3.2.86 \(\int \frac {\csc (c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [186]

Optimal. Leaf size=264 \[ -\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {a^{2/3}-b^{2/3}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a d}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 a \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} d}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 a \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} d} \]

[Out]

-arctanh(cos(d*x+c))/a/d-2/3*b^(1/3)*arctan((b^(1/3)+a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)-b^(2/3))^(1/2))/a/d/
(a^(2/3)-b^(2/3))^(1/2)+2/3*b^(1/3)*arctanh((b^(1/3)+(-1)^(2/3)*a^(1/3)*tan(1/2*d*x+1/2*c))/((-1)^(1/3)*a^(2/3
)+b^(2/3))^(1/2))/a/d/((-1)^(1/3)*a^(2/3)+b^(2/3))^(1/2)+2/3*b^(1/3)*arctanh((b^(1/3)-(-1)^(1/3)*a^(1/3)*tan(1
/2*d*x+1/2*c))/(-(-1)^(2/3)*a^(2/3)+b^(2/3))^(1/2))/a/d/(-(-1)^(2/3)*a^(2/3)+b^(2/3))^(1/2)

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Rubi [A]
time = 0.26, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3299, 3855, 2739, 632, 210, 212} \begin {gather*} -\frac {2 \sqrt [3]{b} \text {ArcTan}\left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^{2/3}-(-1)^{2/3} a^{2/3}}}\right )}{3 a d \sqrt {b^{2/3}-(-1)^{2/3} a^{2/3}}}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 a d \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}-\frac {\tanh ^{-1}(\cos (c+d x))}{a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]/(a + b*Sin[c + d*x]^3),x]

[Out]

(-2*b^(1/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - b^(2/3)]])/(3*a*Sqrt[a^(2/3) - b^(2/3)]
*d) - ArcTanh[Cos[c + d*x]]/(a*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) - (-1)^(1/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[-(
(-1)^(2/3)*a^(2/3)) + b^(2/3)]])/(3*a*Sqrt[-((-1)^(2/3)*a^(2/3)) + b^(2/3)]*d) + (2*b^(1/3)*ArcTanh[(b^(1/3) +
(-1)^(2/3)*a^(1/3)*Tan[(c + d*x)/2])/Sqrt[(-1)^(1/3)*a^(2/3) + b^(2/3)]])/(3*a*Sqrt[(-1)^(1/3)*a^(2/3) + b^(2
/3)]*d)

Rule 210

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

Rule 212

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && 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*} \int \frac {\csc (c+d x)}{a+b \sin ^3(c+d x)} \, dx &=\int \left (\frac {\csc (c+d x)}{a}-\frac {b \sin ^2(c+d x)}{a \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx\\ &=\frac {\int \csc (c+d x) \, dx}{a}-\frac {b \int \frac {\sin ^2(c+d x)}{a+b \sin ^3(c+d x)} \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {b \int \left (\frac {1}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac {1}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}+\frac {1}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{a}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a}-\frac {\sqrt [3]{b} \int \frac {1}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a}-\frac {\sqrt [3]{b} \int \frac {1}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a d}-\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a d}-\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt [3]{-1} \sqrt [3]{a}+2 \sqrt [3]{b} x-\sqrt [3]{-1} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a d}-\frac {\left (2 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{(-1)^{2/3} \sqrt [3]{a}+2 \sqrt [3]{b} x+(-1)^{2/3} \sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a d}\\ &=-\frac {\tanh ^{-1}(\cos (c+d x))}{a d}+\frac {\left (4 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a d}+\frac {\left (4 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{-4 \left ((-1)^{2/3} a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}-2 \sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a d}+\frac {\left (4 \sqrt [3]{b}\right ) \text {Subst}\left (\int \frac {1}{4 \left (\sqrt [3]{-1} a^{2/3}+b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 (-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a d}\\ &=-\frac {2 \sqrt [3]{b} \tan ^{-1}\left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a \sqrt {a^{2/3}-b^{2/3}} d}-\frac {\tanh ^{-1}(\cos (c+d x))}{a d}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}}}\right )}{3 a \sqrt {-(-1)^{2/3} a^{2/3}+b^{2/3}} d}+\frac {2 \sqrt [3]{b} \tanh ^{-1}\left (\frac {\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}}}\right )}{3 a \sqrt {\sqrt [3]{-1} a^{2/3}+b^{2/3}} d}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.16, size = 264, normalized size = 1.00 \begin {gather*} -\frac {6 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-6 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+i b \text {RootSum}\left [-i b+3 i b \text {$\#$1}^2+8 a \text {$\#$1}^3-3 i b \text {$\#$1}^4+i b \text {$\#$1}^6\&,\frac {2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )-4 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2+2 i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2+2 \tan ^{-1}\left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^4-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^4}{b \text {$\#$1}-4 i a \text {$\#$1}^2-2 b \text {$\#$1}^3+b \text {$\#$1}^5}\&\right ]}{6 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]/(a + b*Sin[c + d*x]^3),x]

