3.1.7 \(\int \csc ^3(x) \sec ^5(x) \, dx\) [7]

Optimal. Leaf size=30 \[ -\frac {1}{2} \cot ^2(x)+3 \log (\tan (x))+\frac {3 \tan ^2(x)}{2}+\frac {\tan ^4(x)}{4} \]

[Out]

-1/2*cot(x)^2+3*ln(tan(x))+3/2*tan(x)^2+1/4*tan(x)^4

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2700, 272, 45} \begin {gather*} \frac {\tan ^4(x)}{4}+\frac {3 \tan ^2(x)}{2}-\frac {\cot ^2(x)}{2}+3 \log (\tan (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3*Sec[x]^5,x]

[Out]

-1/2*Cot[x]^2 + 3*Log[Tan[x]] + (3*Tan[x]^2)/2 + Tan[x]^4/4

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 2700

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rubi steps

\begin {gather*} \begin {aligned} \text {Integral} &=\text {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^3} \, dx,x,\tan (x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(1+x)^3}{x^2} \, dx,x,\tan ^2(x)\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (3+\frac {1}{x^2}+\frac {3}{x}+x\right ) \, dx,x,\tan ^2(x)\right )\\ &=-\frac {1}{2} \cot ^2(x)+3 \log (\tan (x))+\frac {3 \tan ^2(x)}{2}+\frac {\tan ^4(x)}{4}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 31, normalized size = 1.03 \begin {gather*} -\frac {1}{2} \csc ^2(x)-3 \log (\cos (x))+3 \log (\sin (x))+\sec ^2(x)+\frac {\sec ^4(x)}{4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3*Sec[x]^5,x]

[Out]

-1/2*Csc[x]^2 - 3*Log[Cos[x]] + 3*Log[Sin[x]] + Sec[x]^2 + Sec[x]^4/4

________________________________________________________________________________________

Maple [A]
time = 0.25, size = 33, normalized size = 1.10

method	result	size
default	\(\frac {1}{4 \sin \left (x \right )^{2} \cos \left (x \right )^{4}}+\frac {3}{4 \sin \left (x \right )^{2} \cos \left (x \right )^{2}}-\frac {3}{2 \sin \left (x \right )^{2}}+3 \ln \left (\tan \left (x \right )\right )\)	\(33\)
parallelrisch	\(-3 \ln \left (1+\csc \left (x \right )-\cot \left (x \right )\right )-3 \ln \left (\csc \left (x \right )-\cot \left (x \right )-1\right )+3 \ln \left (-\cot \left (x \right )+\csc \left (x \right )\right )+\frac {\left (8 \left (\sec ^{4}\left (x \right )\right )+24 \left (\sec ^{2}\left (x \right )\right )-39\right ) \left (\csc ^{2}\left (x \right )\right )}{32}-\frac {9 \left (\cot ^{2}\left (x \right )\right )}{32}\)	\(60\)
norman	\(\frac {-\frac {1}{8}-10 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\frac {57 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{8}+\frac {57 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{8}-\frac {\left (\tan ^{12}\left (\frac {x}{2}\right )\right )}{8}}{\tan \left (\frac {x}{2}\right )^{2} \left (\tan ^{2}\left (\frac {x}{2}\right )-1\right )^{4}}+3 \ln \left (\tan \left (\frac {x}{2}\right )\right )-3 \ln \left (\tan \left (\frac {x}{2}\right )-1\right )-3 \ln \left (1+\tan \left (\frac {x}{2}\right )\right )\)	\(78\)
risch	\(\frac {6 \,{\mathrm e}^{10 i x}+12 \,{\mathrm e}^{8 i x}-4 \,{\mathrm e}^{6 i x}+12 \,{\mathrm e}^{4 i x}+6 \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{4} \left ({\mathrm e}^{2 i x}-1\right )^{2}}+3 \ln \left ({\mathrm e}^{2 i x}-1\right )-3 \ln \left ({\mathrm e}^{2 i x}+1\right )\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(x)^3/cos(x)^5,x,method=_RETURNVERBOSE)

[Out]

