\(\int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx\) [174]

Optimal result

Integrand size = 21, antiderivative size = 134 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {3 a \left (a^2+4 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a \left (a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f} \]

[Out]

-3/8*a*(a^2+4*b^2)*arctanh(cos(f*x+e))/f-b*(2*a^2+b^2)*cot(f*x+e)/f-3/8*a*(a^2+4*b^2)*cot(f*x+e)*csc(f*x+e)/f-
3/4*a^2*b*cot(f*x+e)*csc(f*x+e)^2/f-1/4*a^2*cot(f*x+e)*csc(f*x+e)^3*(a+b*sin(f*x+e))/f

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2871, 3100, 2827, 3853, 3855, 3852, 8} \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=-\frac {3 a \left (a^2+4 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a \left (a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f} \]

[In]

Int[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^3,x]

[Out]

(-3*a*(a^2 + 4*b^2)*ArcTanh[Cos[e + f*x]])/(8*f) - (b*(2*a^2 + b^2)*Cot[e + f*x])/f - (3*a*(a^2 + 4*b^2)*Cot[e
+ f*x]*Csc[e + f*x])/(8*f) - (3*a^2*b*Cot[e + f*x]*Csc[e + f*x]^2)/(4*f) - (a^2*Cot[e + f*x]*Csc[e + f*x]^3*(
a + b*Sin[e + f*x]))/(4*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2827

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2871

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-(b^2*c^2 - 2*a*b*c*d + a^2*d^2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/
(d*f*(n + 1)*(c^2 - d^2))), x] + Dist[1/(d*(n + 1)*(c^2 - d^2)), Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e
+ f*x])^(n + 1)*Simp[b*(m - 2)*(b*c - a*d)^2 + a*d*(n + 1)*(c*(a^2 + b^2) - 2*a*b*d) + (b*(n + 1)*(a*b*c^2 +
c*d*(a^2 + b^2) - 3*a*b*d^2) - a*(n + 2)*(b*c - a*d)^2)*Sin[e + f*x] + b*(b^2*(c^2 - d^2) - m*(b*c - a*d)^2 +
d*n*(2*a*b*c - d*(a^2 + b^2)))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && LtQ[n, -1] && (IntegerQ[m] || IntegersQ[2*m, 2*n])

Rule 3100

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f
_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m
+ 1)*(a^2 - b^2))), x] + Dist[1/(b*(m + 1)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*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}& = -\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}+\frac {1}{4} \int \csc ^4(e+f x) \left (9 a^2 b+3 a \left (a^2+4 b^2\right ) \sin (e+f x)+2 b \left (a^2+2 b^2\right ) \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}+\frac {1}{12} \int \csc ^3(e+f x) \left (9 a \left (a^2+4 b^2\right )+12 b \left (2 a^2+b^2\right ) \sin (e+f x)\right ) \, dx \\ & = -\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}+\left (b \left (2 a^2+b^2\right )\right ) \int \csc ^2(e+f x) \, dx+\frac {1}{4} \left (3 a \left (a^2+4 b^2\right )\right ) \int \csc ^3(e+f x) \, dx \\ & = -\frac {3 a \left (a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f}+\frac {1}{8} \left (3 a \left (a^2+4 b^2\right )\right ) \int \csc (e+f x) \, dx-\frac {\left (b \left (2 a^2+b^2\right )\right ) \text {Subst}(\int 1 \, dx,x,\cot (e+f x))}{f} \\ & = -\frac {3 a \left (a^2+4 b^2\right ) \text {arctanh}(\cos (e+f x))}{8 f}-\frac {b \left (2 a^2+b^2\right ) \cot (e+f x)}{f}-\frac {3 a \left (a^2+4 b^2\right ) \cot (e+f x) \csc (e+f x)}{8 f}-\frac {3 a^2 b \cot (e+f x) \csc ^2(e+f x)}{4 f}-\frac {a^2 \cot (e+f x) \csc ^3(e+f x) (a+b \sin (e+f x))}{4 f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(322\) vs. \(2(134)=268\).

