3.66 \(\int \csc (c+d x) (a+b \sin ^2(c+d x)) \, dx\)

Optimal. Leaf size=26 \[ -\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b \cos (c+d x)}{d} \]

[Out]

-((a*ArcTanh[Cos[c + d*x]])/d) - (b*Cos[c + d*x])/d

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Rubi [A]  time = 0.0248953, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3014, 3770} \[ -\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*ArcTanh[Cos[c + d*x]])/d) - (b*Cos[c + d*x])/d

Rule 3014

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Simp[(C*Cos[
e + f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 2)), x] + Dist[(A*(m + 2) + C*(m + 1))/(m + 2), Int[(b*Sin[e + f*
x])^m, x], x] /; FreeQ[{b, e, f, A, C, m}, x] &&  !LtQ[m, -1]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \csc (c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{b \cos (c+d x)}{d}+a \int \csc (c+d x) \, dx\\ &=-\frac{a \tanh ^{-1}(\cos (c+d x))}{d}-\frac{b \cos (c+d x)}{d}\\ \end{align*}

Mathematica [B]  time = 0.0319162, size = 63, normalized size = 2.42 \[ \frac{a \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{a \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}+\frac{b \sin (c) \sin (d x)}{d}-\frac{b \cos (c) \cos (d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.046, size = 35, normalized size = 1.4 \begin{align*} -{\frac{b\cos \left ( dx+c \right ) }{d}}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*cos(d*x+c)/d+1/d*a*ln(csc(d*x+c)-cot(d*x+c))

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Maxima [A]  time = 0.94894, size = 51, normalized size = 1.96 \begin{align*} -\frac{2 \, b \cos \left (d x + c\right ) + a \log \left (\cos \left (d x + c\right ) + 1\right ) - a \log \left (\cos \left (d x + c\right ) - 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*cos(d*x + c) + a*log(cos(d*x + c) + 1) - a*log(cos(d*x + c) - 1))/d

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Fricas [A]  time = 1.73244, size = 124, normalized size = 4.77 \begin{align*} -\frac{2 \, b \cos \left (d x + c\right ) + a \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) - a \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*b*cos(d*x + c) + a*log(1/2*cos(d*x + c) + 1/2) - a*log(-1/2*cos(d*x + c) + 1/2))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)*(a+b*sin(d*x+c)**2),x)

[Out]

Integral((a + b*sin(c + d*x)**2)*csc(c + d*x), x)

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Giac [B]  time = 1.15154, size = 78, normalized size = 3. \begin{align*} \frac{a \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac{4 \, b}{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(a*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) + 4*b/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1))/d