\(\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [200]
Optimal result
Integrand size = 22, antiderivative size = 136 \[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}+\sqrt {b}} d}
\]
[Out]
-arctanh(cos(d*x+c))/a/d-1/2*b^(1/4)*arctan(b^(1/4)*cos(d*x+c)/(a^(1/2)-b^(1/2))^(1/2))/a/d/(a^(1/2)-b^(1/2))^
(1/2)+1/2*b^(1/4)*arctanh(b^(1/4)*cos(d*x+c)/(a^(1/2)+b^(1/2))^(1/2))/a/d/(a^(1/2)+b^(1/2))^(1/2)
Rubi [A] (verified)
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3294, 1184, 213, 1180, 211,
214} \[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {\text {arctanh}(\cos (c+d x))}{a d}
\]
[In]
Int[Csc[c + d*x]/(a - b*Sin[c + d*x]^4),x]
[Out]
-1/2*(b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(a*Sqrt[Sqrt[a] - Sqrt[b]]*d) - ArcTanh[
Cos[c + d*x]]/(a*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*a*Sqrt[Sqrt[a] + Sq
rt[b]]*d)
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
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 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 1180
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]
Rule 1184
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x
^2)^q/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a
*e^2, 0] && IntegerQ[q]
Rule 3294
Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]
Rubi steps \begin{align*}
\text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )} \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {1}{a \left (-1+x^2\right )}+\frac {b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{a d}-\frac {\text {Subst}\left (\int \frac {b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{a d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a d}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a d} \\ & = -\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}-\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}-\sqrt {b}} d}-\frac {\text {arctanh}(\cos (c+d x))}{a d}+\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \cos (c+d x)}{\sqrt {\sqrt {a}+\sqrt {b}}}\right )}{2 a \sqrt {\sqrt {a}+\sqrt {b}} d} \\
\end{align*}
Mathematica [C] (verified)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.69 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.34
\[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {8 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-8 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+i b \text {RootSum}\left [b-4 b \text {$\#$1}^2-16 a \text {$\#$1}^4+6 b \text {$\#$1}^4-4 b \text {$\#$1}^6+b \text {$\#$1}^8\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2-6 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4+3 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^6-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^6}{-b \text {$\#$1}-8 a \text {$\#$1}^3+3 b \text {$\#$1}^3-3 b \text {$\#$1}^5+b \text {$\#$1}^7}\&\right ]}{8 a d}
\]
[In]
Integrate[Csc[c + d*x]/(a - b*Sin[c + d*x]^4),x]
[Out]
-1/8*(8*Log[Cos[(c + d*x)/2]] - 8*Log[Sin[(c + d*x)/2]] + I*b*RootSum[b - 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*
b*#1^6 + b*#1^8 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 6*Arc
Tan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - 6*ArcTan[Sin[c + d
*x]/(Cos[c + d*x] - #1)]*#1^4 + (3*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + 2*ArcTan[Sin[c + d*x]/(Cos[c +
d*x] - #1)]*#1^6 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^
7) & ])/(a*d)
Maple [A] (verified)
Time = 0.94 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.88
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {-\frac {b^{2} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a}}{d}\) |
\(119\) |
| default |
\(\frac {-\frac {b^{2} \left (\frac {\arctan \left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}-b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}-b \right ) b}}-\frac {\operatorname {arctanh}\left (\frac {\cos \left (d x +c \right ) b}{\sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{2 b \sqrt {\left (\sqrt {a b}+b \right ) b}}\right )}{a}+\frac {\ln \left (\cos \left (d x +c \right )-1\right )}{2 a}-\frac {\ln \left (1+\cos \left (d x +c \right )\right )}{2 a}}{d}\) |
\(119\) |
| risch |
\(\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{d a}+2 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 a^{5} d^{4}-4096 a^{4} b \,d^{4}\right ) \textit {\_Z}^{4}-128 a^{2} b \,d^{2} \textit {\_Z}^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {1024 i d^{3} a^{4}}{b}+1024 i a^{3} d^{3}\right ) \textit {\_R}^{3}+32 i a d \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}+1\right )\right )\) |
\(143\) |
| | |
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[In]
int(csc(d*x+c)/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/a*b^2*(1/2/b/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/2/b/(((a*b)^(1
/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2)))+1/2/a*ln(cos(d*x+c)-1)-1/2/a*ln(1+cos(d*x+c))
)
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 773 vs. \(2 (100) = 200\).