Time = 7.77 (sec) , antiderivative size = 322, normalized size of antiderivative = 2.40 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {\left (-2 a^2 b \cos \left (\frac {1}{2} (e+f x)\right )-b^3 \cos \left (\frac {1}{2} (e+f x)\right )\right ) \csc \left (\frac {1}{2} (e+f x)\right )}{2 f}-\frac {3 \left (a^3+4 a b^2\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}-\frac {a^2 b \cot \left (\frac {1}{2} (e+f x)\right ) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{8 f}-\frac {a^3 \csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}-\frac {3 \left (a^3+4 a b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {3 \left (a^3+4 a b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f}+\frac {3 \left (a^3+4 a b^2\right ) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 f}+\frac {a^3 \sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 f}+\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (2 a^2 b \sin \left (\frac {1}{2} (e+f x)\right )+b^3 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{2 f}+\frac {a^2 b \sec ^2\left (\frac {1}{2} (e+f x)\right ) \tan \left (\frac {1}{2} (e+f x)\right )}{8 f} \]

[In]

Integrate[Csc[e + f*x]^5*(a + b*Sin[e + f*x])^3,x]

[Out]

((-2*a^2*b*Cos[(e + f*x)/2] - b^3*Cos[(e + f*x)/2])*Csc[(e + f*x)/2])/(2*f) - (3*(a^3 + 4*a*b^2)*Csc[(e + f*x)
/2]^2)/(32*f) - (a^2*b*Cot[(e + f*x)/2]*Csc[(e + f*x)/2]^2)/(8*f) - (a^3*Csc[(e + f*x)/2]^4)/(64*f) - (3*(a^3
+ 4*a*b^2)*Log[Cos[(e + f*x)/2]])/(8*f) + (3*(a^3 + 4*a*b^2)*Log[Sin[(e + f*x)/2]])/(8*f) + (3*(a^3 + 4*a*b^2)
*Sec[(e + f*x)/2]^2)/(32*f) + (a^3*Sec[(e + f*x)/2]^4)/(64*f) + (Sec[(e + f*x)/2]*(2*a^2*b*Sin[(e + f*x)/2] +
b^3*Sin[(e + f*x)/2]))/(2*f) + (a^2*b*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/(8*f)

Maple [A] (verified)

Time = 2.65 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {a^{3} \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{8}\right )+3 a^{2} b \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+3 a \,b^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-b^{3} \cot \left (f x +e \right )}{f}\)	\(129\)
default	\(\frac {a^{3} \left (\left (-\frac {\left (\csc ^{3}\left (f x +e \right )\right )}{4}-\frac {3 \csc \left (f x +e \right )}{8}\right ) \cot \left (f x +e \right )+\frac {3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{8}\right )+3 a^{2} b \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (f x +e \right )\right )}{3}\right ) \cot \left (f x +e \right )+3 a \,b^{2} \left (-\frac {\csc \left (f x +e \right ) \cot \left (f x +e \right )}{2}+\frac {\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )}{2}\right )-b^{3} \cot \left (f x +e \right )}{f}\)	\(129\)
parallelrisch	\(\frac {\left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}-\left (\cot ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}+8 \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2} b -8 \left (\cot ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{2} b +8 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}+24 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a \,b^{2}-8 \left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a^{3}-24 \left (\cot ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a \,b^{2}+72 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b +32 \tan \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{3}+24 a^{3} \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )+96 \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right ) a \,b^{2}-72 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) a^{2} b -32 \cot \left (\frac {f x}{2}+\frac {e}{2}\right ) b^{3}}{64 f}\)	\(227\)
risch	\(-\frac {i \left (3 i a^{3} {\mathrm e}^{7 i \left (f x +e \right )}+12 i a \,b^{2} {\mathrm e}^{7 i \left (f x +e \right )}-11 i a^{3} {\mathrm e}^{5 i \left (f x +e \right )}-12 i a \,b^{2} {\mathrm e}^{5 i \left (f x +e \right )}+8 b^{3} {\mathrm e}^{6 i \left (f x +e \right )}-11 i a^{3} {\mathrm e}^{3 i \left (f x +e \right )}-12 i a \,b^{2} {\mathrm e}^{3 i \left (f x +e \right )}-48 a^{2} b \,{\mathrm e}^{4 i \left (f x +e \right )}-24 b^{3} {\mathrm e}^{4 i \left (f x +e \right )}+3 i a^{3} {\mathrm e}^{i \left (f x +e \right )}+12 i b^{2} a \,{\mathrm e}^{i \left (f x +e \right )}+64 a^{2} b \,{\mathrm e}^{2 i \left (f x +e \right )}+24 b^{3} {\mathrm e}^{2 i \left (f x +e \right )}-16 a^{2} b -8 b^{3}\right )}{4 f \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )^{4}}-\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )}{8 f}-\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) b^{2}}{2 f}+\frac {3 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{8 f}+\frac {3 a \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) b^{2}}{2 f}\)	\(311\)
norman	\(\frac {-\frac {a^{3}}{64 f}+\frac {a^{3} \left (\tan ^{14}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {\left (8 a^{3}+21 a \,b^{2}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {\left (27 a^{3}+72 a \,b^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {\left (49 a^{3}+132 a \,b^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{32 f}-\frac {a \left (11 a^{2}+24 b^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}+\frac {a \left (11 a^{2}+24 b^{2}\right ) \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{64 f}-\frac {a^{2} b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}+\frac {a^{2} b \left (\tan ^{13}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}-\frac {b \left (3 a^{2}+b^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {b \left (3 a^{2}+b^{2}\right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {b \left (21 a^{2}+8 b^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {b \left (21 a^{2}+8 b^{2}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}+\frac {3 a \left (a^{2}+4 b^{2}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}\)	\(368\)

