*c
+ 2*I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b) + 6*sqrt(d*x + c)*d*e^((6*I*(d*x + c)*b - 6*I*b*c + 6*
I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^((2*I*(d*x + c)*b - 2*I*b*c + 2*I*a*d)/d)/b - 54*sqrt(d*x + c)*d*e^((-2*I*(
d*x + c)*b + 2*I*b*c - 2*I*a*d)/d)/b + 6*sqrt(d*x + c)*d*e^((-6*I*(d*x + c)*b + 6*I*b*c - 6*I*a*d)/d)/b)/d