3.200 \(\int (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=351 \[ \frac{\sqrt{\frac{\pi }{3}} d^{3/2} \sin \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \sin \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{512 b^{5/2}}+\frac{\sqrt{\frac{\pi }{3}} d^{3/2} \cos \left (6 a-\frac{6 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \cos \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{512 b^{5/2}}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b} \]

[Out]

(-3*(c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(64*b) + ((c + d*x)^(3/2)*Cos[6*a + 6*b*x])/(192*b) + (d^(3/2)*Sqrt[Pi/3
]*Cos[6*a - (6*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1536*b^(5/2)) - (9*d^(3/2)*Sqr
t[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(512*b^(5/2)) + (d^(3/2)*Sq
rt[Pi/3]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(1536*b^(5/2)) - (9*d^(3
/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(512*b^(5/2)) + (9*d
*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(256*b^2) - (d*Sqrt[c + d*x]*Sin[6*a + 6*b*x])/(768*b^2)

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Rubi [A]  time = 0.55918, antiderivative size = 351, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ \frac{\sqrt{\frac{\pi }{3}} d^{3/2} \sin \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{3}{\pi }} \sqrt{b} \sqrt{c+d x}}{\sqrt{d}}\right )}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \sin \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{\pi } \sqrt{d}}\right )}{512 b^{5/2}}+\frac{\sqrt{\frac{\pi }{3}} d^{3/2} \cos \left (6 a-\frac{6 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}}{\sqrt{d}}\right )}{1536 b^{5/2}}-\frac{9 \sqrt{\pi } d^{3/2} \cos \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{512 b^{5/2}}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(-3*(c + d*x)^(3/2)*Cos[2*a + 2*b*x])/(64*b) + ((c + d*x)^(3/2)*Cos[6*a + 6*b*x])/(192*b) + (d^(3/2)*Sqrt[Pi/3
]*Cos[6*a - (6*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(1536*b^(5/2)) - (9*d^(3/2)*Sqr
t[Pi]*Cos[2*a - (2*b*c)/d]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/(512*b^(5/2)) + (d^(3/2)*Sq
rt[Pi/3]*FresnelC[(2*Sqrt[b]*Sqrt[3/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[6*a - (6*b*c)/d])/(1536*b^(5/2)) - (9*d^(3
/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(512*b^(5/2)) + (9*d
*Sqrt[c + d*x]*Sin[2*a + 2*b*x])/(256*b^2) - (d*Sqrt[c + d*x]*Sin[6*a + 6*b*x])/(768*b^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 (c+d x)^{3/2} \cos ^3(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac{3}{32} (c+d x)^{3/2} \sin (2 a+2 b x)-\frac{1}{32} (c+d x)^{3/2} \sin (6 a+6 b x)\right ) \, dx\\ &=-\left (\frac{1}{32} \int (c+d x)^{3/2} \sin (6 a+6 b x) \, dx\right )+\frac{3}{32} \int (c+d x)^{3/2} \sin (2 a+2 b x) \, dx\\ &=-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}-\frac{d \int \sqrt{c+d x} \cos (6 a+6 b x) \, dx}{128 b}+\frac{(9 d) \int \sqrt{c+d x} \cos (2 a+2 b x) \, dx}{128 b}\\ &=-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac{d^2 \int \frac{\sin (6 a+6 b x)}{\sqrt{c+d x}} \, dx}{1536 b^2}-\frac{\left (9 d^2\right ) \int \frac{\sin (2 a+2 b x)}{\sqrt{c+d x}} \, dx}{512 b^2}\\ &=-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac{\left (d^2 \cos \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}{1536 b^2}-\frac{\left (9 d^2 \cos \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}{512 b^2}+\frac{\left (d^2 \sin \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}{1536 b^2}-\frac{\left (9 d^2 \sin \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}{512 b^2}\\ &=-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}+\frac{\left (d \cos \left (6 a-\frac{6 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \sin \left (\frac{6 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{768 b^2}-\frac{\left (9 d \cos \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 )}{256 b^2}+\frac{\left (d \sin \left (6 a-\frac{6 b c}{d}\right )\right ) \operatorname{Subst}\left (\int \cos \left (\frac{6 b x^2}{d}\right ) \, dx,x,\sqrt{c+d x}\right )}{768 b^2}-\frac{\left (9 d \sin \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 )}{256 b^2}\\ &=-\frac{3 (c+d x)^{3/2} \cos (2 a+2 b x)}{64 b}+\frac{(c+d x)^{3/2} \cos (6 a+6 b x)}{192 b}+\frac{d^{3/2} \sqrt{\frac{\pi }{3}} \cos \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 )}{1536 b^{5/2}}-\frac{9 d^{3/2} \sqrt{\pi } \cos \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right )}{512 b^{5/2}}+\frac{d^{3/2} \sqrt{\frac{\pi }{3}} C\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 )}{1536 b^{5/2}}-\frac{9 d^{3/2} \sqrt{\pi } C\left (\frac{2 \sqrt{b} \sqrt{c+d x}}{\sqrt{d} \sqrt{\pi }}\right ) \sin \left (2 a-\frac{2 b c}{d}\right )}{512 b^{5/2}}+\frac{9 d \sqrt{c+d x} \sin (2 a+2 b x)}{256 b^2}-\frac{d \sqrt{c+d x} \sin (6 a+6 b x)}{768 b^2}\\ \end{align*}

