Integrand size = 23, antiderivative size = 81 \[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=\sqrt {\cot ^2(x)} \sin (x)+\frac {1}{3} \cos ^2(x) \sqrt {\cot ^2(x)} \sin (x)+\frac {1}{5} \cos ^4(x) \sqrt {\cot ^2(x)} \sin (x)+\frac {1}{7} \cos ^6(x) \sqrt {\cot ^2(x)} \sin (x)-\text {arctanh}(\cos (x)) \sqrt {\cot ^2(x)} \tan (x) \] Output:
(cot(x)^2)^(1/2)*sin(x)+1/3*cos(x)^2*(cot(x)^2)^(1/2)*sin(x)+1/5*cos(x)^4* (cot(x)^2)^(1/2)*sin(x)+1/7*cos(x)^6*(cot(x)^2)^(1/2)*sin(x)-arctanh(cos(x ))*(cot(x)^2)^(1/2)*tan(x)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.49 \[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\frac {\sqrt {\cot ^2(x)} \operatorname {Hypergeometric2F1}\left (-\frac {7}{2},-\frac {7}{2},-\frac {5}{2},\csc ^2(x)\right ) \sin ^7(x)}{7 \sqrt {-\cot ^2(x)}} \] Input:
Integrate[Cos[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]
Output:
-1/7*(Sqrt[Cot[x]^2]*Hypergeometric2F1[-7/2, -7/2, -5/2, Csc[x]^2]*Sin[x]^ 7)/Sqrt[-Cot[x]^2]
Time = 0.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3042, 3654, 3042, 4609, 3042, 4141, 3042, 25, 3072, 254, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (1-\sin ^2(x)\right )^3 \cos (x) \sqrt {\csc ^2(x)-1} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (1-\sin (x)^2\right )^3 \cos (x) \sqrt {\csc (x)^2-1}dx\) |
| \(\Big \downarrow \) 3654 |
| \(\displaystyle \int \cos ^7(x) \sqrt {\csc ^2(x)-1}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (x+\frac {\pi }{2}\right )^7 \sqrt {\sec \left (x+\frac {\pi }{2}\right )^2-1}dx\) |
| \(\Big \downarrow \) 4609 |
| \(\displaystyle \int \cos ^7(x) \sqrt {\cot ^2(x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (x+\frac {\pi }{2}\right )^7 \sqrt {\tan \left (x+\frac {\pi }{2}\right )^2}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \tan (x) \sqrt {\cot ^2(x)} \int \cos ^7(x) \cot (x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \tan (x) \sqrt {\cot ^2(x)} \int -\sin \left (x+\frac {\pi }{2}\right )^7 \tan \left (x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \int \sin \left (x+\frac {\pi }{2}\right )^7 \tan \left (x+\frac {\pi }{2}\right )dx\) |
| \(\Big \downarrow \) 3072 |
| \(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \int \frac {\cos ^8(x)}{1-\cos ^2(x)}d\cos (x)\) |
| \(\Big \downarrow \) 254 |
| \(\displaystyle \tan (x) \left (-\sqrt {\cot ^2(x)}\right ) \int \left (-\cos ^6(x)-\cos ^4(x)-\cos ^2(x)+\frac {1}{1-\cos ^2(x)}-1\right )d\cos (x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\left (\tan (x) \sqrt {\cot ^2(x)} \left (\text {arctanh}(\cos (x))-\frac {1}{7} \cos ^7(x)-\frac {\cos ^5(x)}{5}-\frac {\cos ^3(x)}{3}-\cos (x)\right )\right )\) |
Input:
Int[Cos[x]*Sqrt[-1 + Csc[x]^2]*(1 - Sin[x]^2)^3,x]
Output:
-((ArcTanh[Cos[x]] - Cos[x] - Cos[x]^3/3 - Cos[x]^5/5 - Cos[x]^7/7)*Sqrt[C ot[x]^2]*Tan[x])
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^2, x], x] /; FreeQ[{a, b}, x] && IGtQ[m, 3]
Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[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]
Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[ 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]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[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]])
Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(b*tan[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Time = 0.50 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.59
| method | result | size |
| default | \(\frac {\tan \left (x \right ) \left (15 \cos \left (x \right )^{7}+21 \cos \left (x \right )^{5}+35 \cos \left (x \right )^{3}+105 \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )+105 \cos \left (x \right )+176\right ) \sqrt {\cot \left (x \right )^{2}}\, \sqrt {4}}{210}\) | \(48\) |
