\(\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx\) [408]
Optimal result
Integrand size = 22, antiderivative size = 117 \[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right ) d}+\frac {\text {arctanh}(\sin (c+d x))}{(a-b) d}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right ) d}
\]
[Out]
arctanh(sin(d*x+c))/(a-b)/d-1/2*b^(1/4)*arctanh(b^(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)/d/(a^(1/2)-b^(1/2))+1/2*b^
(1/4)*arctan(b^(1/4)*sin(d*x+c)/a^(1/4))/a^(3/4)/d/(a^(1/2)+b^(1/2))
Rubi [A] (verified)
Time = 0.18 (sec) , antiderivative size = 117, 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 = {3302, 1185, 213, 1181, 211,
214} \[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}+\sqrt {b}\right )}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} d \left (\sqrt {a}-\sqrt {b}\right )}+\frac {\text {arctanh}(\sin (c+d x))}{d (a-b)}
\]
[In]
Int[Sec[c + d*x]/(a - b*Sin[c + d*x]^4),x]
[Out]
(b^(1/4)*ArcTan[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] + Sqrt[b])*d) + ArcTanh[Sin[c + d*x]]/((a
- b)*d) - (b^(1/4)*ArcTanh[(b^(1/4)*Sin[c + d*x])/a^(1/4)])/(2*a^(3/4)*(Sqrt[a] - Sqrt[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 1181
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q))
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]
Rule 1185
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + c*x^
4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
Rule 3302
Int[cos[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With
[{ff = FreeFactors[Sin[e + f*x], x]}, Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b*(c*ff*x)^n)^p, x]
, x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[(m - 1)/2] && (EqQ[n, 4] || GtQ[m, 0
] || IGtQ[p, 0] || IntegersQ[m, p])
Rubi steps \begin{align*}
\text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b x^4\right )} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{(a-b) \left (-1+x^2\right )}-\frac {b \left (1+x^2\right )}{(a-b) \left (a-b x^4\right )}\right ) \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sin (c+d x)\right )}{(a-b) d}-\frac {b \text {Subst}\left (\int \frac {1+x^2}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{(a-b) d} \\ & = \frac {\text {arctanh}(\sin (c+d x))}{(a-b) d}-\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} \left (\sqrt {a}-\sqrt {b}\right ) d}-\frac {b \text {Subst}\left (\int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt {a} \left (\sqrt {a}+\sqrt {b}\right ) d} \\ & = \frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}+\sqrt {b}\right ) d}+\frac {\text {arctanh}(\sin (c+d x))}{(a-b) d}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \left (\sqrt {a}-\sqrt {b}\right ) d} \\
\end{align*}
Mathematica [C] (verified)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.57
\[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {4 a^{3/4} \text {arctanh}(\sin (c+d x))+\sqrt [4]{b} \left (\left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} \sin (c+d x)\right )+i \left (\left (\sqrt {a}-\sqrt {b}\right ) \log \left (\sqrt [4]{a}-i \sqrt [4]{b} \sin (c+d x)\right )+\left (-\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}+i \sqrt [4]{b} \sin (c+d x)\right )+i \left (\sqrt {a}+\sqrt {b}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} \sin (c+d x)\right )\right )\right )}{4 a^{3/4} (a-b) d}
\]
[In]
Integrate[Sec[c + d*x]/(a - b*Sin[c + d*x]^4),x]
[Out]
(4*a^(3/4)*ArcTanh[Sin[c + d*x]] + b^(1/4)*((Sqrt[a] + Sqrt[b])*Log[a^(1/4) - b^(1/4)*Sin[c + d*x]] + I*((Sqrt
[a] - Sqrt[b])*Log[a^(1/4) - I*b^(1/4)*Sin[c + d*x]] + (-Sqrt[a] + Sqrt[b])*Log[a^(1/4) + I*b^(1/4)*Sin[c + d*
x]] + I*(Sqrt[a] + Sqrt[b])*Log[a^(1/4) + b^(1/4)*Sin[c + d*x]])))/(4*a^(3/4)*(a - b)*d)
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(89)=178\).