Mathematica [A]  time = 0.216276, size = 391, normalized size = 1.11 \[ \frac{\sqrt{3 \pi } d \sin \left (6 a-\frac{6 b c}{d}\right ) \text{FresnelC}\left (2 \sqrt{\frac{3}{\pi }} \sqrt{\frac{b}{d}} \sqrt{c+d x}\right )-81 \sqrt{\pi } d \sin \left (2 a-\frac{2 b c}{d}\right ) \text{FresnelC}\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+\sqrt{3 \pi } d \cos \left (6 a-\frac{6 b c}{d}\right ) S\left (2 \sqrt{\frac{b}{d}} \sqrt{\frac{3}{\pi }} \sqrt{c+d x}\right )-81 \sqrt{\pi } d \cos \left (2 a-\frac{2 b c}{d}\right ) S\left (\frac{2 \sqrt{\frac{b}{d}} \sqrt{c+d x}}{\sqrt{\pi }}\right )+162 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (2 (a+b x))-6 d \sqrt{\frac{b}{d}} \sqrt{c+d x} \sin (6 (a+b x))-216 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))-216 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (2 (a+b x))+24 b d x \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))+24 b c \sqrt{\frac{b}{d}} \sqrt{c+d x} \cos (6 (a+b x))}{4608 b^2 \sqrt{\frac{b}{d}}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^(3/2)*Cos[a + b*x]^3*Sin[a + b*x]^3,x]

[Out]

(-216*b*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[2*(a + b*x)] - 216*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[2*(a + b*x)] + 24*b
*c*Sqrt[b/d]*Sqrt[c + d*x]*Cos[6*(a + b*x)] + 24*b*Sqrt[b/d]*d*x*Sqrt[c + d*x]*Cos[6*(a + b*x)] + d*Sqrt[3*Pi]
*Cos[6*a - (6*b*c)/d]*FresnelS[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]] - 81*d*Sqrt[Pi]*Cos[2*a - (2*b*c)/d]*Fres
nelS[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]] + d*Sqrt[3*Pi]*FresnelC[2*Sqrt[b/d]*Sqrt[3/Pi]*Sqrt[c + d*x]]*Sin[6
*a - (6*b*c)/d] - 81*d*Sqrt[Pi]*FresnelC[(2*Sqrt[b/d]*Sqrt[c + d*x])/Sqrt[Pi]]*Sin[2*a - (2*b*c)/d] + 162*Sqrt
[b/d]*d*Sqrt[c + d*x]*Sin[2*(a + b*x)] - 6*Sqrt[b/d]*d*Sqrt[c + d*x]*Sin[6*(a + b*x)])/(4608*b^2*Sqrt[b/d])

