Integrand size = 18, antiderivative size = 354 \[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx=-\frac {2 (c+d x)^{3/2} \cos (a+b x)}{3 b}-\frac {9 d^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{8 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{24 b^{5/2}}+\frac {d^{3/2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (3 a-\frac {3 b c}{d}\right )}{24 b^{5/2}}-\frac {9 d^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right ) \sin \left (a-\frac {b c}{d}\right )}{8 b^{5/2}}+\frac {d \sqrt {c+d x} \sin (a+b x)}{b^2}-\frac {(c+d x)^{3/2} \cos (a+b x) \sin ^2(a+b x)}{3 b}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2} \] Output:
-2/3*(d*x+c)^(3/2)*cos(b*x+a)/b-9/16*d^(3/2)*2^(1/2)*Pi^(1/2)*cos(a-b*c/d) *FresnelS(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))/b^(5/2)+1/144*d^ (3/2)*6^(1/2)*Pi^(1/2)*cos(3*a-3*b*c/d)*FresnelS(b^(1/2)*6^(1/2)/Pi^(1/2)* (d*x+c)^(1/2)/d^(1/2))/b^(5/2)+1/144*d^(3/2)*6^(1/2)*Pi^(1/2)*FresnelC(b^( 1/2)*6^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/d^(1/2))*sin(3*a-3*b*c/d)/b^(5/2)-9/16 *d^(3/2)*2^(1/2)*Pi^(1/2)*FresnelC(b^(1/2)*2^(1/2)/Pi^(1/2)*(d*x+c)^(1/2)/ d^(1/2))*sin(a-b*c/d)/b^(5/2)+d*(d*x+c)^(1/2)*sin(b*x+a)/b^2-1/3*(d*x+c)^( 3/2)*cos(b*x+a)*sin(b*x+a)^2/b+1/6*d*(d*x+c)^(1/2)*sin(b*x+a)^3/b^2
Result contains complex when optimal does not.
Time = 1.46 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.72 \[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx=\frac {i e^{-\frac {3 i (b c+a d)}{d}} (c+d x)^{5/2} \left (-81 e^{2 i \left (2 a+\frac {b c}{d}\right )} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {i b (c+d x)}{d}\right )+81 e^{2 i a+\frac {4 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {i b (c+d x)}{d}\right )+\sqrt {3} \left (e^{6 i a} \sqrt {\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},-\frac {3 i b (c+d x)}{d}\right )-e^{\frac {6 i b c}{d}} \sqrt {-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {5}{2},\frac {3 i b (c+d x)}{d}\right )\right )\right )}{216 d \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{3/2}} \] Input:
Integrate[(c + d*x)^(3/2)*Sin[a + b*x]^3,x]
Output:
((I/216)*(c + d*x)^(5/2)*(-81*E^((2*I)*(2*a + (b*c)/d))*Sqrt[(I*b*(c + d*x ))/d]*Gamma[5/2, ((-I)*b*(c + d*x))/d] + 81*E^((2*I)*a + ((4*I)*b*c)/d)*Sq rt[((-I)*b*(c + d*x))/d]*Gamma[5/2, (I*b*(c + d*x))/d] + Sqrt[3]*(E^((6*I) *a)*Sqrt[(I*b*(c + d*x))/d]*Gamma[5/2, ((-3*I)*b*(c + d*x))/d] - E^(((6*I) *b*c)/d)*Sqrt[((-I)*b*(c + d*x))/d]*Gamma[5/2, ((3*I)*b*(c + d*x))/d])))/( d*E^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d^2)^(3/2))
Time = 2.17 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.42, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {3042, 3792, 3042, 3777, 3042, 3777, 25, 3042, 3787, 3042, 3785, 3786, 3793, 2009, 3832, 3833}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (c+d x)^{3/2} \sin (a+b x)^3dx\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin ^3(a+b x)}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \int (c+d x)^{3/2} \sin (a+b x)dx+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \int (c+d x)^{3/2} \sin (a+b x)dx+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \int \sqrt {c+d x} \cos (a+b x)dx}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \int \sqrt {c+d x} \sin \left (a+b x+\frac {\pi }{2}\right )dx}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3777 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {d \int -\frac {\sin (a+b x)}{\sqrt {c+d x}}dx}{2 b}+\frac {\sqrt {c+d x} \sin (a+b x)}{b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \int \frac {\sin (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \int \frac {\sin (a+b x)}{\sqrt {c+d x}}dx}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3787 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\sin \left (a-\frac {b c}{d}\right ) \int \frac {\cos \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\sin \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x+\frac {\pi }{2}\right )}{\sqrt {c+d x}}dx+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3785 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}+\cos \left (a-\frac {b c}{d}\right ) \int \frac {\sin \left (\frac {b c}{d}+b x\right )}{\sqrt {c+d x}}dx\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3786 |