[In]

int(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(a^3*((-1/4*csc(f*x+e)^3-3/8*csc(f*x+e))*cot(f*x+e)+3/8*ln(-cot(f*x+e)+csc(f*x+e)))+3*a^2*b*(-2/3-1/3*csc(
f*x+e)^2)*cot(f*x+e)+3*a*b^2*(-1/2*csc(f*x+e)*cot(f*x+e)+1/2*ln(-cot(f*x+e)+csc(f*x+e)))-b^3*cot(f*x+e))

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.78 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {6 \, {\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{3} - 2 \, {\left (5 \, a^{3} + 12 \, a b^{2}\right )} \cos \left (f x + e\right ) - 3 \, {\left ({\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{3} + 4 \, a b^{2} - 2 \, {\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 3 \, {\left ({\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{4} + a^{3} + 4 \, a b^{2} - 2 \, {\left (a^{3} + 4 \, a b^{2}\right )} \cos \left (f x + e\right )^{2}\right )} \log \left (-\frac {1}{2} \, \cos \left (f x + e\right ) + \frac {1}{2}\right ) + 16 \, {\left ({\left (2 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{16 \, {\left (f \cos \left (f x + e\right )^{4} - 2 \, f \cos \left (f x + e\right )^{2} + f\right )}} \]

[In]

integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

1/16*(6*(a^3 + 4*a*b^2)*cos(f*x + e)^3 - 2*(5*a^3 + 12*a*b^2)*cos(f*x + e) - 3*((a^3 + 4*a*b^2)*cos(f*x + e)^4
+ a^3 + 4*a*b^2 - 2*(a^3 + 4*a*b^2)*cos(f*x + e)^2)*log(1/2*cos(f*x + e) + 1/2) + 3*((a^3 + 4*a*b^2)*cos(f*x
+ e)^4 + a^3 + 4*a*b^2 - 2*(a^3 + 4*a*b^2)*cos(f*x + e)^2)*log(-1/2*cos(f*x + e) + 1/2) + 16*((2*a^2*b + b^3)*
cos(f*x + e)^3 - (3*a^2*b + b^3)*cos(f*x + e))*sin(f*x + e))/(f*cos(f*x + e)^4 - 2*f*cos(f*x + e)^2 + f)

Sympy [F(-1)]

Timed out. \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\text {Timed out} \]

[In]

integrate(csc(f*x+e)**5*(a+b*sin(f*x+e))**3,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.21 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} {\left (\frac {2 \, {\left (3 \, \cos \left (f x + e\right )^{3} - 5 \, \cos \left (f x + e\right )\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1} - 3 \, \log \left (\cos \left (f x + e\right ) + 1\right ) + 3 \, \log \left (\cos \left (f x + e\right ) - 1\right )\right )} + 12 \, a b^{2} {\left (\frac {2 \, \cos \left (f x + e\right )}{\cos \left (f x + e\right )^{2} - 1} - \log \left (\cos \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right ) - 1\right )\right )} - \frac {16 \, b^{3}}{\tan \left (f x + e\right )} - \frac {16 \, {\left (3 \, \tan \left (f x + e\right )^{2} + 1\right )} a^{2} b}{\tan \left (f x + e\right )^{3}}}{16 \, f} \]

