\(\int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx\) [758]

Optimal result

Integrand size = 27, antiderivative size = 91 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \]

[Out]

-3/2*a*arctanh(cos(d*x+c))/d-2*a*cot(d*x+c)/d-1/3*a*cot(d*x+c)^3/d+3/2*a*sec(d*x+c)/d-1/2*a*csc(d*x+c)^2*sec(d
*x+c)/d+a*tan(d*x+c)/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2917, 2700, 276, 2702, 294, 327, 213} \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}+\frac {a \tan (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}-\frac {2 a \cot (c+d x)}{d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d} \]

[In]

Int[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

(-3*a*ArcTanh[Cos[c + d*x]])/(2*d) - (2*a*Cot[c + d*x])/d - (a*Cot[c + d*x]^3)/(3*d) + (3*a*Sec[c + d*x])/(2*d
) - (a*Csc[c + d*x]^2*Sec[c + d*x])/(2*d) + (a*Tan[c + d*x])/d

Rule 213

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

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 294

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^
n)^(p + 1)/(b*n*(p + 1))), x] - Dist[c^n*((m - n + 1)/(b*n*(p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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]

Rule 2702

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 2917

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rubi steps \begin{align*} \text {integral}& = a \int \csc ^3(c+d x) \sec ^2(c+d x) \, dx+a \int \csc ^4(c+d x) \sec ^2(c+d x) \, dx \\ & = \frac {a \text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^4} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a \text {Subst}\left (\int \left (1+\frac {1}{x^4}+\frac {2}{x^2}\right ) \, dx,x,\tan (c+d x)\right )}{d}+\frac {(3 a) \text {Subst}\left (\int \frac {x^2}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {2 a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a \tan (c+d x)}{d}+\frac {(3 a) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {3 a \text {arctanh}(\cos (c+d x))}{2 d}-\frac {2 a \cot (c+d x)}{d}-\frac {a \cot ^3(c+d x)}{3 d}+\frac {3 a \sec (c+d x)}{2 d}-\frac {a \csc ^2(c+d x) \sec (c+d x)}{2 d}+\frac {a \tan (c+d x)}{d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(205\) vs. \(2(91)=182\).

Time = 4.14 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.25 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {5 a \cot (c+d x)}{3 d}-\frac {a \csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \cot (c+d x) \csc ^2(c+d x)}{3 d}-\frac {3 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {3 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}+\frac {a \sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 d}+\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}-\frac {a \sin \left (\frac {1}{2} (c+d x)\right )}{d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {a \tan (c+d x)}{d} \]

[In]

Integrate[Csc[c + d*x]^4*Sec[c + d*x]^2*(a + a*Sin[c + d*x]),x]

[Out]

(-5*a*Cot[c + d*x])/(3*d) - (a*Csc[(c + d*x)/2]^2)/(8*d) - (a*Cot[c + d*x]*Csc[c + d*x]^2)/(3*d) - (3*a*Log[Co
s[(c + d*x)/2]])/(2*d) + (3*a*Log[Sin[(c + d*x)/2]])/(2*d) + (a*Sec[(c + d*x)/2]^2)/(8*d) + (a*Sin[(c + d*x)/2
])/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (a*Sin[(c + d*x)/2])/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))
+ (a*Tan[c + d*x])/d

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.12

method	result	size
derivativedivides	\(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )}{d}\)	\(102\)
default	\(\frac {a \left (-\frac {1}{2 \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )}+\frac {3}{2 \cos \left (d x +c \right )}+\frac {3 \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{2}\right )+a \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )}{d}\)	\(102\)
parallelrisch	\(\frac {\left (-90+36 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )+\cot ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )+2 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\cot ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+18 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{24 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\)	\(117\)
risch	\(\frac {-9 i a \,{\mathrm e}^{5 i \left (d x +c \right )}+9 a \,{\mathrm e}^{6 i \left (d x +c \right )}+24 i a \,{\mathrm e}^{3 i \left (d x +c \right )}-24 a \,{\mathrm e}^{4 i \left (d x +c \right )}-7 i a \,{\mathrm e}^{i \left (d x +c \right )}+39 a \,{\mathrm e}^{2 i \left (d x +c \right )}-16 a}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d}\)	\(149\)
norman	\(\frac {\frac {a}{24 d}+\frac {a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {7 a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {35 a \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {9 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {35 a \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {7 a \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {2 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {3 a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}\)	\(218\)

[In]

int(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(a*(-1/2/sin(d*x+c)^2/cos(d*x+c)+3/2/cos(d*x+c)+3/2*ln(csc(d*x+c)-cot(d*x+c)))+a*(-1/3/sin(d*x+c)^3/cos(d*
x+c)+4/3/sin(d*x+c)/cos(d*x+c)-8/3*cot(d*x+c)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (83) = 166\).