Time = 0.37 (sec) , antiderivative size = 773, normalized size of antiderivative = 5.68
\[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) - a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) - a d \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (-b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - a b d\right )} \sqrt {-\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) + a d \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}} \log \left (-b \cos \left (d x + c\right ) - {\left ({\left (a^{4} - a^{3} b\right )} d^{3} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} + a b d\right )} \sqrt {\frac {{\left (a^{3} - a^{2} b\right )} d^{2} \sqrt {\frac {b}{{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} d^{4}}} - b}{{\left (a^{3} - a^{2} b\right )} d^{2}}}\right ) - 2 \, \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, a d}
\]
[In]
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
[Out]
1/4*(a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(b*cos(
d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - a*b*d)*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b
/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))) - a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^
4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b
+ a^3*b^2)*d^4)) + a*b*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*
d^2))) - a*d*sqrt(-((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))*log(-b
*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - a*b*d)*sqrt(-((a^3 - a^2*b)*d^2*s
qrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) + b)/((a^3 - a^2*b)*d^2))) + a*d*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 -
2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a^2*b)*d^2))*log(-b*cos(d*x + c) - ((a^4 - a^3*b)*d^3*sqrt(b/((a^5 - 2*
a^4*b + a^3*b^2)*d^4)) + a*b*d)*sqrt(((a^3 - a^2*b)*d^2*sqrt(b/((a^5 - 2*a^4*b + a^3*b^2)*d^4)) - b)/((a^3 - a
^2*b)*d^2))) - 2*log(1/2*cos(d*x + c) + 1/2) + 2*log(-1/2*cos(d*x + c) + 1/2))/(a*d)
Sympy [F]
\[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\csc {\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx
\]
[In]
integrate(csc(d*x+c)/(a-b*sin(d*x+c)**4),x)
[Out]
Integral(csc(c + d*x)/(a - b*sin(c + d*x)**4), x)
Maxima [F]
\[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )}{b \sin \left (d x + c\right )^{4} - a} \,d x }
\]
[In]
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
[Out]
-1/2*(2*a*d*integrate(-2*(12*b^2*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) - 4*b^2*cos(d*x + c)*sin(2*d*x + 2*c) + 4*b
^2*cos(2*d*x + 2*c)*sin(d*x + c) - b^2*sin(d*x + c) + (b^2*sin(7*d*x + 7*c) - 3*b^2*sin(5*d*x + 5*c) + 3*b^2*s
in(3*d*x + 3*c) - b^2*sin(d*x + c))*cos(8*d*x + 8*c) + 2*(2*b^2*sin(6*d*x + 6*c) + 2*b^2*sin(2*d*x + 2*c) + (8
*a*b - 3*b^2)*sin(4*d*x + 4*c))*cos(7*d*x + 7*c) + 4*(3*b^2*sin(5*d*x + 5*c) - 3*b^2*sin(3*d*x + 3*c) + b^2*si
n(d*x + c))*cos(6*d*x + 6*c) - 6*(2*b^2*sin(2*d*x + 2*c) + (8*a*b - 3*b^2)*sin(4*d*x + 4*c))*cos(5*d*x + 5*c)
- 2*(3*(8*a*b - 3*b^2)*sin(3*d*x + 3*c) - (8*a*b - 3*b^2)*sin(d*x + c))*cos(4*d*x + 4*c) - (b^2*cos(7*d*x + 7*
c) - 3*b^2*cos(5*d*x + 5*c) + 3*b^2*cos(3*d*x + 3*c) - b^2*cos(d*x + c))*sin(8*d*x + 8*c) - (4*b^2*cos(6*d*x +
6*c) + 4*b^2*cos(2*d*x + 2*c) - b^2 + 2*(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*sin(7*d*x + 7*c) - 4*(3*b^2*cos(5*d
*x + 5*c) - 3*b^2*cos(3*d*x + 3*c) + b^2*cos(d*x + c))*sin(6*d*x + 6*c) + 3*(4*b^2*cos(2*d*x + 2*c) - b^2 + 2*
(8*a*b - 3*b^2)*cos(4*d*x + 4*c))*sin(5*d*x + 5*c) + 2*(3*(8*a*b - 3*b^2)*cos(3*d*x + 3*c) - (8*a*b - 3*b^2)*c
os(d*x + c))*sin(4*d*x + 4*c) - 3*(4*b^2*cos(2*d*x + 2*c) - b^2)*sin(3*d*x + 3*c))/(a*b^2*cos(8*d*x + 8*c)^2 +
16*a*b^2*cos(6*d*x + 6*c)^2 + 16*a*b^2*cos(2*d*x + 2*c)^2 + a*b^2*sin(8*d*x + 8*c)^2 + 16*a*b^2*sin(6*d*x + 6
*c)^2 + 16*a*b^2*sin(2*d*x + 2*c)^2 - 8*a*b^2*cos(2*d*x + 2*c) + a*b^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*cos(4
*d*x + 4*c)^2 + 4*(64*a^3 - 48*a^2*b + 9*a*b^2)*sin(4*d*x + 4*c)^2 + 16*(8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c)*s
in(2*d*x + 2*c) - 2*(4*a*b^2*cos(6*d*x + 6*c) + 4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*cos(4
*d*x + 4*c))*cos(8*d*x + 8*c) + 8*(4*a*b^2*cos(2*d*x + 2*c) - a*b^2 + 2*(8*a^2*b - 3*a*b^2)*cos(4*d*x + 4*c))*