1/4/sin(x)^2/cos(x)^4+3/4/sin(x)^2/cos(x)^2-3/2/sin(x)^2+3*ln(tan(x))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (24) = 48\).
time = 0.35, size = 50, normalized size = 1.67 \begin {gather*} -\frac {6 \, \sin \left (x\right )^{4} - 9 \, \sin \left (x\right )^{2} + 2}{4 \, {\left (\sin \left (x\right )^{6} - 2 \, \sin \left (x\right )^{4} + \sin \left (x\right )^{2}\right )}} - \frac {3}{2} \, \log \left (\sin \left (x\right )^{2} - 1\right ) + \frac {3}{2} \, \log \left (\sin \left (x\right )^{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^3/cos(x)^5,x, algorithm="maxima")

[Out]

-1/4*(6*sin(x)^4 - 9*sin(x)^2 + 2)/(sin(x)^6 - 2*sin(x)^4 + sin(x)^2) - 3/2*log(sin(x)^2 - 1) + 3/2*log(sin(x)
^2)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (24) = 48\).
time = 0.62, size = 69, normalized size = 2.30 \begin {gather*} \frac {6 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} - 6 \, {\left (\cos \left (x\right )^{6} - \cos \left (x\right )^{4}\right )} \log \left (\cos \left (x\right )^{2}\right ) + 6 \, {\left (\cos \left (x\right )^{6} - \cos \left (x\right )^{4}\right )} \log \left (-\frac {1}{4} \, \cos \left (x\right )^{2} + \frac {1}{4}\right ) - 1}{4 \, {\left (\cos \left (x\right )^{6} - \cos \left (x\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^3/cos(x)^5,x, algorithm="fricas")

[Out]

1/4*(6*cos(x)^4 - 3*cos(x)^2 - 6*(cos(x)^6 - cos(x)^4)*log(cos(x)^2) + 6*(cos(x)^6 - cos(x)^4)*log(-1/4*cos(x)
^2 + 1/4) - 1)/(cos(x)^6 - cos(x)^4)

________________________________________________________________________________________

Sympy [A]
time = 0.07, size = 46, normalized size = 1.53 \begin {gather*} \frac {3 \log {\left (\cos ^{2}{\left (x \right )} - 1 \right )}}{2} - 3 \log {\left (\cos {\left (x \right )} \right )} - \frac {- 6 \cos ^{4}{\left (x \right )} + 3 \cos ^{2}{\left (x \right )} + 1}{4 \cos ^{6}{\left (x \right )} - 4 \cos ^{4}{\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)**3/cos(x)**5,x)

[Out]

3*log(cos(x)**2 - 1)/2 - 3*log(cos(x)) - (-6*cos(x)**4 + 3*cos(x)**2 + 1)/(4*cos(x)**6 - 4*cos(x)**4)

________________________________________________________________________________________

Giac [A]
time = 0.39, size = 46, normalized size = 1.53 \begin {gather*} \frac {6 \, \cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{2} - 1}{4 \, {\left (\cos \left (x\right )^{2} - 1\right )} \cos \left (x\right )^{4}} + \frac {3}{2} \, \log \left (-\cos \left (x\right )^{2} + 1\right ) - 3 \, \log \left ({\left | \cos \left (x\right ) \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/sin(x)^3/cos(x)^5,x, algorithm="giac")

[Out]

1/4*(6*cos(x)^4 - 3*cos(x)^2 - 1)/((cos(x)^2 - 1)*cos(x)^4) + 3/2*log(-cos(x)^2 + 1) - 3*log(abs(cos(x)))

________________________________________________________________________________________

Mupad [B]
time = 0.17, size = 30, normalized size = 1.00 \begin {gather*} 3\,\ln \left (\mathrm {tan}\left (x\right )\right )+\frac {\frac {3}{4\,{\cos \left (x\right )}^2}+\frac {1}{4\,{\cos \left (x\right )}^4}}{{\sin \left (x\right )}^2}-\frac {3}{2\,{\sin \left (x\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(x)^5*sin(x)^3),x)

[Out]

3*log(tan(x)) + (3/(4*cos(x)^2) + 1/(4*cos(x)^4))/sin(x)^2 - 3/(2*sin(x)^2)

________________________________________________________________________________________

Chatgpt [F] Failed to verify
time = 1.00, size = 6, normalized size = 0.20 \begin {gather*} \frac {8}{15 \cos \left (x \right )^{3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

int(1/sin(x)^3/cos(x)^5,x)

[Out]

8/15/cos(x)^3

________________________________________________________________________________________