\(\int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx\) [74]

Optimal result

Integrand size = 16, antiderivative size = 182 \[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=-\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}+\frac {9 i b e^{i \left (a-\frac {b c}{d}\right )} \sqrt [3]{-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},-\frac {i b (c+d x)}{d}\right )}{8 d^2 \sqrt [3]{c+d x}}-\frac {9 i b e^{-i \left (a-\frac {b c}{d}\right )} \sqrt [3]{\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},\frac {i b (c+d x)}{d}\right )}{8 d^2 \sqrt [3]{c+d x}}+\frac {9 b \sin (a+b x)}{4 d^2 \sqrt [3]{c+d x}} \] Output:

-3/4*cos(b*x+a)/d/(d*x+c)^(4/3)+9/8*I*b*exp(I*(a-b*c/d))*(-I*b*(d*x+c)/d)^ 
(1/3)*GAMMA(2/3,-I*b*(d*x+c)/d)/d^2/(d*x+c)^(1/3)-9/8*I*b*(I*b*(d*x+c)/d)^ 
(1/3)*GAMMA(2/3,I*b*(d*x+c)/d)/d^2/exp(I*(a-b*c/d))/(d*x+c)^(1/3)+9/4*b*si 
n(b*x+a)/d^2/(d*x+c)^(1/3)
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.69 \[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=\frac {i b e^{-\frac {i (b c+a d)}{d}} \left (e^{2 i a} \sqrt [3]{-\frac {i b (c+d x)}{d}} \Gamma \left (-\frac {4}{3},-\frac {i b (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \sqrt [3]{\frac {i b (c+d x)}{d}} \Gamma \left (-\frac {4}{3},\frac {i b (c+d x)}{d}\right )\right )}{2 d^2 \sqrt [3]{c+d x}} \] Input:

Integrate[Cos[a + b*x]/(c + d*x)^(7/3),x]
 

Output:

((I/2)*b*(E^((2*I)*a)*(((-I)*b*(c + d*x))/d)^(1/3)*Gamma[-4/3, ((-I)*b*(c 
+ d*x))/d] - E^(((2*I)*b*c)/d)*((I*b*(c + d*x))/d)^(1/3)*Gamma[-4/3, (I*b* 
(c + d*x))/d]))/(d^2*E^((I*(b*c + a*d))/d)*(c + d*x)^(1/3))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 3778, 25, 3042, 3778, 3042, 3788, 26, 2612}

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.

\(\displaystyle \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{(c+d x)^{7/3}}dx\)

\(\Big \downarrow \) 3778

\(\displaystyle \frac {3 b \int -\frac {\sin (a+b x)}{(c+d x)^{4/3}}dx}{4 d}-\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 b \int \frac {\sin (a+b x)}{(c+d x)^{4/3}}dx}{4 d}-\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 b \int \frac {\sin (a+b x)}{(c+d x)^{4/3}}dx}{4 d}-\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}\)

\(\Big \downarrow \) 3778

\(\displaystyle -\frac {3 b \left (\frac {3 b \int \frac {\cos (a+b x)}{\sqrt [3]{c+d x}}dx}{d}-\frac {3 \sin (a+b x)}{d \sqrt [3]{c+d x}}\right )}{4 d}-\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 b \left (\frac {3 b \int \frac {\sin \left (a+b x+\frac {\pi }{2}\right )}{\sqrt [3]{c+d x}}dx}{d}-\frac {3 \sin (a+b x)}{d \sqrt [3]{c+d x}}\right )}{4 d}-\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}\)

\(\Big \downarrow \) 3788

\(\displaystyle -\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}-\frac {3 b \left (-\frac {3 \sin (a+b x)}{d \sqrt [3]{c+d x}}+\frac {3 b \left (\frac {1}{2} i \int -\frac {i e^{-i (a+b x)}}{\sqrt [3]{c+d x}}dx-\frac {1}{2} i \int \frac {i e^{i (a+b x)}}{\sqrt [3]{c+d x}}dx\right )}{d}\right )}{4 d}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}-\frac {3 b \left (-\frac {3 \sin (a+b x)}{d \sqrt [3]{c+d x}}+\frac {3 b \left (\frac {1}{2} \int \frac {e^{-i (a+b x)}}{\sqrt [3]{c+d x}}dx+\frac {1}{2} \int \frac {e^{i (a+b x)}}{\sqrt [3]{c+d x}}dx\right )}{d}\right )}{4 d}\)

\(\Big \downarrow \) 2612

\(\displaystyle -\frac {3 \cos (a+b x)}{4 d (c+d x)^{4/3}}-\frac {3 b \left (-\frac {3 \sin (a+b x)}{d \sqrt [3]{c+d x}}+\frac {3 b \left (\frac {i e^{-i \left (a-\frac {b c}{d}\right )} \sqrt [3]{\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},\frac {i b (c+d x)}{d}\right )}{2 b \sqrt [3]{c+d x}}-\frac {i e^{i \left (a-\frac {b c}{d}\right )} \sqrt [3]{-\frac {i b (c+d x)}{d}} \Gamma \left (\frac {2}{3},-\frac {i b (c+d x)}{d}\right )}{2 b \sqrt [3]{c+d x}}\right )}{d}\right )}{4 d}\)

Input:

Int[Cos[a + b*x]/(c + d*x)^(7/3),x]
 

Output:

(-3*Cos[a + b*x])/(4*d*(c + d*x)^(4/3)) - (3*b*((3*b*(((-1/2*I)*E^(I*(a - 
(b*c)/d))*(((-I)*b*(c + d*x))/d)^(1/3)*Gamma[2/3, ((-I)*b*(c + d*x))/d])/( 
b*(c + d*x)^(1/3)) + ((I/2)*((I*b*(c + d*x))/d)^(1/3)*Gamma[2/3, (I*b*(c + 
 d*x))/d])/(b*E^(I*(a - (b*c)/d))*(c + d*x)^(1/3))))/d - (3*Sin[a + b*x])/ 
(d*(c + d*x)^(1/3))))/(4*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2612

