\(\int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx\) [8]

Optimal result

Integrand size = 32, antiderivative size = 61 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot (e+f x) \csc (e+f x)}{2 f} \]

[Out]

1/2*a^2*c*arctanh(cos(f*x+e))/f-1/3*a^2*c*cot(f*x+e)^3/f-1/2*a^2*c*cot(f*x+e)*csc(f*x+e)/f

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3029, 2785, 2687, 30, 2691, 3855} \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \text {arctanh}(\cos (e+f x))}{2 f}-\frac {a^2 c \cot ^3(e+f x)}{3 f}-\frac {a^2 c \cot (e+f x) \csc (e+f x)}{2 f} \]

[In]

Int[Csc[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]

[Out]

(a^2*c*ArcTanh[Cos[e + f*x]])/(2*f) - (a^2*c*Cot[e + f*x]^3)/(3*f) - (a^2*c*Cot[e + f*x]*Csc[e + f*x])/(2*f)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

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

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2785

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 3029

Int[sin[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*
(x_)])^(n_.), x_Symbol] :> Dist[a^n*c^n, Int[Tan[e + f*x]^p*(a + b*Sin[e + f*x])^(m - n), x], x] /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && EqQ[p + 2*n, 0] && IntegerQ[n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(61)=122\).

Time = 0.20 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.82 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=a^2 c \left (\frac {\cot \left (\frac {1}{2} (e+f x)\right )}{6 f}-\frac {\csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}-\frac {\cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{24 f}+\frac {\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}-\frac {\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}-\frac {\tan \left (\frac {1}{2} (e+f x)\right )}{6 f}+\frac {\sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{24 f}\right ) \]

[In]

Integrate[Csc[e + f*x]^4*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]

[Out]

a^2*c*(Cot[(e + f*x)/2]/(6*f) - Csc[(e + f*x)/2]^2/(8*f) - (Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/(24*f) + Log[
Cos[(e + f*x)/2]]/(2*f) - Log[Sin[(e + f*x)/2]]/(2*f) + Sec[(e + f*x)/2]^2/(8*f) - Tan[(e + f*x)/2]/(6*f) + (S
ec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(24*f))

Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.56

method	result	size
parallelrisch	\(\frac {a^{2} c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )-\left (\cot ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-3 \left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-3 \tan \left (\frac {f x}{2}+\frac {e}{2}\right )-12 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \cot \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}\)	\(95\)
derivativedivides	\(\frac {-a^{2} c \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+a^{2} c \cot \left (f x +e \right )+a^{2} c \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )}{f}\)	\(100\)
default	\(\frac {-a^{2} c \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )+a^{2} c \cot \left (f x +e \right )+a^{2} c \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )}{2}\right )+a^{2} c \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )}{f}\)	\(100\)
risch	\(\frac {a^{2} c \left (6 i {\mathrm e}^{4 i \left (f x +e \right )}+3 \,{\mathrm e}^{5 i \left (f x +e \right )}+2 i-3 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{3}}+\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{2 f}-\frac {a^{2} c \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{2 f}\)	\(103\)
norman	\(\frac {-\frac {a^{2} c}{24 f}-\frac {3 a^{2} c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {7 a^{2} c \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {11 a^{2} c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {a^{2} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {a^{2} c \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {a^{2} c \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {a^{2} c \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {a^{2} c \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}-\frac {a^{2} c \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}\)	\(216\)

[In]

int(csc(f*x+e)^4*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/24*a^2*c*(tan(1/2*f*x+1/2*e)^3-cot(1/2*f*x+1/2*e)^3+3*tan(1/2*f*x+1/2*e)^2-3*cot(1/2*f*x+1/2*e)^2-3*tan(1/2*
f*x+1/2*e)-12*ln(tan(1/2*f*x+1/2*e))+3*cot(1/2*f*x+1/2*e))/f

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 137 vs. \(2 (55) = 110\).

Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.25 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {4 \, a^{2} c \cos \left (f x + e\right )^{3} + 6 \, a^{2} c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 3 \, {\left (a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right ) - 3 \, {\left (a^{2} c \cos \left (f x + e\right )^{2} - a^{2} c\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) \sin \left (f x + e\right )}{12 \, {\left (f \cos \left (f x + e\right )^{2} - f\right )} \sin \left (f x + e\right )} \]

[In]

integrate(csc(f*x+e)^4*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm="fricas")

[Out]

1/12*(4*a^2*c*cos(f*x + e)^3 + 6*a^2*c*cos(f*x + e)*sin(f*x + e) + 3*(a^2*c*cos(f*x + e)^2 - a^2*c)*log(1/2*co
s(f*x + e) + 1/2)*sin(f*x + e) - 3*(a^2*c*cos(f*x + e)^2 - a^2*c)*log(-1/2*cos(f*x + e) + 1/2)*sin(f*x + e))/(
(f*cos(f*x + e)^2 - f)*sin(f*x + e))

Sympy [F]

\[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=- a^{2} c \left (\int \left (- \sin {\left (e + f x \right )} \csc ^{4}{\left (e + f x \right )}\right )\, dx + \int \sin ^{2}{\left (e + f x \right )} \csc ^{4}{\left (e + f x \right )}\, dx + \int \sin ^{3}{\left (e + f x \right )} \csc ^{4}{\left (e + f x \right )}\, dx + \int \left (- \csc ^{4}{\left (e + f x \right )}\right )\, dx\right ) \]

[In]

integrate(csc(f*x+e)**4*(a+a*sin(f*x+e))**2*(c-c*sin(f*x+e)),x)

[Out]

-a**2*c*(Integral(-sin(e + f*x)*csc(e + f*x)**4, x) + Integral(sin(e + f*x)**2*csc(e + f*x)**4, x) + Integral(
sin(e + f*x)**3*csc(e + f*x)**4, x) + Integral(-csc(e + f*x)**4, x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 120 vs. \(2 (55) = 110\).

Time = 0.20 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.97 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {3 \, a^{2} c {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + 6 \, a^{2} c {\left (\log \left (\cos \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + \frac {12 \, a^{2} c}{\tan \left (f x + e\right )} - \frac {4 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} c}{\tan \left (f x + e\right )^{3}}}{12 \, f} \]

[In]

integrate(csc(f*x+e)^4*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (55) = 110\).

Time = 0.31 (sec) , antiderivative size = 139, normalized size of antiderivative = 2.28 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{2} c \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + \frac {22 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, a^{2} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{2} c}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3}}}{24 \, f} \]

[In]

integrate(csc(f*x+e)^4*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm="giac")

[Out]

1/24*(a^2*c*tan(1/2*f*x + 1/2*e)^3 + 3*a^2*c*tan(1/2*f*x + 1/2*e)^2 - 12*a^2*c*log(abs(tan(1/2*f*x + 1/2*e)))
- 3*a^2*c*tan(1/2*f*x + 1/2*e) + (22*a^2*c*tan(1/2*f*x + 1/2*e)^3 + 3*a^2*c*tan(1/2*f*x + 1/2*e)^2 - 3*a^2*c*t
an(1/2*f*x + 1/2*e) - a^2*c)/tan(1/2*f*x + 1/2*e)^3)/f

Mupad [B] (verification not implemented)

Time = 11.69 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.16 \[ \int \csc ^4(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{8\,f}-\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{8\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-c\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+c\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {c\,a^2}{3}\right )}{8\,f}+\frac {a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{24\,f}-\frac {a^2\,c\,\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{2\,f} \]

[In]

int(((a + a*sin(e + f*x))^2*(c - c*sin(e + f*x)))/sin(e + f*x)^4,x)

[Out]

(a^2*c*tan(e/2 + (f*x)/2)^2)/(8*f) - (a^2*c*tan(e/2 + (f*x)/2))/(8*f) - (cot(e/2 + (f*x)/2)^3*((a^2*c)/3 + a^2
*c*tan(e/2 + (f*x)/2) - a^2*c*tan(e/2 + (f*x)/2)^2))/(8*f) + (a^2*c*tan(e/2 + (f*x)/2)^3)/(24*f) - (a^2*c*log(
tan(e/2 + (f*x)/2)))/(2*f)