Integrand size = 18, antiderivative size = 247 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}-\frac {128 b^{7/2} \sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{105 d^{9/2}}+\frac {128 b^{7/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{105 d^{9/2}}-\frac {8 b \cos (a+b x) \sin (a+b x)}{35 d^2 (c+d x)^{5/2}}+\frac {128 b^3 \cos (a+b x) \sin (a+b x)}{105 d^4 \sqrt {c+d x}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}+\frac {32 b^2 \sin ^2(a+b x)}{105 d^3 (c+d x)^{3/2}} \] Output:
-16/105*b^2/d^3/(d*x+c)^(3/2)-128/105*b^(7/2)*Pi^(1/2)*cos(2*a-2*b*c/d)*Fr esnelC(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))/d^(9/2)+128/105*b^(7/2)*P i^(1/2)*FresnelS(2*b^(1/2)*(d*x+c)^(1/2)/d^(1/2)/Pi^(1/2))*sin(2*a-2*b*c/d )/d^(9/2)-8/35*b*cos(b*x+a)*sin(b*x+a)/d^2/(d*x+c)^(5/2)+128/105*b^3*cos(b *x+a)*sin(b*x+a)/d^4/(d*x+c)^(1/2)-2/7*sin(b*x+a)^2/d/(d*x+c)^(7/2)+32/105 *b^2*sin(b*x+a)^2/d^3/(d*x+c)^(3/2)
Result contains complex when optimal does not.
Time = 1.42 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.12 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=\frac {-30 d^3+e^{2 i a} \left (15 d^3 e^{2 i b x}-4 i b (c+d x) \left (-3 d^2 e^{2 i b x}+4 b e^{-\frac {2 i b c}{d}} (c+d x) \left (e^{\frac {2 i b (c+d x)}{d}} (-i d+4 b (c+d x))-4 i \sqrt {2} d \left (-\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {2 i b (c+d x)}{d}\right )\right )\right )\right )+e^{-2 i (a+b x)} \left (15 d^3+4 i b (c+d x) \left (-3 d^2-2 i b (c+d x) \left (-2 d+8 i b (c+d x)-8 \sqrt {2} d e^{\frac {2 i b (c+d x)}{d}} \left (\frac {i b (c+d x)}{d}\right )^{3/2} \Gamma \left (\frac {1}{2},\frac {2 i b (c+d x)}{d}\right )\right )\right )\right )}{210 d^4 (c+d x)^{7/2}} \] Input:
Integrate[Sin[a + b*x]^2/(c + d*x)^(9/2),x]
Output:
(-30*d^3 + E^((2*I)*a)*(15*d^3*E^((2*I)*b*x) - (4*I)*b*(c + d*x)*(-3*d^2*E ^((2*I)*b*x) + (4*b*(c + d*x)*(E^(((2*I)*b*(c + d*x))/d)*((-I)*d + 4*b*(c + d*x)) - (4*I)*Sqrt[2]*d*(((-I)*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((-2*I)* b*(c + d*x))/d]))/E^(((2*I)*b*c)/d))) + (15*d^3 + (4*I)*b*(c + d*x)*(-3*d^ 2 - (2*I)*b*(c + d*x)*(-2*d + (8*I)*b*(c + d*x) - 8*Sqrt[2]*d*E^(((2*I)*b* (c + d*x))/d)*((I*b*(c + d*x))/d)^(3/2)*Gamma[1/2, ((2*I)*b*(c + d*x))/d]) ))/E^((2*I)*(a + b*x)))/(210*d^4*(c + d*x)^(7/2))
Time = 0.81 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.20, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3795, 17, 3042, 3795, 17, 3042, 3793, 2009}
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.