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Maple [A]  time = 0.039, size = 383, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d} \left ( -{\frac{3\,d \left ( dx+c \right ) ^{3/2}}{128\,b}\cos \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }+{\frac{9\,d}{128\,b} \left ( 1/4\,{\frac{d\sqrt{dx+c}}{b}\sin \left ( 2\,{\frac{ \left ( dx+c \right ) b}{d}}+2\,{\frac{ad-bc}{d}} \right ) }-1/8\,{\frac{d\sqrt{\pi }}{b} \left ( \cos \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 ) +\sin \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 ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) }+{\frac{d \left ( dx+c \right ) ^{3/2}}{384\,b}\cos \left ( 6\,{\frac{ \left ( dx+c \right ) b}{d}}+6\,{\frac{ad-bc}{d}} \right ) }-{\frac{d}{128\,b} \left ( 1/12\,{\frac{d\sqrt{dx+c}}{b}\sin \left ( 6\,{\frac{ \left ( dx+c \right ) b}{d}}+6\,{\frac{ad-bc}{d}} \right ) }-{\frac{d\sqrt{2}\sqrt{\pi }\sqrt{6}}{144\,b} \left ( \cos \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 ) +\sin \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 ) \right ){\frac{1}{\sqrt{{\frac{b}{d}}}}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-3/128/b*d*(d*x+c)^(3/2)*cos(2/d*(d*x+c)*b+2*(a*d-b*c)/d)+9/128/b*d*(1/4/b*d*(d*x+c)^(1/2)*sin(2/d*(d*x+c
)*b+2*(a*d-b*c)/d)-1/8/b*d*Pi^(1/2)/(b/d)^(1/2)*(cos(2*(a*d-b*c)/d)*FresnelS(2/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1
/2)*b/d)+sin(2*(a*d-b*c)/d)*FresnelC(2/Pi^(1/2)/(b/d)^(1/2)*(d*x+c)^(1/2)*b/d)))+1/384/b*d*(d*x+c)^(3/2)*cos(6
/d*(d*x+c)*b+6*(a*d-b*c)/d)-1/128/b*d*(1/12/b*d*(d*x+c)^(1/2)*sin(6/d*(d*x+c)*b+6*(a*d-b*c)/d)-1/144/b*d*2^(1/
2)*Pi^(1/2)*6^(1/2)/(b/d)^(1/2)*(cos(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)+sin(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))))

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Maxima [C]  time = 2.21766, size = 1829, normalized size = 5.21 \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)^(3/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="maxima")

[Out]

1/73728*sqrt(6)*sqrt(2)*(32*sqrt(6)*sqrt(2)*(d*x + c)^(3/2)*b*d*abs(b)*cos(6*((d*x + c)*b - b*c + a*d)/d)/abs(
d) - 288*sqrt(6)*sqrt(2)*(d*x + c)^(3/2)*b*d*abs(b)*cos(2*((d*x + c)*b - b*c + a*d)/d)/abs(d) - 8*sqrt(6)*sqrt
(2)*sqrt(d*x + c)*d^2*abs(b)*sin(6*((d*x + c)*b - b*c + a*d)/d)/abs(d) + 216*sqrt(6)*sqrt(2)*sqrt(d*x + c)*d^2
*abs(b)*sin(2*((d*x + c)*b - b*c + a*d)/d)/abs(d) - (sqrt(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(p
i)*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^2*sqrt(abs(b)/abs(d))*cos(-6*(b*c - a*d)/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*p
i + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*sqrt(abs(b)/abs(d))*sin(-6*(b*c - a*d)/d))*erf(sqrt(
d*x + c)*sqrt(6*I*b/d)) - (sqrt(6)*(27*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))
) + 27*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*sin(1/4*pi + 1/
2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 27*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0,
d/sqrt(d^2))))*d^2*sqrt(abs(b)/abs(d))*cos(-2*(b*c - a*d)/d) + sqrt(6)*(27*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0
, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2)
)) - 27*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*I*sqrt(pi)*sin(-1/4*pi +
1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*sqrt(abs(b)/abs(d))*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x
+ c)*sqrt(2*I*b/d)) - (sqrt(6)*(-27*I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2)))
- 27*I*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*sin(1/4*pi + 1/2*
arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 27*sqrt(pi)*sin(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/
sqrt(d^2))))*d^2*sqrt(abs(b)/abs(d))*cos(-2*(b*c - a*d)/d) + sqrt(6)*(27*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0,
b) + 1/2*arctan2(0, d/sqrt(d^2))) + 27*sqrt(pi)*cos(-1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2)))
+ 27*I*sqrt(pi)*sin(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) - 27*I*sqrt(pi)*sin(-1/4*pi + 1
/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))))*d^2*sqrt(abs(b)/abs(d))*sin(-2*(b*c - a*d)/d))*erf(sqrt(d*x +
c)*sqrt(-2*I*b/d)) - (sqrt(2)*(I*sqrt(pi)*cos(1/4*pi + 1/2*arctan2(0, b) + 1/2*arctan2(0, d/sqrt(d^2))) + I*s
qrt(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
^2*sqrt(abs(b)/abs(d))*cos(-6*(b*c - a*d)/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*arc
tan2(0, d/sqrt(d^2))))*d^2*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^2*d*abs(b))

