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

Optimal. Leaf size=419 \[ -\frac{i e^{i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i b (c+d x)}{d}\right )}{16 b}+\frac{i 3^{-m-1} e^{3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i b (c+d x)}{d}\right )}{32 b}+\frac{i 5^{-m-1} e^{5 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{5 i b (c+d x)}{d}\right )}{32 b}+\frac{i e^{-i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i b (c+d x)}{d}\right )}{16 b}-\frac{i 3^{-m-1} e^{-3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 i b (c+d x)}{d}\right )}{32 b}-\frac{i 5^{-m-1} e^{-5 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{5 i b (c+d x)}{d}\right )}{32 b} \]

[Out]

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

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Rubi [A]  time = 0.434733, antiderivative size = 419, normalized size of antiderivative = 1., 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, 3307, 2181} \[ -\frac{i e^{i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i b (c+d x)}{d}\right )}{16 b}+\frac{i 3^{-m-1} e^{3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 i b (c+d x)}{d}\right )}{32 b}+\frac{i 5^{-m-1} e^{5 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{5 i b (c+d x)}{d}\right )}{32 b}+\frac{i e^{-i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i b (c+d x)}{d}\right )}{16 b}-\frac{i 3^{-m-1} e^{-3 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 i b (c+d x)}{d}\right )}{32 b}-\frac{i 5^{-m-1} e^{-5 i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{5 i b (c+d x)}{d}\right )}{32 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 3307

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

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]

Rubi steps

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

Mathematica [A]  time = 0.605986, size = 409, normalized size = 0.98 \[ -\frac{i 3^{-m-1} e^{-\frac{3 i (a d+b c)}{d}} (c+d x)^m \left (\frac{b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{\frac{6 i b c}{d}} \left (-\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{3 i b (c+d x)}{d}\right )-e^{6 i a} \left (\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{3 i b (c+d x)}{d}\right )\right )}{32 b}-\frac{i 5^{-m-1} e^{-\frac{5 i (a d+b c)}{d}} (c+d x)^m \left (\frac{b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{\frac{10 i b c}{d}} \left (-\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{5 i b (c+d x)}{d}\right )-e^{10 i a} \left (\frac{i b (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{5 i b (c+d x)}{d}\right )\right )}{32 b}-\frac{i e^{-\frac{i (a d+b c)}{d}} (c+d x)^m \left (e^{2 i a} \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i b (c+d x)}{d}\right )-e^{\frac{2 i b c}{d}} \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i b (c+d x)}{d}\right )\right )}{16 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-I/16)*(c + d*x)^m*((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))/(b*E^((I*(b*c + a*d))/d)) - ((I/32)*3^(-1 - m)
*(c + d*x)^m*(-(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]))/(b*E^(((3*I)*(b*c + a*d))/d)*((b^2*(c + d*x)^2)/d
^2)^m) - ((I/32)*5^(-1 - m)*(c + d*x)^m*(-(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]))/(b*E^(((5*I)*(b*c +
a*d))/d)*((b^2*(c + d*x)^2)/d^2)^m)

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Maple [F]  time = 0.331, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( \cos \left ( bx+a \right ) \right ) ^{3} \left ( \sin \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.577308, size = 726, normalized size = 1.73 \begin{align*} \frac{-3 i \, 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 i \, 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 i \, 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 i \, 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 i \, 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 i \, 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(-3*I*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*I*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*I*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*I*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*I*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*I*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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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)**m*cos(b*x+a)**3*sin(b*x+a)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \cos \left (b x + a\right )^{3} \sin \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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