3.88 \(\int (c+d x)^m \cos ^2(a+b x) \sin ^3(a+b x) \, dx\)

Optimal. Leaf size=407 \[ -\frac {e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i b (c+d x)}{d}\right )}{16 b}-\frac {3^{-m-1} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 i b (c+d x)}{d}\right )}{32 b}+\frac {5^{-m-1} e^{5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {5 i b (c+d x)}{d}\right )}{32 b}-\frac {e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i b (c+d x)}{d}\right )}{16 b}-\frac {3^{-m-1} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 i b (c+d x)}{d}\right )}{32 b}+\frac {5^{-m-1} e^{-5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {5 i b (c+d x)}{d}\right )}{32 b} \]

[Out]

-1/16*exp(I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-I*b*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)-1/16*(d*x+c)^m*GAMMA(1+m,I*b
*(d*x+c)/d)/b/exp(I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)-1/32*3^(-1-m)*exp(3*I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-3*I*b
*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)-1/32*3^(-1-m)*(d*x+c)^m*GAMMA(1+m,3*I*b*(d*x+c)/d)/b/exp(3*I*(a-b*c/d))/((I
*b*(d*x+c)/d)^m)+1/32*5^(-1-m)*exp(5*I*(a-b*c/d))*(d*x+c)^m*GAMMA(1+m,-5*I*b*(d*x+c)/d)/b/((-I*b*(d*x+c)/d)^m)
+1/32*5^(-1-m)*(d*x+c)^m*GAMMA(1+m,5*I*b*(d*x+c)/d)/b/exp(5*I*(a-b*c/d))/((I*b*(d*x+c)/d)^m)

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Rubi [A]  time = 0.40, antiderivative size = 407, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4406, 3308, 2181} \[ -\frac {e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {i b (c+d x)}{d}\right )}{16 b}-\frac {3^{-m-1} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {3 i b (c+d x)}{d}\right )}{32 b}+\frac {5^{-m-1} e^{5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {5 i b (c+d x)}{d}\right )}{32 b}-\frac {e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {i b (c+d x)}{d}\right )}{16 b}-\frac {3^{-m-1} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {3 i b (c+d x)}{d}\right )}{32 b}+\frac {5^{-m-1} e^{-5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {5 i b (c+d x)}{d}\right )}{32 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(16*b*(((-I)*b*(c + d*x))/d)^m) - ((c +
d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(16*b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) - (3^(-1 - m)*E^((3*I
)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d])/(32*b*(((-I)*b*(c + d*x))/d)^m) - (3^(-1 -
m)*(c + d*x)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d])/(32*b*E^((3*I)*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + (5^
(-1 - m)*E^((5*I)*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-5*I)*b*(c + d*x))/d])/(32*b*(((-I)*b*(c + d*x))/d
)^m) + (5^(-1 - m)*(c + d*x)^m*Gamma[1 + m, ((5*I)*b*(c + d*x))/d])/(32*b*E^((5*I)*(a - (b*c)/d))*((I*b*(c + d
*x))/d)^m)

Rule 2181

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

Rule 3308

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

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]

Rubi steps

\begin {align*} \int (c+d x)^m \cos ^2(a+b x) \sin ^3(a+b x) \, dx &=\int \left (\frac {1}{8} (c+d x)^m \sin (a+b x)+\frac {1}{16} (c+d x)^m \sin (3 a+3 b x)-\frac {1}{16} (c+d x)^m \sin (5 a+5 b x)\right ) \, dx\\ &=\frac {1}{16} \int (c+d x)^m \sin (3 a+3 b x) \, dx-\frac {1}{16} \int (c+d x)^m \sin (5 a+5 b x) \, dx+\frac {1}{8} \int (c+d x)^m \sin (a+b x) \, dx\\ &=\frac {1}{32} i \int e^{-i (3 a+3 b x)} (c+d x)^m \, dx-\frac {1}{32} i \int e^{i (3 a+3 b x)} (c+d x)^m \, dx-\frac {1}{32} i \int e^{-i (5 a+5 b x)} (c+d x)^m \, dx+\frac {1}{32} i \int e^{i (5 a+5 b x)} (c+d x)^m \, dx+\frac {1}{16} i \int e^{-i (a+b x)} (c+d x)^m \, dx-\frac {1}{16} i \int e^{i (a+b x)} (c+d x)^m \, dx\\ &=-\frac {e^{i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {i b (c+d x)}{d}\right )}{16 b}-\frac {e^{-i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {i b (c+d x)}{d}\right )}{16 b}-\frac {3^{-1-m} e^{3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 i b (c+d x)}{d}\right )}{32 b}-\frac {3^{-1-m} e^{-3 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 i b (c+d x)}{d}\right )}{32 b}+\frac {5^{-1-m} e^{5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {5 i b (c+d x)}{d}\right )}{32 b}+\frac {5^{-1-m} e^{-5 i \left (a-\frac {b c}{d}\right )} (c+d x)^m \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {5 i b (c+d x)}{d}\right )}{32 b}\\ \end {align*}