| \(\displaystyle \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (a+b x)^2}{(c+d x)^{9/2}}dx\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {16 b^2 \int \frac {\sin ^2(a+b x)}{(c+d x)^{5/2}}dx}{35 d^2}+\frac {8 b^2 \int \frac {1}{(c+d x)^{5/2}}dx}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {16 b^2 \int \frac {\sin ^2(a+b x)}{(c+d x)^{5/2}}dx}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {16 b^2 \int \frac {\sin (a+b x)^2}{(c+d x)^{5/2}}dx}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {16 b^2 \left (-\frac {16 b^2 \int \frac {\sin ^2(a+b x)}{\sqrt {c+d x}}dx}{3 d^2}+\frac {8 b^2 \int \frac {1}{\sqrt {c+d x}}dx}{3 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sin ^2(a+b x)}{3 d (c+d x)^{3/2}}\right )}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {16 b^2 \left (-\frac {16 b^2 \int \frac {\sin ^2(a+b x)}{\sqrt {c+d x}}dx}{3 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sin ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {16 b^2 \sqrt {c+d x}}{3 d^3}\right )}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {16 b^2 \left (-\frac {16 b^2 \int \frac {\sin (a+b x)^2}{\sqrt {c+d x}}dx}{3 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sin ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {16 b^2 \sqrt {c+d x}}{3 d^3}\right )}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle -\frac {16 b^2 \left (-\frac {16 b^2 \int \left (\frac {1}{2 \sqrt {c+d x}}-\frac {\cos (2 a+2 b x)}{2 \sqrt {c+d x}}\right )dx}{3 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sin ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {16 b^2 \sqrt {c+d x}}{3 d^3}\right )}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {16 b^2 \left (-\frac {16 b^2 \left (-\frac {\sqrt {\pi } \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {\pi } \sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {b} \sqrt {c+d x}}{\sqrt {d} \sqrt {\pi }}\right )}{2 \sqrt {b} \sqrt {d}}+\frac {\sqrt {c+d x}}{d}\right )}{3 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{3 d^2 \sqrt {c+d x}}-\frac {2 \sin ^2(a+b x)}{3 d (c+d x)^{3/2}}+\frac {16 b^2 \sqrt {c+d x}}{3 d^3}\right )}{35 d^2}-\frac {8 b \sin (a+b x) \cos (a+b x)}{35 d^2 (c+d x)^{5/2}}-\frac {2 \sin ^2(a+b x)}{7 d (c+d x)^{7/2}}-\frac {16 b^2}{105 d^3 (c+d x)^{3/2}}\) |
Input:
Int[Sin[a + b*x]^2/(c + d*x)^(9/2),x]
Output:
(-16*b^2)/(105*d^3*(c + d*x)^(3/2)) - (8*b*Cos[a + b*x]*Sin[a + b*x])/(35* d^2*(c + d*x)^(5/2)) - (2*Sin[a + b*x]^2)/(7*d*(c + d*x)^(7/2)) - (16*b^2* ((16*b^2*Sqrt[c + d*x])/(3*d^3) - (16*b^2*(Sqrt[c + d*x]/d - (Sqrt[Pi]*Cos [2*a - (2*b*c)/d]*FresnelC[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[Pi])])/ (2*Sqrt[b]*Sqrt[d]) + (Sqrt[Pi]*FresnelS[(2*Sqrt[b]*Sqrt[c + d*x])/(Sqrt[d ]*Sqrt[Pi])]*Sin[2*a - (2*b*c)/d])/(2*Sqrt[b]*Sqrt[d])))/(3*d^2) - (8*b*Co s[a + b*x]*Sin[a + b*x])/(3*d^2*Sqrt[c + d*x]) - (2*Sin[a + b*x]^2)/(3*d*( c + d*x)^(3/2))))/(35*d^2)
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[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[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 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] && LtQ[m, -2]