[Out]

-1/6*(6*Log[Cos[(c + d*x)/2]] - 6*Log[Sin[(c + d*x)/2]] + I*b*RootSum[(-I)*b + (3*I)*b*#1^2 + 8*a*#1^3 - (3*I)
*b*#1^4 + I*b*#1^6 & , (2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - 4*A
rcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 + (2*I)*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 + 2*ArcTan[Sin[c +
d*x]/(Cos[c + d*x] - #1)]*#1^4 - I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4)/(b*#1 - (4*I)*a*#1^2 - 2*b*#1^3 +
b*#1^5) & ])/(a*d)

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.75, size = 96, normalized size = 0.36

method	result	size
derivativedivides	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\)	\(96\)
default	\(\frac {\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\)	\(96\)
risch	\(2 i \left (\munderset {\textit {\_R} =\RootOf \left (\left (46656 a^{8} d^{6}-46656 b^{2} a^{6} d^{6}\right ) \textit {\_Z}^{6}-3888 b^{2} a^{4} d^{4} \textit {\_Z}^{4}-108 a^{2} b^{2} d^{2} \textit {\_Z}^{2}-b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (-\frac {7776 i a^{7} d^{5}}{b^{2}}+7776 i a^{5} d^{5}\right ) \textit {\_R}^{5}+\left (-\frac {1296 i a^{5} d^{4}}{b}+1296 i a^{3} b \,d^{4}\right ) \textit {\_R}^{4}+648 i a^{3} d^{3} \textit {\_R}^{3}+\left (\frac {36 i a^{3} d^{2}}{b}+72 i a b \,d^{2}\right ) \textit {\_R}^{2}+18 i a d \textit {\_R} +\frac {i b}{a}\right )\right )+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{a d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{a d}\)	\(222\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/a*ln(tan(1/2*d*x+1/2*c))-4/3/a*b*sum(_R^2/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=
RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="maxima")

[Out]

-1/2*(2*a*d*integrate(2*(16*a*b*cos(3*d*x + 3*c)^2 + 16*a*b*sin(3*d*x + 3*c)^2 + 3*b^2*cos(d*x + c)*sin(2*d*x
+ 2*c) - 3*b^2*cos(2*d*x + 2*c)*sin(d*x + c) + b^2*sin(d*x + c) - (b^2*sin(5*d*x + 5*c) - 2*b^2*sin(3*d*x + 3*
c) + b^2*sin(d*x + c))*cos(6*d*x + 6*c) - (8*a*b*cos(3*d*x + 3*c) + 3*b^2*sin(4*d*x + 4*c) - 3*b^2*sin(2*d*x +
2*c))*cos(5*d*x + 5*c) - 3*(2*b^2*sin(3*d*x + 3*c) - b^2*sin(d*x + c))*cos(4*d*x + 4*c) - 2*(4*a*b*cos(d*x +
c) + 3*b^2*sin(2*d*x + 2*c))*cos(3*d*x + 3*c) + (b^2*cos(5*d*x + 5*c) - 2*b^2*cos(3*d*x + 3*c) + b^2*cos(d*x +
c))*sin(6*d*x + 6*c) + (3*b^2*cos(4*d*x + 4*c) - 3*b^2*cos(2*d*x + 2*c) - 8*a*b*sin(3*d*x + 3*c) + b^2)*sin(5
*d*x + 5*c) + 3*(2*b^2*cos(3*d*x + 3*c) - b^2*cos(d*x + c))*sin(4*d*x + 4*c) + 2*(3*b^2*cos(2*d*x + 2*c) - 4*a
*b*sin(d*x + c) - b^2)*sin(3*d*x + 3*c))/(a*b^2*cos(6*d*x + 6*c)^2 + 9*a*b^2*cos(4*d*x + 4*c)^2 + 64*a^3*cos(3
*d*x + 3*c)^2 + 9*a*b^2*cos(2*d*x + 2*c)^2 + a*b^2*sin(6*d*x + 6*c)^2 + 9*a*b^2*sin(4*d*x + 4*c)^2 + 64*a^3*si
n(3*d*x + 3*c)^2 - 48*a^2*b*cos(3*d*x + 3*c)*sin(2*d*x + 2*c) + 9*a*b^2*sin(2*d*x + 2*c)^2 - 6*a*b^2*cos(2*d*x
+ 2*c) + a*b^2 - 2*(3*a*b^2*cos(4*d*x + 4*c) - 3*a*b^2*cos(2*d*x + 2*c) - 8*a^2*b*sin(3*d*x + 3*c) + a*b^2)*c
os(6*d*x + 6*c) - 6*(3*a*b^2*cos(2*d*x + 2*c) + 8*a^2*b*sin(3*d*x + 3*c) - a*b^2)*cos(4*d*x + 4*c) - 2*(8*a^2*
b*cos(3*d*x + 3*c) + 3*a*b^2*sin(4*d*x + 4*c) - 3*a*b^2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 6*(8*a^2*b*cos(3*
d*x + 3*c) - 3*a*b^2*sin(2*d*x + 2*c))*sin(4*d*x + 4*c) + 16*(3*a^2*b*cos(2*d*x + 2*c) - a^2*b)*sin(3*d*x + 3*
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) + cos(c)^2 + sin(d*x)^2 + 2*sin(d*x)*sin(c) + sin(c)^2))/(a*d)