| \(\displaystyle -\frac {d^2 \int \frac {\sin (a+b x)^3}{\sqrt {c+d x}}dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}+\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {d^2 \int \left (\frac {3 \sin (a+b x)}{4 \sqrt {c+d x}}-\frac {\sin (3 a+3 b x)}{4 \sqrt {c+d x}}\right )dx}{12 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}+\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}+\frac {2 \cos \left (a-\frac {b c}{d}\right ) \int \sin \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )-\frac {d^2 \left (-\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{12 b^2}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3832 |
| \(\displaystyle \frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {2 \sin \left (a-\frac {b c}{d}\right ) \int \cos \left (\frac {b (c+d x)}{d}\right )d\sqrt {c+d x}}{d}+\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )-\frac {d^2 \left (-\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{12 b^2}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
| \(\Big \downarrow \) 3833 |
| \(\displaystyle -\frac {d^2 \left (-\frac {\sqrt {\frac {\pi }{6}} \sin \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\frac {\pi }{2}} \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {3 \sqrt {\frac {\pi }{2}} \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}-\frac {\sqrt {\frac {\pi }{6}} \cos \left (3 a-\frac {3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {6}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{2 \sqrt {b} \sqrt {d}}\right )}{12 b^2}+\frac {d \sqrt {c+d x} \sin ^3(a+b x)}{6 b^2}+\frac {2}{3} \left (\frac {3 d \left (\frac {\sqrt {c+d x} \sin (a+b x)}{b}-\frac {d \left (\frac {\sqrt {2 \pi } \sin \left (a-\frac {b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}+\frac {\sqrt {2 \pi } \cos \left (a-\frac {b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {b} \sqrt {\frac {2}{\pi }} \sqrt {c+d x}}{\sqrt {d}}\right )}{\sqrt {b} \sqrt {d}}\right )}{2 b}\right )}{2 b}-\frac {(c+d x)^{3/2} \cos (a+b x)}{b}\right )-\frac {(c+d x)^{3/2} \sin ^2(a+b x) \cos (a+b x)}{3 b}\) |
Input:
Int[(c + d*x)^(3/2)*Sin[a + b*x]^3,x]
Output:
-1/12*(d^2*((3*Sqrt[Pi/2]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sq rt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[d]) - (Sqrt[Pi/6]*Cos[3*a - (3*b*c) /d]*FresnelS[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]])/(2*Sqrt[b]*Sqrt[ d]) - (Sqrt[Pi/6]*FresnelC[(Sqrt[b]*Sqrt[6/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin [3*a - (3*b*c)/d])/(2*Sqrt[b]*Sqrt[d]) + (3*Sqrt[Pi/2]*FresnelC[(Sqrt[b]*S qrt[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(2*Sqrt[b]*Sqrt[d])))/ b^2 - ((c + d*x)^(3/2)*Cos[a + b*x]*Sin[a + b*x]^2)/(3*b) + (d*Sqrt[c + d* x]*Sin[a + b*x]^3)/(6*b^2) + (2*(-(((c + d*x)^(3/2)*Cos[a + b*x])/b) + (3* d*(-1/2*(d*((Sqrt[2*Pi]*Cos[a - (b*c)/d]*FresnelS[(Sqrt[b]*Sqrt[2/Pi]*Sqrt [c + d*x])/Sqrt[d]])/(Sqrt[b]*Sqrt[d]) + (Sqrt[2*Pi]*FresnelC[(Sqrt[b]*Sqr t[2/Pi]*Sqrt[c + d*x])/Sqrt[d]]*Sin[a - (b*c)/d])/(Sqrt[b]*Sqrt[d])))/b + (Sqrt[c + d*x]*Sin[a + b*x])/b))/(2*b)))/3
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[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]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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]
Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos [(d*e - c*f)/d] Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
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]
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]
Time = 1.50 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{4 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{12 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{4 b}}{d}\) | \(384\) |
| default | \(\frac {-\frac {3 d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{4 b}+\frac {9 d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {b \left (d x +c \right )}{d}+\frac {a d -b c}{d}\right )}{2 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \left (\cos \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {a d -b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{4 b \sqrt {\frac {b}{d}}}\right )}{4 b}+\frac {d \left (d x +c \right )^{\frac {3}{2}} \cos \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{12 b}-\frac {d \left (\frac {d \sqrt {d x +c}\, \sin \left (\frac {3 b \left (d x +c \right )}{d}+\frac {3 a d -3 b c}{d}\right )}{6 b}-\frac {d \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \left (\cos \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )+\sin \left (\frac {3 a d -3 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{36 b \sqrt {\frac {b}{d}}}\right )}{4 b}}{d}\) | \(384\) |
