3.259 \(\int (c+d x)^m \sin (a+b x) \tan ^2(a+b x) \, dx\)

Optimal. Leaf size=149 \[ \text{CannotIntegrate}\left (\tan (a+b x) \sec (a+b x) (c+d x)^m,x\right )+\frac{e^{i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{i b (c+d x)}{d}\right )}{2 b}+\frac{e^{-i \left (a-\frac{b c}{d}\right )} (c+d x)^m \left (\frac{i b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{i b (c+d x)}{d}\right )}{2 b} \]

[Out]

CannotIntegrate[(c + d*x)^m*Sec[a + b*x]*Tan[a + b*x], x] + (E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I
)*b*(c + d*x))/d])/(2*b*(((-I)*b*(c + d*x))/d)^m) + ((c + d*x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(2*b*E^(I*(a
- (b*c)/d))*((I*b*(c + d*x))/d)^m)

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Rubi [A]  time = 0.157146, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (c+d x)^m \sin (a+b x) \tan ^2(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(E^(I*(a - (b*c)/d))*(c + d*x)^m*Gamma[1 + m, ((-I)*b*(c + d*x))/d])/(2*b*(((-I)*b*(c + d*x))/d)^m) + ((c + d*
x)^m*Gamma[1 + m, (I*b*(c + d*x))/d])/(2*b*E^(I*(a - (b*c)/d))*((I*b*(c + d*x))/d)^m) + Defer[Int][(c + d*x)^m
*Sec[a + b*x]*Tan[a + b*x], x]

Rubi steps

\begin{align*} \int (c+d x)^m \sin (a+b x) \tan ^2(a+b x) \, dx &=-\int (c+d x)^m \sin (a+b x) \, dx+\int (c+d x)^m \sec (a+b x) \tan (a+b x) \, dx\\ &=-\left (\frac{1}{2} i \int e^{-i (a+b x)} (c+d x)^m \, dx\right )+\frac{1}{2} i \int e^{i (a+b x)} (c+d x)^m \, dx+\int (c+d x)^m \sec (a+b x) \tan (a+b x) \, 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 )}{2 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 )}{2 b}+\int (c+d x)^m \sec (a+b x) \tan (a+b x) \, dx\\ \end{align*}

Mathematica [A]  time = 5.09666, size = 0, normalized size = 0. \[ \int (c+d x)^m \sin (a+b x) \tan ^2(a+b x) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.17, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m}\sin \left ( bx+a \right ) \left ( \tan \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*sin(b*x+a)*tan(b*x+a)^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \sin \left (b x + a\right ) \tan \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*sin(b*x+a)*tan(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (d x + c\right )}^{m} \sin \left (b x + a\right ) \tan \left (b x + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((d*x + c)^m*sin(b*x + a)*tan(b*x + a)^2, x)

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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*sin(b*x+a)*tan(b*x+a)**2,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \sin \left (b x + a\right ) \tan \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*sin(b*x+a)*tan(b*x+a)^2,x, algorithm="giac")

[Out]

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