3.19 \(\int \csc ^4(c+d x) (a+b \tan (c+d x)) \, dx\)

Optimal. Leaf size=57 \[ -\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d} \]

[Out]

-((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d) + (b*Log[Tan[c + d*x]])/d

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Rubi [A]  time = 0.0879247, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {766} \[ -\frac{a \cot ^3(c+d x)}{3 d}-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^2(c+d x)}{2 d}+\frac{b \log (\tan (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^4*(a + b*Tan[c + d*x]),x]

[Out]

-((a*Cot[c + d*x])/d) - (b*Cot[c + d*x]^2)/(2*d) - (a*Cot[c + d*x]^3)/(3*d) + (b*Log[Tan[c + d*x]])/d

Rule 766

Int[((e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(e*x
)^m*(f + g*x)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, e, f, g, m}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \csc ^4(c+d x) (a+b \tan (c+d x)) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x) \left (1+x^2\right )}{x^4} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^4}+\frac{b}{x^3}+\frac{a}{x^2}+\frac{b}{x}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{a \cot (c+d x)}{d}-\frac{b \cot ^2(c+d x)}{2 d}-\frac{a \cot ^3(c+d x)}{3 d}+\frac{b \log (\tan (c+d x))}{d}\\ \end{align*}

Mathematica [A]  time = 0.244585, size = 72, normalized size = 1.26 \[ -\frac{2 a \cot (c+d x)}{3 d}-\frac{a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac{b \left (\csc ^2(c+d x)-2 \log (\sin (c+d x))+2 \log (\cos (c+d x))\right )}{2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^4*(a + b*Tan[c + d*x]),x]

[Out]

(-2*a*Cot[c + d*x])/(3*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (b*(Csc[c + d*x]^2 + 2*Log[Cos[c + d*x]] -
2*Log[Sin[c + d*x]]))/(2*d)

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Maple [A]  time = 0.081, size = 60, normalized size = 1.1 \begin{align*} -{\frac{b}{2\,d \left ( \sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{b\ln \left ( \tan \left ( dx+c \right ) \right ) }{d}}-{\frac{2\,\cot \left ( dx+c \right ) a}{3\,d}}-{\frac{\cot \left ( dx+c \right ) a \left ( \csc \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^4*(a+b*tan(d*x+c)),x)

[Out]

-1/2/d*b/sin(d*x+c)^2+b*ln(tan(d*x+c))/d-2/3*a*cot(d*x+c)/d-1/3/d*a*cot(d*x+c)*csc(d*x+c)^2

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Maxima [A]  time = 1.12848, size = 68, normalized size = 1.19 \begin{align*} \frac{6 \, b \log \left (\tan \left (d x + c\right )\right ) - \frac{6 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right ) + 2 \, a}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*(a+b*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/6*(6*b*log(tan(d*x + c)) - (6*a*tan(d*x + c)^2 + 3*b*tan(d*x + c) + 2*a)/tan(d*x + c)^3)/d

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Fricas [B]  time = 2.79451, size = 316, normalized size = 5.54 \begin{align*} -\frac{4 \, a \cos \left (d x + c\right )^{3} + 3 \,{\left (b \cos \left (d x + c\right )^{2} - b\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) \sin \left (d x + c\right ) - 3 \,{\left (b \cos \left (d x + c\right )^{2} - b\right )} \log \left (-\frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4}\right ) \sin \left (d x + c\right ) - 6 \, a \cos \left (d x + c\right ) - 3 \, b \sin \left (d x + c\right )}{6 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*(a+b*tan(d*x+c)),x, algorithm="fricas")

[Out]

-1/6*(4*a*cos(d*x + c)^3 + 3*(b*cos(d*x + c)^2 - b)*log(cos(d*x + c)^2)*sin(d*x + c) - 3*(b*cos(d*x + c)^2 - b
)*log(-1/4*cos(d*x + c)^2 + 1/4)*sin(d*x + c) - 6*a*cos(d*x + c) - 3*b*sin(d*x + c))/((d*cos(d*x + c)^2 - d)*s
in(d*x + c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (c + d x \right )}\right ) \csc ^{4}{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**4*(a+b*tan(d*x+c)),x)

[Out]

Integral((a + b*tan(c + d*x))*csc(c + d*x)**4, x)

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Giac [A]  time = 1.36171, size = 84, normalized size = 1.47 \begin{align*} \frac{6 \, b \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac{11 \, b \tan \left (d x + c\right )^{3} + 6 \, a \tan \left (d x + c\right )^{2} + 3 \, b \tan \left (d x + c\right ) + 2 \, a}{\tan \left (d x + c\right )^{3}}}{6 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^4*(a+b*tan(d*x+c)),x, algorithm="giac")

[Out]

1/6*(6*b*log(abs(tan(d*x + c))) - (11*b*tan(d*x + c)^3 + 6*a*tan(d*x + c)^2 + 3*b*tan(d*x + c) + 2*a)/tan(d*x
+ c)^3)/d