3.67 \(\int \csc ^3(c+d x) (a+b \sin ^2(c+d x)) \, dx\)

Optimal. Leaf size=40 \[ -\frac{(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d} \]

[Out]

-((a + 2*b)*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)

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Rubi [A]  time = 0.0308706, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3012, 3770} \[ -\frac{(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a + 2*b)*ArcTanh[Cos[c + d*x]])/(2*d) - (a*Cot[c + d*x]*Csc[c + d*x])/(2*d)

Rule 3012

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*Cos[e
+ f*x]*(b*Sin[e + f*x])^(m + 1))/(b*f*(m + 1)), x] + Dist[(A*(m + 2) + C*(m + 1))/(b^2*(m + 1)), Int[(b*Sin[e
+ f*x])^(m + 2), x], x] /; FreeQ[{b, e, f, A, C}, 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 ^3(c+d x) \left (a+b \sin ^2(c+d x)\right ) \, dx &=-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}+\frac{1}{2} (a+2 b) \int \csc (c+d x) \, dx\\ &=-\frac{(a+2 b) \tanh ^{-1}(\cos (c+d x))}{2 d}-\frac{a \cot (c+d x) \csc (c+d x)}{2 d}\\ \end{align*}

Mathematica [B]  time = 0.0373156, size = 118, normalized size = 2.95 \[ -\frac{a \csc ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a \sec ^2\left (\frac{1}{2} (c+d x)\right )}{8 d}+\frac{a \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}-\frac{a \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 d}+\frac{b \log \left (\sin \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d}-\frac{b \log \left (\cos \left (\frac{c}{2}+\frac{d x}{2}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*Csc[(c + d*x)/2]^2)/(8*d) - (b*Log[Cos[c/2 + (d*x)/2]])/d - (a*Log[Cos[(c + d*x)/2]])/(2*d) + (b*Log[Sin[c
/2 + (d*x)/2]])/d + (a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d)

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Maple [A]  time = 0.055, size = 63, normalized size = 1.6 \begin{align*} -{\frac{\cot \left ( dx+c \right ) a\csc \left ( dx+c \right ) }{2\,d}}+{\frac{a\ln \left ( \csc \left ( dx+c \right ) -\cot \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{b\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)^3*(a+sin(d*x+c)^2*b),x)

[Out]

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

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Maxima [A]  time = 0.936022, size = 78, normalized size = 1.95 \begin{align*} -\frac{{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right ) -{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right ) - \frac{2 \, a \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{2} - 1}}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.70168, size = 246, normalized size = 6.15 \begin{align*} \frac{2 \, a \cos \left (d x + c\right ) -{\left ({\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{2} - a - 2 \, b\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) +{\left ({\left (a + 2 \, b\right )} \cos \left (d x + c\right )^{2} - a - 2 \, b\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{4 \,{\left (d \cos \left (d x + c\right )^{2} - d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(csc(d*x+c)**3*(a+b*sin(d*x+c)**2),x)

[Out]

Timed out

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Giac [B]  time = 1.18255, size = 163, normalized size = 4.08 \begin{align*} \frac{2 \,{\left (a + 2 \, b\right )} \log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) + \frac{{\left (a - \frac{2 \, a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{4 \, b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}{\left (\cos \left (d x + c\right ) + 1\right )}}{\cos \left (d x + c\right ) - 1} - \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*(a + 2*b)*log(abs(-cos(d*x + c) + 1)/abs(cos(d*x + c) + 1)) + (a - 2*a*(cos(d*x + c) - 1)/(cos(d*x + c)
+ 1) - 4*b*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))*(cos(d*x + c) + 1)/(cos(d*x + c) - 1) - a*(cos(d*x + c) - 1
)/(cos(d*x + c) + 1))/d