| risch | \(\frac {i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) {\mathrm e}^{2 i x}}{1+{\mathrm e}^{2 i x}}-\frac {i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) {\mathrm e}^{2 i x}}{1+{\mathrm e}^{2 i x}}-\frac {i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, {\mathrm e}^{9 i x}}{896 \left (1+{\mathrm e}^{2 i x}\right )}+\frac {121 i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, {\mathrm e}^{-i x}}{192 \left (1+{\mathrm e}^{2 i x}\right )}-\frac {i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right )}{1+{\mathrm e}^{2 i x}}+\frac {i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right )}{1+{\mathrm e}^{2 i x}}-\frac {53 i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \cos \left (7 x \right )}{4480 \left (1+{\mathrm e}^{2 i x}\right )}+\frac {9 \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \sin \left (7 x \right )}{640 \left (1+{\mathrm e}^{2 i x}\right )}-\frac {233 i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \cos \left (5 x \right )}{3360 \left (1+{\mathrm e}^{2 i x}\right )}+\frac {2 \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \sin \left (5 x \right )}{21 \left (1+{\mathrm e}^{2 i x}\right )}-\frac {263 i \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \cos \left (3 x \right )}{480 \left (1+{\mathrm e}^{2 i x}\right )}+\frac {57 \sqrt {-\frac {\left (1+{\mathrm e}^{2 i x}\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}}\, \sin \left (3 x \right )}{80 \left (1+{\mathrm e}^{2 i x}\right )}\) | \(483\) |
Input:
int(cos(x)*(-1+csc(x)^2)^(1/2)*(1-sin(x)^2)^3,x,method=_RETURNVERBOSE)
Output:
1/210*tan(x)*(15*cos(x)^7+21*cos(x)^5+35*cos(x)^3+105*ln(csc(x)-cot(x))+10 5*cos(x)+176)*(cot(x)^2)^(1/2)*4^(1/2)
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.51 \[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\frac {1}{7} \, \cos \left (x\right )^{7} - \frac {1}{5} \, \cos \left (x\right )^{5} - \frac {1}{3} \, \cos \left (x\right )^{3} - \cos \left (x\right ) + \frac {1}{2} \, \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - \frac {1}{2} \, \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) \] Input:
integrate(cos(x)*(-1+csc(x)^2)^(1/2)*(1-sin(x)^2)^3,x, algorithm="fricas")
Output:
-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)
Timed out. \[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=\text {Timed out} \] Input:
integrate(cos(x)*(-1+csc(x)**2)**(1/2)*(1-sin(x)**2)**3,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.06 \[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=\frac {1}{7} \, {\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac {7}{2}} \sin \left (x\right )^{7} + \frac {1}{5} \, {\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac {5}{2}} \sin \left (x\right )^{5} + \frac {1}{3} \, {\left (\frac {1}{\sin \left (x\right )^{2}} - 1\right )}^{\frac {3}{2}} \sin \left (x\right )^{3} + \sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1} \sin \left (x\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1} \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, \log \left (\sqrt {\frac {1}{\sin \left (x\right )^{2}} - 1} \sin \left (x\right ) - 1\right ) \] Input:
integrate(cos(x)*(-1+csc(x)^2)^(1/2)*(1-sin(x)^2)^3,x, algorithm="maxima")
Output:
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)
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.54 \[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=\frac {1}{210} \, {\left (30 \, \cos \left (x\right )^{7} + 42 \, \cos \left (x\right )^{5} + 70 \, \cos \left (x\right )^{3} + 210 \, \cos \left (x\right ) - 105 \, \log \left (\cos \left (x\right ) + 1\right ) + 105 \, \log \left (-\cos \left (x\right ) + 1\right )\right )} \mathrm {sgn}\left (\sin \left (x\right )\right ) \] Input:
integrate(cos(x)*(-1+csc(x)^2)^(1/2)*(1-sin(x)^2)^3,x, algorithm="giac")
Output:
1/210*(30*cos(x)^7 + 42*cos(x)^5 + 70*cos(x)^3 + 210*cos(x) - 105*log(cos( x) + 1) + 105*log(-cos(x) + 1))*sgn(sin(x))
Timed out. \[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\int \cos \left (x\right )\,\sqrt {\frac {1}{{\sin \left (x\right )}^2}-1}\,{\left ({\sin \left (x\right )}^2-1\right )}^3 \,d x \] Input:
int(-cos(x)*(1/sin(x)^2 - 1)^(1/2)*(sin(x)^2 - 1)^3,x)
Output:
-int(cos(x)*(1/sin(x)^2 - 1)^(1/2)*(sin(x)^2 - 1)^3, x)
\[ \int \cos (x) \sqrt {-1+\csc ^2(x)} \left (1-\sin ^2(x)\right )^3 \, dx=-\left (\int \sqrt {\csc \left (x \right )^{2}-1}\, \cos \left (x \right ) \sin \left (x \right )^{6}d x \right )+3 \left (\int \sqrt {\csc \left (x \right )^{2}-1}\, \cos \left (x \right ) \sin \left (x \right )^{4}d x \right )-3 \left (\int \sqrt {\csc \left (x \right )^{2}-1}\, \cos \left (x \right ) \sin \left (x \right )^{2}d x \right )+\int \sqrt {\csc \left (x \right )^{2}-1}\, \cos \left (x \right )d x \] Input:
int(cos(x)*(-1+csc(x)^2)^(1/2)*(1-sin(x)^2)^3,x)
Output:
- int(sqrt(csc(x)**2 - 1)*cos(x)*sin(x)**6,x) + 3*int(sqrt(csc(x)**2 - 1) *cos(x)*sin(x)**4,x) - 3*int(sqrt(csc(x)**2 - 1)*cos(x)*sin(x)**2,x) + int (sqrt(csc(x)**2 - 1)*cos(x),x)