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Fricas [C] Result contains complex when optimal does not.
time = 4.00, size = 29139, normalized size = 110.38 \begin {gather*} \text {too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="fricas")

[Out]

1/12*(sqrt(2/3)*sqrt(1/6)*a*d*sqrt(((a^4 - a^2*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d
^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^
2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/
(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6
- a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))*d
^2 + 3*sqrt(1/3)*(a^4 - a^2*b^2)*d^2*sqrt(-((a^8 - 2*a^6*b^2 + a^4*b^4)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/
(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^
2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/
3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) -
1/1458*b^2 ...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\csc {\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sin(d*x+c)**3),x)

[Out]

Integral(csc(c + d*x)/(a + b*sin(c + d*x)**3), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)/(a+b*sin(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(csc(d*x + c)/(b*sin(d*x + c)^3 + a), x)

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Mupad [B]
time = 15.66, size = 1439, normalized size = 5.45 \begin {gather*} \frac {\sum _{k=1}^6\ln \left (98304\,b^5+\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1048576-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^2\,a^2\,b^5\,98304+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^3\,a^3\,b^5\,5898240-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^4\,a^4\,b^5\,7962624-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^4\,a^6\,b^3\,663552-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^5\,a^5\,b^5\,5308416+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^5\,a^7\,b^3\,10616832+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^6\,a^6\,b^5\,7962624-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^6\,a^8\,b^3\,9953280-\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )\,a\,b^5\,589824-\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )\,a^2\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,24576-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^2\,a\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3145728+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^2\,a^3\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,466944-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^3\,a^2\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,18874368-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^3\,a^4\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,3981312+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^4\,a^3\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,56623104+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^4\,a^5\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,20791296+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^5\,a^4\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,84934656-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^5\,a^6\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,78962688-{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^6\,a^5\,b^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,254803968+{\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}^6\,a^7\,b^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,252813312\right )\,\mathrm {root}\left (729\,a^6\,b^2\,z^6-729\,a^8\,z^6-243\,a^4\,b^2\,z^4+27\,a^2\,b^2\,z^2-b^2,z,k\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(c + d*x)*(a + b*sin(c + d*x)^3)),x)

[Out]

symsum(log(98304*b^5 + 1048576*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z,
k)*b^6*tan(c/2 + (d*x)/2) - 98304*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2
, z, k)^2*a^2*b^5 + 5898240*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)
^3*a^3*b^5 - 7962624*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^4*a^4*
b^5 - 663552*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^4*a^6*b^3 - 53
08416*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^5*a^5*b^5 + 10616832*
root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^5*a^7*b^3 + 7962624*root(72
9*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^6*a^6*b^5 - 9953280*root(729*a^6*b
^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^6*a^8*b^3 - 589824*root(729*a^6*b^2*z^6 -
729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)*a*b^5 - 24576*root(729*a^6*b^2*z^6 - 729*a^8*z^6
- 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)*a^2*b^4*tan(c/2 + (d*x)/2) - 3145728*root(729*a^6*b^2*z^6 - 72
9*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^2*a*b^6*tan(c/2 + (d*x)/2) + 466944*root(729*a^6*b^2
*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^2*a^3*b^4*tan(c/2 + (d*x)/2) - 18874368*roo
t(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^3*a^2*b^6*tan(c/2 + (d*x)/2) -
3981312*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^3*a^4*b^4*tan(c/2
+ (d*x)/2) + 56623104*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^4*a^3
*b^6*tan(c/2 + (d*x)/2) + 20791296*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2
, z, k)^4*a^5*b^4*tan(c/2 + (d*x)/2) + 84934656*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*
b^2*z^2 - b^2, z, k)^5*a^4*b^6*tan(c/2 + (d*x)/2) - 78962688*root(729*a^6*b^2*z^6 - 729*a^8*z^6 - 243*a^4*b^2*
z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^5*a^6*b^4*tan(c/2 + (d*x)/2) - 254803968*root(729*a^6*b^2*z^6 - 729*a^8*z^6
- 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^6*a^5*b^6*tan(c/2 + (d*x)/2) + 252813312*root(729*a^6*b^2*z^6
- 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k)^6*a^7*b^4*tan(c/2 + (d*x)/2))*root(729*a^6*b^2*z
^6 - 729*a^8*z^6 - 243*a^4*b^2*z^4 + 27*a^2*b^2*z^2 - b^2, z, k), k, 1, 6)/d + log(tan(c/2 + (d*x)/2))/(a*d)

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