Time = 1.22 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.58
| | |
| method | result | size |
| | |
| derivativedivides |
\(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a -2 b}+\frac {b \left (-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a -b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}}{d}\) |
\(185\) |
| default |
\(\frac {-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{2 a -2 b}+\frac {b \left (-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {2 \arctan \left (\frac {\sin \left (d x +c \right )}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\sin \left (d x +c \right )+\left (\frac {a}{b}\right )^{\frac {1}{4}}}{\sin \left (d x +c \right )-\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{a -b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{2 a -2 b}}{d}\) |
\(185\) |
| risch |
\(\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \left (a -b \right )}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \left (a -b \right )}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (4096 a^{5} d^{4}-8192 a^{4} b \,d^{4}+4096 a^{3} b^{2} d^{4}\right ) \textit {\_Z}^{4}-256 a^{2} b \,d^{2} \textit {\_Z}^{2}-b \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\left (-\frac {1024 i d^{3} a^{5}}{a b +b^{2}}+\frac {2048 i d^{3} a^{4} b}{a b +b^{2}}-\frac {1024 i d^{3} a^{3} b^{2}}{a b +b^{2}}\right ) \textit {\_R}^{3}+\left (\frac {48 i d \,a^{2} b}{a b +b^{2}}+\frac {16 i a \,b^{2} d}{a b +b^{2}}\right ) \textit {\_R} \right ) {\mathrm e}^{i \left (d x +c \right )}-\frac {a b}{a b +b^{2}}-\frac {b^{2}}{a b +b^{2}}\right )\right )\) |
\(257\) |
| | |
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[In]
int(sec(d*x+c)/(a-b*sin(d*x+c)^4),x,method=_RETURNVERBOSE)
[Out]
1/d*(-1/(2*a-2*b)*ln(sin(d*x+c)-1)+b/(a-b)*(-1/4*(1/b*a)^(1/4)/a*(ln((sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1
/b*a)^(1/4)))+2*arctan(sin(d*x+c)/(1/b*a)^(1/4)))+1/4/b/(1/b*a)^(1/4)*(2*arctan(sin(d*x+c)/(1/b*a)^(1/4))-ln((
sin(d*x+c)+(1/b*a)^(1/4))/(sin(d*x+c)-(1/b*a)^(1/4)))))+1/(2*a-2*b)*ln(1+sin(d*x+c)))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1329 vs. \(2 (89) = 178\).
Time = 0.42 (sec) , antiderivative size = 1329, normalized size of antiderivative = 11.36
\[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="fricas")
[Out]
-1/4*((a - b)*d*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4
*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 -
2*a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4))
- (a^2*b + a*b^2)*d)*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^
2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) - (a - b)*d*sqrt(-((a^3 - 2*a^2*b + a*b^
2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*
a^2*b + a*b^2)*d^2))*log(1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b
^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + (a^2*b + a*b^2)*d)*sqrt(-((a^3 - 2*a^2*b
+ a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) - 2*b)/((a^
3 - 2*a^2*b + a*b^2)*d^2))) - (a - b)*d*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 -
4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(-1/2*(a*b + b^2)*s
in(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*
a^4*b^3 + a^3*b^4)*d^4)) - (a^2*b + a*b^2)*d)*sqrt(((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/(
(a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) + (a - b)*d*sqr
t(-((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4
)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))*log(-1/2*(a*b + b^2)*sin(d*x + c) + 1/2*((a^5 - 2*a^4*b + a^3*b^
2)*d^3*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d^4)) + (a^2*b + a*b^2)
*d)*sqrt(-((a^3 - 2*a^2*b + a*b^2)*d^2*sqrt((a^2*b + 2*a*b^2 + b^3)/((a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 +
a^3*b^4)*d^4)) - 2*b)/((a^3 - 2*a^2*b + a*b^2)*d^2))) - 2*log(sin(d*x + c) + 1) + 2*log(-sin(d*x + c) + 1))/((
a - b)*d)
Sympy [F]
\[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\int \frac {\sec {\left (c + d x \right )}}{a - b \sin ^{4}{\left (c + d x \right )}}\, dx
\]
[In]
integrate(sec(d*x+c)/(a-b*sin(d*x+c)**4),x)
[Out]
Integral(sec(c + d*x)/(a - b*sin(c + d*x)**4), x)
Maxima [A] (verification not implemented)
none
Time = 0.31 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.43
\[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\frac {\frac {b {\left (\frac {2 \, {\left (\sqrt {a} - \sqrt {b}\right )} \arctan \left (\frac {\sqrt {b} \sin \left (d x + c\right )}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (\sqrt {a} + \sqrt {b}\right )} \log \left (\frac {\sqrt {b} \sin \left (d x + c\right ) - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} \sin \left (d x + c\right ) + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}\right )}}{a - b} + \frac {2 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} - \frac {2 \, \log \left (\sin \left (d x + c\right ) - 1\right )}{a - b}}{4 \, d}
\]
[In]
integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="maxima")
[Out]
1/4*(b*(2*(sqrt(a) - sqrt(b))*arctan(sqrt(b)*sin(d*x + c)/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b)
)*sqrt(b)) + (sqrt(a) + sqrt(b))*log((sqrt(b)*sin(d*x + c) - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*sin(d*x + c) + sq
rt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/(a - b) + 2*log(sin(d*x + c) + 1)/(a - b) - 2*l
og(sin(d*x + c) - 1)/(a - b))/d
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (89) = 178\).