Input:
int((d*x+c)^(3/2)*sin(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
2/d*(-3/8/b*d*(d*x+c)^(3/2)*cos(b*(d*x+c)/d+(a*d-b*c)/d)+9/8/b*d*(1/2/b*d* (d*x+c)^(1/2)*sin(b*(d*x+c)/d+(a*d-b*c)/d)-1/4/b*d*2^(1/2)*Pi^(1/2)/(b/d)^ (1/2)*(cos((a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1 /2)/d)+sin((a*d-b*c)/d)*FresnelC(2^(1/2)/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1 /2)/d)))+1/24/b*d*(d*x+c)^(3/2)*cos(3*b*(d*x+c)/d+3*(a*d-b*c)/d)-1/8/b*d*( 1/6/b*d*(d*x+c)^(1/2)*sin(3*b*(d*x+c)/d+3*(a*d-b*c)/d)-1/36/b*d*2^(1/2)*Pi ^(1/2)*3^(1/2)/(b/d)^(1/2)*(cos(3*(a*d-b*c)/d)*FresnelS(2^(1/2)/Pi^(1/2)*3 ^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d)+sin(3*(a*d-b*c)/d)*FresnelC(2^(1/2)/ Pi^(1/2)*3^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))
Time = 0.10 (sec) , antiderivative size = 300, normalized size of antiderivative = 0.85 \[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx=\frac {\sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {S}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 81 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {S}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 81 \, \sqrt {2} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {2} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + \sqrt {6} \pi d^{2} \sqrt {\frac {b}{\pi d}} \operatorname {C}\left (\sqrt {6} \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {3 \, {\left (b c - a d\right )}}{d}\right ) + 24 \, {\left (2 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right )^{3} - 6 \, {\left (b^{2} d x + b^{2} c\right )} \cos \left (b x + a\right ) - {\left (b d \cos \left (b x + a\right )^{2} - 7 \, b d\right )} \sin \left (b x + a\right )\right )} \sqrt {d x + c}}{144 \, b^{3}} \] Input:
integrate((d*x+c)^(3/2)*sin(b*x+a)^3,x, algorithm="fricas")
Output:
1/144*(sqrt(6)*pi*d^2*sqrt(b/(pi*d))*cos(-3*(b*c - a*d)/d)*fresnel_sin(sqr t(6)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 81*sqrt(2)*pi*d^2*sqrt(b/(pi*d))*cos( -(b*c - a*d)/d)*fresnel_sin(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d))) - 81*sqr t(2)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(2)*sqrt(d*x + c)*sqrt(b/(pi*d) ))*sin(-(b*c - a*d)/d) + sqrt(6)*pi*d^2*sqrt(b/(pi*d))*fresnel_cos(sqrt(6) *sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-3*(b*c - a*d)/d) + 24*(2*(b^2*d*x + b^ 2*c)*cos(b*x + a)^3 - 6*(b^2*d*x + b^2*c)*cos(b*x + a) - (b*d*cos(b*x + a) ^2 - 7*b*d)*sin(b*x + a))*sqrt(d*x + c))/b^3
\[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx=\int \left (c + d x\right )^{\frac {3}{2}} \sin ^{3}{\left (a + b x \right )}\, dx \] Input:
integrate((d*x+c)**(3/2)*sin(b*x+a)**3,x)
Output:
Integral((c + d*x)**(3/2)*sin(a + b*x)**3, x)
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.41 \[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx =\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)*sin(b*x+a)^3,x, algorithm="maxima")
Output:
1/576*(48*(d*x + c)^(3/2)*b^3*cos(3*((d*x + c)*b - b*c + a*d)/d)/d - 432*( d*x + c)^(3/2)*b^3*cos(((d*x + c)*b - b*c + a*d)/d)/d - 24*sqrt(d*x + c)*b ^2*sin(3*((d*x + c)*b - b*c + a*d)/d) + 648*sqrt(d*x + c)*b^2*sin(((d*x + c)*b - b*c + a*d)/d) - (-(I + 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1 /4)*cos(-3*(b*c - a*d)/d) + (I - 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2) ^(1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(3*I*b/d)) + 81*(-(I + 1)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) + (I - 1)*sqr t(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d))*erf(sqrt(d*x + c)*s qrt(I*b/d)) + 81*((I - 1)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*cos(-(b*c - a*d)/d) - (I + 1)*sqrt(2)*sqrt(pi)*b*d*(b^2/d^2)^(1/4)*sin(-(b*c - a*d)/d ))*erf(sqrt(d*x + c)*sqrt(-I*b/d)) - ((I - 1)*9^(1/4)*sqrt(2)*sqrt(pi)*b*d *(b^2/d^2)^(1/4)*cos(-3*(b*c - a*d)/d) - (I + 1)*9^(1/4)*sqrt(2)*sqrt(pi)* b*d*(b^2/d^2)^(1/4)*sin(-3*(b*c - a*d)/d))*erf(sqrt(d*x + c)*sqrt(-3*I*b/d )))*d/b^4
Result contains complex when optimal does not.
