3.867 \(\int \cos (x) \sqrt{-1+\csc ^2(x)} (1-\sin ^2(x))^3 \, dx\)

Optimal. Leaf size=81 \[ \sin (x) \sqrt{\cot ^2(x)}+\frac{1}{7} \sin (x) \cos ^6(x) \sqrt{\cot ^2(x)}+\frac{1}{5} \sin (x) \cos ^4(x) \sqrt{\cot ^2(x)}+\frac{1}{3} \sin (x) \cos ^2(x) \sqrt{\cot ^2(x)}-\tan (x) \sqrt{\cot ^2(x)} \tanh ^{-1}(\cos (x)) \]

[Out]

Sqrt[Cot[x]^2]*Sin[x] + (Cos[x]^2*Sqrt[Cot[x]^2]*Sin[x])/3 + (Cos[x]^4*Sqrt[Cot[x]^2]*Sin[x])/5 + (Cos[x]^6*Sq
rt[Cot[x]^2]*Sin[x])/7 - ArcTanh[Cos[x]]*Sqrt[Cot[x]^2]*Tan[x]

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Rubi [A]  time = 0.158987, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3175, 4121, 3658, 2592, 302, 206} \[ \sin (x) \sqrt{\cot ^2(x)}+\frac{1}{7} \sin (x) \cos ^6(x) \sqrt{\cot ^2(x)}+\frac{1}{5} \sin (x) \cos ^4(x) \sqrt{\cot ^2(x)}+\frac{1}{3} \sin (x) \cos ^2(x) \sqrt{\cot ^2(x)}-\tan (x) \sqrt{\cot ^2(x)} \tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]

[Out]

Sqrt[Cot[x]^2]*Sin[x] + (Cos[x]^2*Sqrt[Cot[x]^2]*Sin[x])/3 + (Cos[x]^4*Sqrt[Cot[x]^2]*Sin[x])/5 + (Cos[x]^6*Sq
rt[Cot[x]^2]*Sin[x])/7 - ArcTanh[Cos[x]]*Sqrt[Cot[x]^2]*Tan[x]

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 4121

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 2592

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> With[{ff = FreeFactors[S
in[e + f*x], x]}, Dist[ff/f, Subst[Int[(ff*x)^(m + n)/(a^2 - ff^2*x^2)^((n + 1)/2), x], x, (a*Sin[e + f*x])/ff
], x]] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \cos (x) \sqrt{-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx &=\int \cos ^7(x) \sqrt{-1+\csc ^2(x)} \, dx\\ &=\int \cos ^7(x) \sqrt{\cot ^2(x)} \, dx\\ &=\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \int \cos ^7(x) \cot (x) \, dx\\ &=-\left (\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \operatorname{Subst}\left (\int \frac{x^8}{1-x^2} \, dx,x,\cos (x)\right )\right )\\ &=-\left (\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \operatorname{Subst}\left (\int \left (-1-x^2-x^4-x^6+\frac{1}{1-x^2}\right ) \, dx,x,\cos (x)\right )\right )\\ &=\sqrt{\cot ^2(x)} \sin (x)+\frac{1}{3} \cos ^2(x) \sqrt{\cot ^2(x)} \sin (x)+\frac{1}{5} \cos ^4(x) \sqrt{\cot ^2(x)} \sin (x)+\frac{1}{7} \cos ^6(x) \sqrt{\cot ^2(x)} \sin (x)-\left (\sqrt{\cot ^2(x)} \tan (x)\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\cos (x)\right )\\ &=\sqrt{\cot ^2(x)} \sin (x)+\frac{1}{3} \cos ^2(x) \sqrt{\cot ^2(x)} \sin (x)+\frac{1}{5} \cos ^4(x) \sqrt{\cot ^2(x)} \sin (x)+\frac{1}{7} \cos ^6(x) \sqrt{\cot ^2(x)} \sin (x)-\tanh ^{-1}(\cos (x)) \sqrt{\cot ^2(x)} \tan (x)\\ \end{align*}

Mathematica [A]  time = 0.0635501, size = 55, normalized size = 0.68 \[ \frac{\tan (x) \sqrt{\cot ^2(x)} \left (9765 \cos (x)+1295 \cos (3 x)+189 \cos (5 x)+15 \cos (7 x)+6720 \log \left (\sin \left (\frac{x}{2}\right )\right )-6720 \log \left (\cos \left (\frac{x}{2}\right )\right )\right )}{6720} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]

[Out]

(Sqrt[Cot[x]^2]*(9765*Cos[x] + 1295*Cos[3*x] + 189*Cos[5*x] + 15*Cos[7*x] - 6720*Log[Cos[x/2]] + 6720*Log[Sin[
x/2]])*Tan[x])/6720