Time = 0.85 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.16
\[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=-\frac {\frac {4 \, {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} + 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} - \sqrt {2} a b^{3}} + \frac {4 \, {\left (\left (-a b^{3}\right )^{\frac {1}{4}} b^{2} + \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} - 2 \, \sin \left (d x + c\right )\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{\sqrt {2} a^{2} b^{2} - \sqrt {2} a b^{3}} + \frac {{\left (\sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a^{2} b^{2} - a b^{3}} - \frac {{\left (\sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} - \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}}\right )} \log \left (\sin \left (d x + c\right )^{2} - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}} \sin \left (d x + c\right ) + \sqrt {-\frac {a}{b}}\right )}{a^{2} b^{2} - a b^{3}} - \frac {4 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a - b} + \frac {4 \, \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a - b}}{8 \, d}
\]
[In]
integrate(sec(d*x+c)/(a-b*sin(d*x+c)^4),x, algorithm="giac")
[Out]
-1/8*(4*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*arctan(1/2*sqrt(2)*(sqrt(2)*(-a/b)^(1/4) + 2*sin(d*x + c))/(-a/b
)^(1/4))/(sqrt(2)*a^2*b^2 - sqrt(2)*a*b^3) + 4*((-a*b^3)^(1/4)*b^2 + (-a*b^3)^(3/4))*arctan(-1/2*sqrt(2)*(sqrt
(2)*(-a/b)^(1/4) - 2*sin(d*x + c))/(-a/b)^(1/4))/(sqrt(2)*a^2*b^2 - sqrt(2)*a*b^3) + (sqrt(2)*(-a*b^3)^(1/4)*b
^2 - sqrt(2)*(-a*b^3)^(3/4))*log(sin(d*x + c)^2 + sqrt(2)*(-a/b)^(1/4)*sin(d*x + c) + sqrt(-a/b))/(a^2*b^2 - a
*b^3) - (sqrt(2)*(-a*b^3)^(1/4)*b^2 - sqrt(2)*(-a*b^3)^(3/4))*log(sin(d*x + c)^2 - sqrt(2)*(-a/b)^(1/4)*sin(d*
x + c) + sqrt(-a/b))/(a^2*b^2 - a*b^3) - 4*log(abs(sin(d*x + c) + 1))/(a - b) + 4*log(abs(sin(d*x + c) - 1))/(
a - b))/d
Mupad [B] (verification not implemented)
Time = 16.54 (sec) , antiderivative size = 3891, normalized size of antiderivative = 33.26
\[
\int \frac {\sec (c+d x)}{a-b \sin ^4(c+d x)} \, dx=\text {Too large to display}
\]
[In]
int(1/(cos(c + d*x)*(a - b*sin(c + d*x)^4)),x)
[Out]
(atan(((b^5*sin(c + d*x)*3i + ((((32*a*b^7 + 64*a^2*b^6 - 224*a^3*b^5 + 128*a^4*b^4 - (sin(c + d*x)*(512*a^2*b
^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4))/(4*(a - b)))/(2*(a - b)) + (sin(c + d*x)*(32*a*b^6 - 16*b^7 + 2
40*a^2*b^5))/2)/(2*(a - b)) - 10*a*b^5 + 2*b^6)*1i)/(2*(a - b)))/(a - b) + (b^5*sin(c + d*x)*3i - ((((32*a*b^7
+ 64*a^2*b^6 - 224*a^3*b^5 + 128*a^4*b^4 + (sin(c + d*x)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b
^4))/(4*(a - b)))/(2*(a - b)) - (sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))/2)/(2*(a - b)) - 10*a*b^5 + 2
*b^6)*1i)/(2*(a - b)))/(a - b))/((3*b^5*sin(c + d*x) + (((32*a*b^7 + 64*a^2*b^6 - 224*a^3*b^5 + 128*a^4*b^4 -
(sin(c + d*x)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4))/(4*(a - b)))/(2*(a - b)) + (sin(c + d*x
)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))/2)/(2*(a - b)) - 10*a*b^5 + 2*b^6)/(2*(a - b)))/(a - b) - (3*b^5*sin(c +
d*x) - (((32*a*b^7 + 64*a^2*b^6 - 224*a^3*b^5 + 128*a^4*b^4 + (sin(c + d*x)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a
^4*b^5 + 512*a^5*b^4))/(4*(a - b)))/(2*(a - b)) - (sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))/2)/(2*(a -
b)) - 10*a*b^5 + 2*b^6)/(2*(a - b)))/(a - b)))*1i)/(d*(a - b)) - (atan(((((((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^
3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(64*a*b^7 + 128*a^2*b^6 - 448*a^3*b^5 + 256*a^4*b^4 + sin(c
+ d*x)*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(512*a^2*b^7 - 512
*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4)) - sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))*((2*a^2*b + a*(a^3*b)
^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 20*a*b^5 + 4*b^6)*((2*a^2*b + a*(a^3*b)^(1/2
) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 6*b^5*sin(c + d*x))*((2*a^2*b + a*(a^3*b)^(1/2) +
b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*1i - (((((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))