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Fricas [A]  time = 0.719563, size = 821, normalized size = 2.34 \begin{align*} \frac{\sqrt{3} \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{6 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{S}\left (2 \, \sqrt{3} \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) + \sqrt{3} \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{C}\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 ) - 81 \, \pi d^{2} \sqrt{\frac{b}{\pi d}} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) \operatorname{S}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) - 81 \, \pi d^{2} \sqrt{\frac{b}{\pi d}} \operatorname{C}\left (2 \, \sqrt{d x + c} \sqrt{\frac{b}{\pi d}}\right ) \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) + 96 \,{\left (8 \,{\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{6} - 12 \,{\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{4} + 2 \, b^{2} d x + 2 \, b^{2} c -{\left (2 \, b d \cos \left (b x + a\right )^{5} - 2 \, b d \cos \left (b x + a\right )^{3} - 3 \, b d \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )\right )} \sqrt{d x + c}}{4608 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4608*(sqrt(3)*pi*d^2*sqrt(b/(pi*d))*cos(-6*(b*c - a*d)/d)*fresnel_sin(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d))
) + sqrt(3)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(2*sqrt(3)*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-6*(b*c - a*d)/d) -
81*pi*d^2*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d))) - 81*pi*d^2*sqrt(b/
(pi*d))*fresnel_cos(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) + 96*(8*(b^2*d*x + b^2*c)*cos(b*x +
a)^6 - 12*(b^2*d*x + b^2*c)*cos(b*x + a)^4 + 2*b^2*d*x + 2*b^2*c - (2*b*d*cos(b*x + a)^5 - 2*b*d*cos(b*x + a)^
3 - 3*b*d*cos(b*x + a))*sin(b*x + a))*sqrt(d*x + c))/b^3

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

[Out]

Timed out

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Giac [C]  time = 2.07807, size = 1481, normalized size = 4.22 \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)^(3/2)*cos(b*x+a)^3*sin(b*x+a)^3,x, algorithm="giac")

[Out]

1/9216*(4*(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*sqr
t(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) - 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)*c
- I*sqrt(3)*sqrt(pi)*(-4*I*b*c*d + d^2)*d*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^2) - I*sqrt(3)*sqrt(pi)*(-4*I*b*c*d - d^2)*d*er
f(-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^2) - 9*I*sqrt(pi)*(12*I*b*c*d - 9*d^2)*d*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^2) - 9*I*sqrt(pi)*(12*I*b*c*d + 9*
d^2)*d*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^2) + 6*I*(-4*I*(d*x + c)^(3/2)*b*d + 4*I*sqrt(d*x + c)*b*c*d + sqrt(d*x + c)*d^2)*e^((6
*I*(d*x + c)*b - 6*I*b*c + 6*I*a*d)/d)/b^2 + 18*I*(12*I*(d*x + c)^(3/2)*b*d - 12*I*sqrt(d*x + c)*b*c*d - 9*sqr
t(d*x + c)*d^2)*e^((2*I*(d*x + c)*b - 2*I*b*c + 2*I*a*d)/d)/b^2 + 18*I*(12*I*(d*x + c)^(3/2)*b*d - 12*I*sqrt(d
*x + c)*b*c*d + 9*sqrt(d*x + c)*d^2)*e^((-2*I*(d*x + c)*b + 2*I*b*c - 2*I*a*d)/d)/b^2 + 6*I*(-4*I*(d*x + c)^(3
/2)*b*d + 4*I*sqrt(d*x + c)*b*c*d - sqrt(d*x + c)*d^2)*e^((-6*I*(d*x + c)*b + 6*I*b*c - 6*I*a*d)/d)/b^2)/d