cos(6*d*x + 6*c) - 4*(8*a^2*b - 3*a*b^2 - 4*(8*a^2*b - 3*a*b^2)*cos(2*d*x + 2*c))*cos(4*d*x + 4*c) - 4*(2*a*b^
2*sin(6*d*x + 6*c) + 2*a*b^2*sin(2*d*x + 2*c) + (8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c))*sin(8*d*x + 8*c) + 16*(2
*a*b^2*sin(2*d*x + 2*c) + (8*a^2*b - 3*a*b^2)*sin(4*d*x + 4*c))*sin(6*d*x + 6*c)), x) + log(cos(d*x)^2 + 2*cos
(d*x)*cos(c) + cos(c)^2 + sin(d*x)^2 - 2*sin(d*x)*sin(c) + sin(c)^2) - log(cos(d*x)^2 - 2*cos(d*x)*cos(c) + co
s(c)^2 + sin(d*x)^2 + 2*sin(d*x)*sin(c) + sin(c)^2))/(a*d)
Giac [F]
\[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int { -\frac {\csc \left (d x + c\right )}{b \sin \left (d x + c\right )^{4} - a} \,d x }
\]
[In]
integrate(csc(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="giac")
[Out]
sage0*x
Mupad [B] (verification not implemented)
Time = 15.64 (sec) , antiderivative size = 2031, normalized size of antiderivative = 14.93
\[
\int \frac {\csc (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
int(1/(sin(c + d*x)*(a - b*sin(c + d*x)^4)),x)
[Out]
- (atan(((((((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*(256*a^4*b^4 - 192*a^3*b^5 + cos(c + d*x)*(768*
a^4*b^5 - 512*a^5*b^4)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b^5*cos(c + d*x))*((a^2*b
+ (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 12*a*b^5)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 6
*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*1i + (((((a^2*b + (a^5*b)^(1/2))/(16*(a^
4*b - a^5)))^(1/2)*(192*a^3*b^5 - 256*a^4*b^4 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b + (a^5*b)^(1/
2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)
- 12*a*b^5)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 6*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/
(16*(a^4*b - a^5)))^(1/2)*1i)/((((((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*(256*a^4*b^4 - 192*a^3*b^
5 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b^5
*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 12*a*b^5)*((a^2*b + (a^5*b)^(1/2))/(16*(a^
4*b - a^5)))^(1/2) + 6*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - (((((a^2*b + (a^
5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*(192*a^3*b^5 - 256*a^4*b^4 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((
a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)) - 144*a^2*b^5*cos(c + d*x))*((a^2*b + (a^5*b)^(1/2))/(16*(a^
4*b - a^5)))^(1/2) - 12*a*b^5)*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 6*b^5*cos(c + d*x))*((a^2*
b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)))*((a^2*b + (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*2i)/d - atan
h((32*b^4*cos(c + d*x))/(32*b^4 - (18*b^5)/a) - (18*b^5*cos(c + d*x))/(32*a*b^4 - 18*b^5))/(a*d) - (atan(((6*b
^5*cos(c + d*x) + (((256*a^4*b^4 - 192*a^3*b^5 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b - (a^5*b)^(1
/2))/(16*(a^4*b - a^5)))^(1/2))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - 144*a^2*b^5*cos(c + d*x))
*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 12*a*b^5)*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(
1/2))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*1i + (6*b^5*cos(c + d*x) + (((192*a^3*b^5 - 256*a^4*b
^4 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2))*((a^2*b - (a
^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - 144*a^2*b^5*cos(c + d*x))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5))
)^(1/2) - 12*a*b^5)*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b -
a^5)))^(1/2)*1i)/((6*b^5*cos(c + d*x) + (((256*a^4*b^4 - 192*a^3*b^5 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4
)*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - 144
*a^2*b^5*cos(c + d*x))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) + 12*a*b^5)*((a^2*b - (a^5*b)^(1/2))
/(16*(a^4*b - a^5)))^(1/2))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - (6*b^5*cos(c + d*x) + (((192*
a^3*b^5 - 256*a^4*b^4 + cos(c + d*x)*(768*a^4*b^5 - 512*a^5*b^4)*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^
(1/2))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2) - 144*a^2*b^5*cos(c + d*x))*((a^2*b - (a^5*b)^(1/2))
/(16*(a^4*b - a^5)))^(1/2) - 12*a*b^5)*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2))*((a^2*b - (a^5*b)^(
1/2))/(16*(a^4*b - a^5)))^(1/2)))*((a^2*b - (a^5*b)^(1/2))/(16*(a^4*b - a^5)))^(1/2)*2i)/d