/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(64*a*b^7 + 128*a^2*b^6 - 448*a^3*b^5 + 256*a^4*b^4 - sin(c + d*x)*((2*
a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(512*a^2*b^7 - 512*a^3*b^6 -
512*a^4*b^5 + 512*a^5*b^4)) + sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))*((2*a^2*b + a*(a^3*b)^(1/2) + b*
(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 20*a*b^5 + 4*b^6)*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*
b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) + 6*b^5*sin(c + d*x))*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^
(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*1i)/((((((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 -
2*a^4*b + a^3*b^2)))^(1/2)*(64*a*b^7 + 128*a^2*b^6 - 448*a^3*b^5 + 256*a^4*b^4 + sin(c + d*x)*((2*a^2*b + a*(
a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5
+ 512*a^5*b^4)) - sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/
2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 20*a*b^5 + 4*b^6)*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(
16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 6*b^5*sin(c + d*x))*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*
(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) + (((((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3
*b^2)))^(1/2)*(64*a*b^7 + 128*a^2*b^6 - 448*a^3*b^5 + 256*a^4*b^4 - sin(c + d*x)*((2*a^2*b + a*(a^3*b)^(1/2) +
b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4
)) + sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*a^2*b^5))*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 -
2*a^4*b + a^3*b^2)))^(1/2) - 20*a*b^5 + 4*b^6)*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^
4*b + a^3*b^2)))^(1/2) + 6*b^5*sin(c + d*x))*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b
+ a^3*b^2)))^(1/2)))*((2*a^2*b + a*(a^3*b)^(1/2) + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*2i)
/d - (atan((((((-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(64*a*b^7
+ 128*a^2*b^6 - 448*a^3*b^5 + 256*a^4*b^4 + sin(c + d*x)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*
(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4)) - sin(c + d*x)*(32*
a*b^6 - 16*b^7 + 240*a^2*b^5))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))
^(1/2) - 20*a*b^5 + 4*b^6)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/
2) - 6*b^5*sin(c + d*x))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)
*1i - ((((-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(64*a*b^7 + 128
*a^2*b^6 - 448*a^3*b^5 + 256*a^4*b^4 - sin(c + d*x)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 -
2*a^4*b + a^3*b^2)))^(1/2)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4)) + sin(c + d*x)*(32*a*b^6
- 16*b^7 + 240*a^2*b^5))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)
- 20*a*b^5 + 4*b^6)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) + 6
*b^5*sin(c + d*x))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*1i)/(
((((-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(64*a*b^7 + 128*a^2*b
^6 - 448*a^3*b^5 + 256*a^4*b^4 + sin(c + d*x)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4
*b + a^3*b^2)))^(1/2)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4)) - sin(c + d*x)*(32*a*b^6 - 16*b
^7 + 240*a^2*b^5))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 20*
a*b^5 + 4*b^6)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 6*b^5*s
in(c + d*x))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) + ((((-(a*(
a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*(64*a*b^7 + 128*a^2*b^6 - 448*
a^3*b^5 + 256*a^4*b^4 - sin(c + d*x)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*
b^2)))^(1/2)*(512*a^2*b^7 - 512*a^3*b^6 - 512*a^4*b^5 + 512*a^5*b^4)) + sin(c + d*x)*(32*a*b^6 - 16*b^7 + 240*
a^2*b^5))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) - 20*a*b^5 + 4
*b^6)*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2) + 6*b^5*sin(c + d*
x))*(-(a*(a^3*b)^(1/2) - 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)))*(-(a*(a^3*b)^(1/2)
- 2*a^2*b + b*(a^3*b)^(1/2))/(16*(a^5 - 2*a^4*b + a^3*b^2)))^(1/2)*2i)/d