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Maple [A]  time = 0.157, size = 65, normalized size = 0.8 \begin{align*}{\frac{\sqrt{4}\sin \left ( x \right ) }{210\,\cos \left ( x \right ) } \left ( 15\, \left ( \cos \left ( x \right ) \right ) ^{7}+21\, \left ( \cos \left ( x \right ) \right ) ^{5}+35\, \left ( \cos \left ( x \right ) \right ) ^{3}+105\,\cos \left ( x \right ) +105\,\ln \left ( -{\frac{-1+\cos \left ( x \right ) }{\sin \left ( x \right ) }} \right ) +176 \right ) \sqrt{-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{2}}{-1+ \left ( \cos \left ( x \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(1-sin(x)^2)^3*(-1+csc(x)^2)^(1/2),x)

[Out]

1/210*4^(1/2)*(15*cos(x)^7+21*cos(x)^5+35*cos(x)^3+105*cos(x)+105*ln(-(-1+cos(x))/sin(x))+176)*sin(x)*(-cos(x)
^2/(-1+cos(x)^2))^(1/2)/cos(x)

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Maxima [A]  time = 0.985823, size = 116, normalized size = 1.43 \begin{align*} \frac{1}{7} \,{\left (\frac{1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac{7}{2}} \sin \left (x\right )^{7} + \frac{1}{5} \,{\left (\frac{1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac{5}{2}} \sin \left (x\right )^{5} + \frac{1}{3} \,{\left (\frac{1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac{3}{2}} \sin \left (x\right )^{3} + \sqrt{\frac{1}{\sin \left (x\right )^{2}} - 1} \sin \left (x\right ) - \frac{1}{2} \, \log \left (\sqrt{\frac{1}{\sin \left (x\right )^{2}} - 1} \sin \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{1}{\sin \left (x\right )^{2}} - 1} \sin \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1-sin(x)^2)^3*(-1+csc(x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/7*(1/sin(x)^2 - 1)^(7/2)*sin(x)^7 + 1/5*(1/sin(x)^2 - 1)^(5/2)*sin(x)^5 + 1/3*(1/sin(x)^2 - 1)^(3/2)*sin(x)^
3 + sqrt(1/sin(x)^2 - 1)*sin(x) - 1/2*log(sqrt(1/sin(x)^2 - 1)*sin(x) + 1) + 1/2*log(sqrt(1/sin(x)^2 - 1)*sin(
x) - 1)

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Fricas [A]  time = 2.18922, size = 150, normalized size = 1.85 \begin{align*} -\frac{1}{7} \, \cos \left (x\right )^{7} - \frac{1}{5} \, \cos \left (x\right )^{5} - \frac{1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) + \frac{1}{2} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) - \frac{1}{2} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1-sin(x)^2)^3*(-1+csc(x)^2)^(1/2),x, algorithm="fricas")

[Out]

-1/7*cos(x)^7 - 1/5*cos(x)^5 - 1/3*cos(x)^3 - cos(x) + 1/2*log(1/2*cos(x) + 1/2) - 1/2*log(-1/2*cos(x) + 1/2)

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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(cos(x)*(1-sin(x)**2)**3*(-1+csc(x)**2)**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 1.09033, size = 138, normalized size = 1.7 \begin{align*} -\frac{1}{210} \,{\left (30 \,{\left (\sin \left (x\right )^{2} - 1\right )}^{3} \sqrt{-\sin \left (x\right )^{2} + 1} - 42 \,{\left (\sin \left (x\right )^{2} - 1\right )}^{2} \sqrt{-\sin \left (x\right )^{2} + 1} - 70 \,{\left (-\sin \left (x\right )^{2} + 1\right )}^{\frac{3}{2}} - 210 \, \sqrt{-\sin \left (x\right )^{2} + 1} + 105 \, \log \left (\sqrt{-\sin \left (x\right )^{2} + 1} + 1\right ) - 105 \, \log \left (-\sqrt{-\sin \left (x\right )^{2} + 1} + 1\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1-sin(x)^2)^3*(-1+csc(x)^2)^(1/2),x, algorithm="giac")

[Out]

-1/210*(30*(sin(x)^2 - 1)^3*sqrt(-sin(x)^2 + 1) - 42*(sin(x)^2 - 1)^2*sqrt(-sin(x)^2 + 1) - 70*(-sin(x)^2 + 1)
^(3/2) - 210*sqrt(-sin(x)^2 + 1) + 105*log(sqrt(-sin(x)^2 + 1) + 1) - 105*log(-sqrt(-sin(x)^2 + 1) + 1))*sgn(s
in(x))