[In]

integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

1/16*(a^3*(2*(3*cos(f*x + e)^3 - 5*cos(f*x + e))/(cos(f*x + e)^4 - 2*cos(f*x + e)^2 + 1) - 3*log(cos(f*x + e)
+ 1) + 3*log(cos(f*x + e) - 1)) + 12*a*b^2*(2*cos(f*x + e)/(cos(f*x + e)^2 - 1) - log(cos(f*x + e) + 1) + log(
cos(f*x + e) - 1)) - 16*b^3/tan(f*x + e) - 16*(3*tan(f*x + e)^2 + 1)*a^2*b/tan(f*x + e)^3)/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (126) = 252\).

Time = 0.34 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.90 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 72 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 32 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, {\left (a^{3} + 4 \, a b^{2}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}\right ) - \frac {50 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 200 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 72 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 32 \, b^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a b^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 8 \, a^{2} b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a^{3}}{\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}}{64 \, f} \]

[In]

integrate(csc(f*x+e)^5*(a+b*sin(f*x+e))^3,x, algorithm="giac")

[Out]

1/64*(a^3*tan(1/2*f*x + 1/2*e)^4 + 8*a^2*b*tan(1/2*f*x + 1/2*e)^3 + 8*a^3*tan(1/2*f*x + 1/2*e)^2 + 24*a*b^2*ta
n(1/2*f*x + 1/2*e)^2 + 72*a^2*b*tan(1/2*f*x + 1/2*e) + 32*b^3*tan(1/2*f*x + 1/2*e) + 24*(a^3 + 4*a*b^2)*log(ab
s(tan(1/2*f*x + 1/2*e))) - (50*a^3*tan(1/2*f*x + 1/2*e)^4 + 200*a*b^2*tan(1/2*f*x + 1/2*e)^4 + 72*a^2*b*tan(1/
2*f*x + 1/2*e)^3 + 32*b^3*tan(1/2*f*x + 1/2*e)^3 + 8*a^3*tan(1/2*f*x + 1/2*e)^2 + 24*a*b^2*tan(1/2*f*x + 1/2*e
)^2 + 8*a^2*b*tan(1/2*f*x + 1/2*e) + a^3)/tan(1/2*f*x + 1/2*e)^4)/f

Mupad [B] (verification not implemented)

Time = 6.67 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.51 \[ \int \csc ^5(e+f x) (a+b \sin (e+f x))^3 \, dx=\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{64\,f}-\frac {{\mathrm {cot}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,a^3+6\,a\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (18\,a^2\,b+8\,b^3\right )+\frac {a^3}{4}+2\,a^2\,b\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}{16\,f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {a^3}{8}+\frac {3\,a\,b^2}{8}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (\frac {3\,a^3}{8}+\frac {3\,a\,b^2}{2}\right )}{f}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {9\,a^2\,b}{8}+\frac {b^3}{2}\right )}{f}+\frac {a^2\,b\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3}{8\,f} \]

[In]

int((a + b*sin(e + f*x))^3/sin(e + f*x)^5,x)

[Out]

(a^3*tan(e/2 + (f*x)/2)^4)/(64*f) - (cot(e/2 + (f*x)/2)^4*(tan(e/2 + (f*x)/2)^2*(6*a*b^2 + 2*a^3) + tan(e/2 +
(f*x)/2)^3*(18*a^2*b + 8*b^3) + a^3/4 + 2*a^2*b*tan(e/2 + (f*x)/2)))/(16*f) + (tan(e/2 + (f*x)/2)^2*((3*a*b^2)
/8 + a^3/8))/f + (log(tan(e/2 + (f*x)/2))*((3*a*b^2)/2 + (3*a^3)/8))/f + (tan(e/2 + (f*x)/2)*((9*a^2*b)/8 + b^
3/2))/f + (a^2*b*tan(e/2 + (f*x)/2)^3)/(8*f)