Time = 0.27 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.38 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=-\frac {32 \, a \cos \left (d x + c\right )^{4} + 14 \, a \cos \left (d x + c\right )^{3} - 48 \, a \cos \left (d x + c\right )^{2} - 18 \, a \cos \left (d x + c\right ) + 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 9 \, {\left (a \cos \left (d x + c\right )^{4} - 2 \, a \cos \left (d x + c\right )^{2} + {\left (a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) + a\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 2 \, {\left (16 \, a \cos \left (d x + c\right )^{3} + 9 \, a \cos \left (d x + c\right )^{2} - 15 \, a \cos \left (d x + c\right ) - 6 \, a\right )} \sin \left (d x + c\right ) + 12 \, a}{12 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right ) - d\right )} \sin \left (d x + c\right ) + d\right )}} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

-1/12*(32*a*cos(d*x + c)^4 + 14*a*cos(d*x + c)^3 - 48*a*cos(d*x + c)^2 - 18*a*cos(d*x + c) + 9*(a*cos(d*x + c)
^4 - 2*a*cos(d*x + c)^2 + (a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos(d*x + c) - a)*sin(d*x + c) + a)*log(1/2
*cos(d*x + c) + 1/2) - 9*(a*cos(d*x + c)^4 - 2*a*cos(d*x + c)^2 + (a*cos(d*x + c)^3 + a*cos(d*x + c)^2 - a*cos
(d*x + c) - a)*sin(d*x + c) + a)*log(-1/2*cos(d*x + c) + 1/2) - 2*(16*a*cos(d*x + c)^3 + 9*a*cos(d*x + c)^2 -
15*a*cos(d*x + c) - 6*a)*sin(d*x + c) + 12*a)/(d*cos(d*x + c)^4 - 2*d*cos(d*x + c)^2 + (d*cos(d*x + c)^3 + d*c
os(d*x + c)^2 - d*cos(d*x + c) - d)*sin(d*x + c) + d)

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**4*sec(d*x+c)**2*(a+a*sin(d*x+c)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.08 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {3 \, a {\left (\frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 2\right )}}{\cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )} - 3 \, \log \left (\cos \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\cos \left (d x + c\right ) - 1\right )\right )} - 4 \, a {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )}}{12 \, d} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/12*(3*a*(2*(3*cos(d*x + c)^2 - 2)/(cos(d*x + c)^3 - cos(d*x + c)) - 3*log(cos(d*x + c) + 1) + 3*log(cos(d*x
+ c) - 1)) - 4*a*((6*tan(d*x + c)^2 + 1)/tan(d*x + c)^3 - 3*tan(d*x + c)))/d

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.43 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 36 \, a \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {48 \, a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1} - \frac {66 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 21 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 3 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{24 \, d} \]

[In]

integrate(csc(d*x+c)^4*sec(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

1/24*(a*tan(1/2*d*x + 1/2*c)^3 + 3*a*tan(1/2*d*x + 1/2*c)^2 + 36*a*log(abs(tan(1/2*d*x + 1/2*c))) + 21*a*tan(1
/2*d*x + 1/2*c) - 48*a/(tan(1/2*d*x + 1/2*c) - 1) - (66*a*tan(1/2*d*x + 1/2*c)^3 + 21*a*tan(1/2*d*x + 1/2*c)^2
+ 3*a*tan(1/2*d*x + 1/2*c) + a)/tan(1/2*d*x + 1/2*c)^3)/d

Mupad [B] (verification not implemented)

Time = 9.48 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.58 \[ \int \csc ^4(c+d x) \sec ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {7\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {-23\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {2\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {a}{3}}{d\,\left (8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\right )}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}+\frac {3\,a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \]

[In]

int((a + a*sin(c + d*x))/(cos(c + d*x)^2*sin(c + d*x)^4),x)

[Out]

(7*a*tan(c/2 + (d*x)/2))/(8*d) - (a/3 + (2*a*tan(c/2 + (d*x)/2))/3 + 6*a*tan(c/2 + (d*x)/2)^2 - 23*a*tan(c/2 +
(d*x)/2)^3)/(d*(8*tan(c/2 + (d*x)/2)^3 - 8*tan(c/2 + (d*x)/2)^4)) + (a*tan(c/2 + (d*x)/2)^2)/(8*d) + (a*tan(c
/2 + (d*x)/2)^3)/(24*d) + (3*a*log(tan(c/2 + (d*x)/2)))/(2*d)