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] 
:> Simp[(-F^(g*(e - c*(f/d))))*((c + d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d) 
)^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m + 1, 
 ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] && 
!IntegerQ[m]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3778

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

rule 3788

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol 
] :> Simp[I/2   Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp 
[I/2   Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e 
, f, m}, x] && IntegerQ[2*k]
 

Maple [F]

\[\int \frac {\cos \left (b x +a \right )}{\left (d x +c \right )^{\frac {7}{3}}}d x\]

Input:

int(cos(b*x+a)/(d*x+c)^(7/3),x)
 

Output:

int(cos(b*x+a)/(d*x+c)^(7/3),x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.42 \[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=-\frac {3 \, {\left (3 \, {\left ({\left (i \, b d^{2} x^{2} + 2 i \, b c d x + i \, b c^{2}\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {i \, b}{d}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, \frac {i \, b d x + i \, b c}{d}\right ) + 3 \, {\left ({\left (-i \, b d^{2} x^{2} - 2 i \, b c d x - i \, b c^{2}\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (-\frac {i \, b}{d}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, \frac {-i \, b d x - i \, b c}{d}\right ) + 2 \, {\left (d x + c\right )}^{\frac {2}{3}} {\left (d \cos \left (b x + a\right ) - 3 \, {\left (b d x + b c\right )} \sin \left (b x + a\right )\right )}\right )}}{8 \, {\left (d^{4} x^{2} + 2 \, c d^{3} x + c^{2} d^{2}\right )}} \] Input:

integrate(cos(b*x+a)/(d*x+c)^(7/3),x, algorithm="fricas")
 

Output:

-3/8*(3*((I*b*d^2*x^2 + 2*I*b*c*d*x + I*b*c^2)*cos(-(b*c - a*d)/d) + (b*d^ 
2*x^2 + 2*b*c*d*x + b*c^2)*sin(-(b*c - a*d)/d))*(I*b/d)^(1/3)*gamma(2/3, ( 
I*b*d*x + I*b*c)/d) + 3*((-I*b*d^2*x^2 - 2*I*b*c*d*x - I*b*c^2)*cos(-(b*c 
- a*d)/d) + (b*d^2*x^2 + 2*b*c*d*x + b*c^2)*sin(-(b*c - a*d)/d))*(-I*b/d)^ 
(1/3)*gamma(2/3, (-I*b*d*x - I*b*c)/d) + 2*(d*x + c)^(2/3)*(d*cos(b*x + a) 
 - 3*(b*d*x + b*c)*sin(b*x + a)))/(d^4*x^2 + 2*c*d^3*x + c^2*d^2)
 

Sympy [F]

\[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=\int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{\frac {7}{3}}}\, dx \] Input:

integrate(cos(b*x+a)/(d*x+c)**(7/3),x)
 

Output:

Integral(cos(a + b*x)/(c + d*x)**(7/3), x)
 

Maxima [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.75 \[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=-\frac {{\left ({\left ({\left (i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {4}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (-i \, \sqrt {3} - 1\right )} \Gamma \left (-\frac {4}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + {\left ({\left (\sqrt {3} + i\right )} \Gamma \left (-\frac {4}{3}, \frac {i \, {\left (d x + c\right )} b}{d}\right ) + {\left (\sqrt {3} - i\right )} \Gamma \left (-\frac {4}{3}, -\frac {i \, {\left (d x + c\right )} b}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )\right )} \left (\frac {{\left (d x + c\right )} b}{d}\right )^{\frac {4}{3}}}{4 \, {\left (d x + c\right )}^{\frac {4}{3}} d} \] Input:

integrate(cos(b*x+a)/(d*x+c)^(7/3),x, algorithm="maxima")
 

Output:

-1/4*(((I*sqrt(3) - 1)*gamma(-4/3, I*(d*x + c)*b/d) + (-I*sqrt(3) - 1)*gam 
ma(-4/3, -I*(d*x + c)*b/d))*cos(-(b*c - a*d)/d) + ((sqrt(3) + I)*gamma(-4/ 
3, I*(d*x + c)*b/d) + (sqrt(3) - I)*gamma(-4/3, -I*(d*x + c)*b/d))*sin(-(b 
*c - a*d)/d))*((d*x + c)*b/d)^(4/3)/((d*x + c)^(4/3)*d)
 

Giac [F]

\[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=\int { \frac {\cos \left (b x + a\right )}{{\left (d x + c\right )}^{\frac {7}{3}}} \,d x } \] Input:

integrate(cos(b*x+a)/(d*x+c)^(7/3),x, algorithm="giac")
 

Output:

integrate(cos(b*x + a)/(d*x + c)^(7/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^{7/3}} \,d x \] Input:

int(cos(a + b*x)/(c + d*x)^(7/3),x)
 

Output:

int(cos(a + b*x)/(c + d*x)^(7/3), x)
 

Reduce [F]

\[ \int \frac {\cos (a+b x)}{(c+d x)^{7/3}} \, dx=\int \frac {\cos \left (b x +a \right )}{\left (d x +c \right )^{\frac {1}{3}} c^{2}+2 \left (d x +c \right )^{\frac {1}{3}} c d x +\left (d x +c \right )^{\frac {1}{3}} d^{2} x^{2}}d x \] Input:

int(cos(b*x+a)/(d*x+c)^(7/3),x)
                                                                                    
                                                                                    
 

Output:

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