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Mathematica [A]  time = 0.63, size = 376, normalized size = 0.92 \[ \frac {e^{-\frac {5 i (a d+b c)}{d}} (c+d x)^m \left (-5\ 3^{-m} e^{\frac {2 i (a d+b c)}{d}} \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{6 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (m+1,-\frac {3 i b (c+d x)}{d}\right )+e^{\frac {6 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {3 i b (c+d x)}{d}\right )\right )+3\ 5^{-m} \left (\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{10 i a} \left (\frac {i b (c+d x)}{d}\right )^m \Gamma \left (m+1,-\frac {5 i b (c+d x)}{d}\right )+e^{\frac {10 i b c}{d}} \left (-\frac {i b (c+d x)}{d}\right )^m \Gamma \left (m+1,\frac {5 i b (c+d x)}{d}\right )\right )+30 e^{\frac {4 i (a d+b c)}{d}} \left (-e^{2 i a} \left (-\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {i b (c+d x)}{d}\right )-e^{\frac {2 i b c}{d}} \left (\frac {i b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {i b (c+d x)}{d}\right )\right )\right )}{480 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c + d*x)^m*(30*E^(((4*I)*(b*c + a*d))/d)*(-((E^((2*I)*a)*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(((-I)*b*(c + d
*x))/d)^m) - (E^(((2*I)*b*c)/d)*Gamma[1 + m, (I*b*(c + d*x))/d])/((I*b*(c + d*x))/d)^m) - (5*E^(((2*I)*(b*c +
a*d))/d)*(E^((6*I)*a)*((I*b*(c + d*x))/d)^m*Gamma[1 + m, ((-3*I)*b*(c + d*x))/d] + E^(((6*I)*b*c)/d)*(((-I)*b*
(c + d*x))/d)^m*Gamma[1 + m, ((3*I)*b*(c + d*x))/d]))/(3^m*((b^2*(c + d*x)^2)/d^2)^m) + (3*(E^((10*I)*a)*((I*b
*(c + d*x))/d)^m*Gamma[1 + m, ((-5*I)*b*(c + d*x))/d] + E^(((10*I)*b*c)/d)*(((-I)*b*(c + d*x))/d)^m*Gamma[1 +
m, ((5*I)*b*(c + d*x))/d]))/(5^m*((b^2*(c + d*x)^2)/d^2)^m)))/(480*b*E^(((5*I)*(b*c + a*d))/d))

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fricas [A]  time = 0.72, size = 276, normalized size = 0.68 \[ \frac {3 \, e^{\left (-\frac {d m \log \left (\frac {5 i \, b}{d}\right ) - 5 i \, b c + 5 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {5 i \, b d x + 5 i \, b c}{d}\right ) - 5 \, e^{\left (-\frac {d m \log \left (\frac {3 i \, b}{d}\right ) - 3 i \, b c + 3 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {3 i \, b d x + 3 i \, b c}{d}\right ) - 30 \, e^{\left (-\frac {d m \log \left (\frac {i \, b}{d}\right ) - i \, b c + i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {i \, b d x + i \, b c}{d}\right ) - 30 \, e^{\left (-\frac {d m \log \left (-\frac {i \, b}{d}\right ) + i \, b c - i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-i \, b d x - i \, b c}{d}\right ) - 5 \, e^{\left (-\frac {d m \log \left (-\frac {3 i \, b}{d}\right ) + 3 i \, b c - 3 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-3 i \, b d x - 3 i \, b c}{d}\right ) + 3 \, e^{\left (-\frac {d m \log \left (-\frac {5 i \, b}{d}\right ) + 5 i \, b c - 5 i \, a d}{d}\right )} \Gamma \left (m + 1, \frac {-5 i \, b d x - 5 i \, b c}{d}\right )}{480 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(3*e^(-(d*m*log(5*I*b/d) - 5*I*b*c + 5*I*a*d)/d)*gamma(m + 1, (5*I*b*d*x + 5*I*b*c)/d) - 5*e^(-(d*m*log(
3*I*b/d) - 3*I*b*c + 3*I*a*d)/d)*gamma(m + 1, (3*I*b*d*x + 3*I*b*c)/d) - 30*e^(-(d*m*log(I*b/d) - I*b*c + I*a*
d)/d)*gamma(m + 1, (I*b*d*x + I*b*c)/d) - 30*e^(-(d*m*log(-I*b/d) + I*b*c - I*a*d)/d)*gamma(m + 1, (-I*b*d*x -
I*b*c)/d) - 5*e^(-(d*m*log(-3*I*b/d) + 3*I*b*c - 3*I*a*d)/d)*gamma(m + 1, (-3*I*b*d*x - 3*I*b*c)/d) + 3*e^(-(
d*m*log(-5*I*b/d) + 5*I*b*c - 5*I*a*d)/d)*gamma(m + 1, (-5*I*b*d*x - 5*I*b*c)/d))/b

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^m*cos(b*x + a)^2*sin(b*x + a)^3, x)

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maple [F]  time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (\cos ^{2}\left (b x +a \right )\right ) \left (\sin ^{3}\left (b x +a \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{2} \sin \left (b x + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((d*x + c)^m*cos(b*x + a)^2*sin(b*x + a)^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\cos \left (a+b\,x\right )}^2\,{\sin \left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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