Time = 0.68 (sec) , antiderivative size = 1521, normalized size of antiderivative = 4.30 \[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx=\text {Too large to display} \] Input:
integrate((d*x+c)^(3/2)*sin(b*x+a)^3,x, algorithm="giac")
Output:
1/288*(12*(9*sqrt(2)*sqrt(pi)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)* (-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqr t(b^2*d^2) + 1)) - sqrt(6)*sqrt(pi)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d* x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I *b*d/sqrt(b^2*d^2) + 1)) + 9*sqrt(2)*sqrt(pi)*d*erf(-1/2*I*sqrt(2)*sqrt(b* d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-I*b*c + I*a*d)/d)/(sqrt (b*d)*(I*b*d/sqrt(b^2*d^2) + 1)) - sqrt(6)*sqrt(pi)*d*erf(1/2*I*sqrt(6)*sq rt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(-I*b*c + I*a*d) /d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)))*c^2 - 4*(27*sqrt(2)*sqrt(pi)*( 2*b*c + I*d)*d*erf(1/2*I*sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2* d^2) + 1)/d)*e^((I*b*c - I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)*b ) - sqrt(6)*sqrt(pi)*(6*b*c - I*d)*d*erf(-1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^(-3*(I*b*c - I*a*d)/d)/(sqrt(b*d)*(I* b*d/sqrt(b^2*d^2) + 1)*b) + 27*sqrt(2)*sqrt(pi)*(2*b*c - I*d)*d*erf(-1/2*I *sqrt(2)*sqrt(b*d)*sqrt(d*x + c)*(I*b*d/sqrt(b^2*d^2) + 1)/d)*e^((-I*b*c + I*a*d)/d)/(sqrt(b*d)*(I*b*d/sqrt(b^2*d^2) + 1)*b) - sqrt(6)*sqrt(pi)*(6*b *c + I*d)*d*erf(1/2*I*sqrt(6)*sqrt(b*d)*sqrt(d*x + c)*(-I*b*d/sqrt(b^2*d^2 ) + 1)/d)*e^(-3*(-I*b*c + I*a*d)/d)/(sqrt(b*d)*(-I*b*d/sqrt(b^2*d^2) + 1)* b) + 54*sqrt(d*x + c)*d*e^((I*(d*x + c)*b - I*b*c + I*a*d)/d)/b - 6*sqrt(d *x + c)*d*e^(-3*(I*(d*x + c)*b - I*b*c + I*a*d)/d)/b + 54*sqrt(d*x + c)...
Timed out. \[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx=\int {\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^{3/2} \,d x \] Input:
int(sin(a + b*x)^3*(c + d*x)^(3/2),x)
Output:
int(sin(a + b*x)^3*(c + d*x)^(3/2), x)
\[ \int (c+d x)^{3/2} \sin ^3(a+b x) \, dx=\left (\int \sqrt {d x +c}\, \sin \left (b x +a \right )^{3} x d x \right ) d +\left (\int \sqrt {d x +c}\, \sin \left (b x +a \right )^{3}d x \right ) c \] Input:
int((d*x+c)^(3/2)*sin(b*x+a)^3,x)
Output:
int(sqrt(c + d*x)*sin(a + b*x)**3*x,x)*d + int(sqrt(c + d*x)*sin(a + b*x)* *3,x)*c