Time = 1.20 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{7 \left (d x +c \right )^{\frac {7}{2}}}+\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{7 \left (d x +c \right )^{\frac {7}{2}}}+\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{5 \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 b \left (-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}\right )}{7 d}}{d}\) | \(273\) |
| default | \(\frac {-\frac {1}{7 \left (d x +c \right )^{\frac {7}{2}}}+\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{7 \left (d x +c \right )^{\frac {7}{2}}}+\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{5 \left (d x +c \right )^{\frac {5}{2}}}+\frac {4 b \left (-\frac {\cos \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{3 \left (d x +c \right )^{\frac {3}{2}}}-\frac {4 b \left (-\frac {\sin \left (\frac {2 b \left (d x +c \right )}{d}+\frac {2 a d -2 b c}{d}\right )}{\sqrt {d x +c}}+\frac {2 b \sqrt {\pi }\, \left (\cos \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {FresnelC}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )-\sin \left (\frac {2 a d -2 b c}{d}\right ) \operatorname {FresnelS}\left (\frac {2 b \sqrt {d x +c}}{\sqrt {\pi }\, \sqrt {\frac {b}{d}}\, d}\right )\right )}{d \sqrt {\frac {b}{d}}}\right )}{3 d}\right )}{5 d}\right )}{7 d}}{d}\) | \(273\) |
Input:
int(sin(b*x+a)^2/(d*x+c)^(9/2),x,method=_RETURNVERBOSE)
Output:
2/d*(-1/14/(d*x+c)^(7/2)+1/14/(d*x+c)^(7/2)*cos(2*b*(d*x+c)/d+2*(a*d-b*c)/ d)+2/7*b/d*(-1/5/(d*x+c)^(5/2)*sin(2*b*(d*x+c)/d+2*(a*d-b*c)/d)+4/5*b/d*(- 1/3/(d*x+c)^(3/2)*cos(2*b*(d*x+c)/d+2*(a*d-b*c)/d)-4/3*b/d*(-1/(d*x+c)^(1/ 2)*sin(2*b*(d*x+c)/d+2*(a*d-b*c)/d)+2*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)*b*(d*x+c)^(1/2)/d)-sin(2*(a*d-b*c )/d)*FresnelS(2/Pi^(1/2)/(b/d)^(1/2)*b*(d*x+c)^(1/2)/d))))))
Leaf count of result is larger than twice the leaf count of optimal. 422 vs. \(2 (195) = 390\).
Time = 0.12 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.71 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=-\frac {2 \, {\left (64 \, {\left (\pi b^{3} d^{4} x^{4} + 4 \, \pi b^{3} c d^{3} x^{3} + 6 \, \pi b^{3} c^{2} d^{2} x^{2} + 4 \, \pi b^{3} c^{3} d x + \pi b^{3} c^{4}\right )} \sqrt {\frac {b}{\pi d}} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {C}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) - 64 \, {\left (\pi b^{3} d^{4} x^{4} + 4 \, \pi b^{3} c d^{3} x^{3} + 6 \, \pi b^{3} c^{2} d^{2} x^{2} + 4 \, \pi b^{3} c^{3} d x + \pi b^{3} c^{4}\right )} \sqrt {\frac {b}{\pi d}} \operatorname {S}\left (2 \, \sqrt {d x + c} \sqrt {\frac {b}{\pi d}}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (8 \, b^{2} d^{3} x^{2} + 16 \, b^{2} c d^{2} x + 8 \, b^{2} c^{2} d - 15 \, d^{3} - {\left (16 \, b^{2} d^{3} x^{2} + 32 \, b^{2} c d^{2} x + 16 \, b^{2} c^{2} d - 15 \, d^{3}\right )} \cos \left (b x + a\right )^{2} + 4 \, {\left (16 \, b^{3} d^{3} x^{3} + 48 \, b^{3} c d^{2} x^{2} + 16 \, b^{3} c^{3} - 3 \, b c d^{2} + 3 \, {\left (16 \, b^{3} c^{2} d - b d^{3}\right )} x\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )\right )} \sqrt {d x + c}\right )}}{105 \, {\left (d^{8} x^{4} + 4 \, c d^{7} x^{3} + 6 \, c^{2} d^{6} x^{2} + 4 \, c^{3} d^{5} x + c^{4} d^{4}\right )}} \] Input:
integrate(sin(b*x+a)^2/(d*x+c)^(9/2),x, algorithm="fricas")
Output:
-2/105*(64*(pi*b^3*d^4*x^4 + 4*pi*b^3*c*d^3*x^3 + 6*pi*b^3*c^2*d^2*x^2 + 4 *pi*b^3*c^3*d*x + pi*b^3*c^4)*sqrt(b/(pi*d))*cos(-2*(b*c - a*d)/d)*fresnel _cos(2*sqrt(d*x + c)*sqrt(b/(pi*d))) - 64*(pi*b^3*d^4*x^4 + 4*pi*b^3*c*d^3 *x^3 + 6*pi*b^3*c^2*d^2*x^2 + 4*pi*b^3*c^3*d*x + pi*b^3*c^4)*sqrt(b/(pi*d) )*fresnel_sin(2*sqrt(d*x + c)*sqrt(b/(pi*d)))*sin(-2*(b*c - a*d)/d) - (8*b ^2*d^3*x^2 + 16*b^2*c*d^2*x + 8*b^2*c^2*d - 15*d^3 - (16*b^2*d^3*x^2 + 32* b^2*c*d^2*x + 16*b^2*c^2*d - 15*d^3)*cos(b*x + a)^2 + 4*(16*b^3*d^3*x^3 + 48*b^3*c*d^2*x^2 + 16*b^3*c^3 - 3*b*c*d^2 + 3*(16*b^3*c^2*d - b*d^3)*x)*co s(b*x + a)*sin(b*x + a))*sqrt(d*x + c))/(d^8*x^4 + 4*c*d^7*x^3 + 6*c^2*d^6 *x^2 + 4*c^3*d^5*x + c^4*d^4)
Timed out. \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=\text {Timed out} \] Input:
integrate(sin(b*x+a)**2/(d*x+c)**(9/2),x)
Output:
Timed out
Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.55 \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=\frac {7 \, \sqrt {2} {\left ({\left (-\left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, -\frac {2 i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + {\left (-\left (i + 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, \frac {2 i \, {\left (d x + c\right )} b}{d}\right ) + \left (i - 1\right ) \, \sqrt {2} \Gamma \left (-\frac {7}{2}, -\frac {2 i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {7}{2}} - 1}{7 \, {\left (d x + c\right )}^{\frac {7}{2}} d} \] Input:
integrate(sin(b*x+a)^2/(d*x+c)^(9/2),x, algorithm="maxima")
Output:
1/7*(7*sqrt(2)*((-(I - 1)*sqrt(2)*gamma(-7/2, 2*I*(d*x + c)*b/d) + (I + 1) *sqrt(2)*gamma(-7/2, -2*I*(d*x + c)*b/d))*cos(-2*(b*c - a*d)/d) + (-(I + 1 )*sqrt(2)*gamma(-7/2, 2*I*(d*x + c)*b/d) + (I - 1)*sqrt(2)*gamma(-7/2, -2* I*(d*x + c)*b/d))*sin(-2*(b*c - a*d)/d))*((d*x + c)*b/d)^(7/2) - 1)/((d*x + c)^(7/2)*d)
\[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=\int { \frac {\sin \left (b x + a\right )^{2}}{{\left (d x + c\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate(sin(b*x+a)^2/(d*x+c)^(9/2),x, algorithm="giac")
Output:
integrate(sin(b*x + a)^2/(d*x + c)^(9/2), x)
Timed out. \[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=\int \frac {{\sin \left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^{9/2}} \,d x \] Input:
int(sin(a + b*x)^2/(c + d*x)^(9/2),x)
Output:
int(sin(a + b*x)^2/(c + d*x)^(9/2), x)
\[ \int \frac {\sin ^2(a+b x)}{(c+d x)^{9/2}} \, dx=\int \frac {\sin \left (b x +a \right )^{2}}{\sqrt {d x +c}\, c^{4}+4 \sqrt {d x +c}\, c^{3} d x +6 \sqrt {d x +c}\, c^{2} d^{2} x^{2}+4 \sqrt {d x +c}\, c \,d^{3} x^{3}+\sqrt {d x +c}\, d^{4} x^{4}}d x \] Input:
int(sin(b*x+a)^2/(d*x+c)^(9/2),x)
Output:
int(sin(a + b*x)**2/(sqrt(c + d*x)*c**4 + 4*sqrt(c + d*x)*c**3*d*x + 6*sqr t(c + d*x)*c**2*d**2*x**2 + 4*sqrt(c + d*x)*c*d**3*x**3 + sqrt(c + d*x)*d* *4*x**4),x)