Integrand size = 10, antiderivative size = 87 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\frac {\arctan \left (\sqrt {1+\sqrt {-\frac {b}{a}}} \tan (x)\right )}{2 a \sqrt {1+\sqrt {-\frac {b}{a}}}}+\frac {\text {arctanh}\left (\sqrt {-1+\sqrt {-\frac {b}{a}}} \tan (x)\right )}{2 a \sqrt {-1+\sqrt {-\frac {b}{a}}}} \] Output:
1/2*arctan((1+(-b/a)^(1/2))^(1/2)*tan(x))/a/(1+(-b/a)^(1/2))^(1/2)+1/2*arc tanh((-1+(-b/a)^(1/2))^(1/2)*tan(x))/a/(-1+(-b/a)^(1/2))^(1/2)
Result contains complex when optimal does not.
Time = 4.68 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.70 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\frac {\left (\sqrt {a}-i \sqrt {b}\right ) \sqrt {a+i \sqrt {a} \sqrt {b}} \arctan \left (\frac {\sqrt {a+i \sqrt {a} \sqrt {b}} \tan (x)}{\sqrt {a}}\right )-\left (\sqrt {a}+i \sqrt {b}\right ) \sqrt {-a+i \sqrt {a} \sqrt {b}} \text {arctanh}\left (\frac {\sqrt {-a+i \sqrt {a} \sqrt {b}} \tan (x)}{\sqrt {a}}\right )}{2 a (a+b)} \] Input:
Integrate[(a + b*Sin[x]^4)^(-1),x]
Output:
((Sqrt[a] - I*Sqrt[b])*Sqrt[a + I*Sqrt[a]*Sqrt[b]]*ArcTan[(Sqrt[a + I*Sqrt [a]*Sqrt[b]]*Tan[x])/Sqrt[a]] - (Sqrt[a] + I*Sqrt[b])*Sqrt[-a + I*Sqrt[a]* Sqrt[b]]*ArcTanh[(Sqrt[-a + I*Sqrt[a]*Sqrt[b]]*Tan[x])/Sqrt[a]])/(2*a*(a + b))
Leaf count is larger than twice the leaf count of optimal. \(541\) vs. \(2(87)=174\).
Time = 1.31 (sec) , antiderivative size = 541, normalized size of antiderivative = 6.22, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.900, Rules used = {3042, 3688, 1483, 1142, 25, 27, 1083, 217, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {1}{a+b \sin ^4(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{a+b \sin (x)^4}dx\) |
| \(\Big \downarrow \) 3688 |
| \(\displaystyle \int \frac {\tan ^2(x)+1}{(a+b) \tan ^4(x)+2 a \tan ^2(x)+a}d\tan (x)\) |
| \(\Big \downarrow \) 1483 |
| \(\displaystyle \frac {\sqrt [4]{a+b} \int \frac {\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \tan (x)}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\sqrt [4]{a+b} \int \frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \tan (x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2} (a+b)^{5/4}}-\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int -\frac {\sqrt {2} \left (\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \tan (x)\right )}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \tan (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \left (\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \tan (x)\right )}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \left (\sqrt {2} \tan (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \tan (x)}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\sqrt [4]{a+b} \left (\frac {\sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2} (a+b)^{5/4}}+\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \tan (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \tan (x)}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2}}-\frac {\sqrt {2} \sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{-\left (2 \tan (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )^2-\frac {2 \sqrt {a} \left (a+\sqrt {a+b} \sqrt {a}+b\right )}{(a+b)^{3/2}}}d\left (2 \tan (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{(a+b)^{5/4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \tan (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2}}-\frac {\sqrt {2} \sqrt [4]{a} \left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \int \frac {1}{-\left (2 \tan (x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )^2-\frac {2 \sqrt {a} \left (a+\sqrt {a+b} \sqrt {a}+b\right )}{(a+b)^{3/2}}}d\left (2 \tan (x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}\right )}{(a+b)^{5/4}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}-\sqrt {2} \tan (x)}{\tan ^2(x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2}}+\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (2 \tan (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\sqrt [4]{a+b} \left (\frac {\left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \int \frac {\sqrt {2} \tan (x)+\frac {\sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b}}{(a+b)^{3/4}}}{\tan ^2(x)+\frac {\sqrt {2} \sqrt [4]{a} \sqrt {a-\sqrt {a+b} \sqrt {a}+b} \tan (x)}{(a+b)^{3/4}}+\frac {\sqrt {a}}{\sqrt {a+b}}}d\tan (x)}{\sqrt {2}}+\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}+2 \tan (x)\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\sqrt [4]{a+b} \left (\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (2 \tan (x)-\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}-\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)-\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \tan (x)+\sqrt {a} \sqrt [4]{a+b}\right )\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}+\frac {\sqrt [4]{a+b} \left (\frac {\left (\sqrt {a+b}+\sqrt {a}\right ) \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \arctan \left (\frac {(a+b)^{3/4} \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}{(a+b)^{3/4}}+2 \tan (x)\right )}{\sqrt {2} \sqrt [4]{a} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}\right )}{\sqrt {a+b} \sqrt {\sqrt {a} \sqrt {a+b}+a+b}}+\frac {1}{2} \left (1-\frac {\sqrt {a}}{\sqrt {a+b}}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)+\sqrt {2} \sqrt [4]{a} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b} \tan (x)+\sqrt {a} \sqrt [4]{a+b}\right )\right )}{2 \sqrt {2} a^{3/4} \sqrt {-\sqrt {a} \sqrt {a+b}+a+b}}\) |
Input:
Int[(a + b*Sin[x]^4)^(-1),x]
Output:
((a + b)^(1/4)*(((Sqrt[a] + Sqrt[a + b])*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]] *ArcTan[((a + b)^(3/4)*(-((Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b ]])/(a + b)^(3/4)) + 2*Tan[x]))/(Sqrt[2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt [a + b]])])/(Sqrt[a + b]*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) - ((1 - Sqrt[a ]/Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/4) - Sqrt[2]*a^(1/4)*Sqrt[a + b - Sq rt[a]*Sqrt[a + b]]*Tan[x] + (a + b)^(3/4)*Tan[x]^2])/2))/(2*Sqrt[2]*a^(3/4 )*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]) + ((a + b)^(1/4)*(((Sqrt[a] + Sqrt[a + b])*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*ArcTan[((a + b)^(3/4)*((Sqrt[2]*a^ (1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])/(a + b)^(3/4) + 2*Tan[x]))/(Sqrt[ 2]*a^(1/4)*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]])])/(Sqrt[a + b]*Sqrt[a + b + Sqrt[a]*Sqrt[a + b]]) + ((1 - Sqrt[a]/Sqrt[a + b])*Log[Sqrt[a]*(a + b)^(1/ 4) + Sqrt[2]*a^(1/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]]*Tan[x] + (a + b)^(3 /4)*Tan[x]^2])/2))/(2*Sqrt[2]*a^(3/4)*Sqrt[a + b - Sqrt[a]*Sqrt[a + b]])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[a/c, 2]}, With[{r = Rt[2*q - b/c, 2]}, Simp[1/(2*c*q*r) In t[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Simp[1/(2*c*q*r) Int[(d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && N eQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + 2*a*ff^2*x^2 + ( a + b)*ff^4*x^4)^p/(1 + ff^2*x^2)^(2*p + 1), x], x, Tan[e + f*x]/ff], x]] / ; FreeQ[{a, b, e, f}, x] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.50 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16
| method | result | size |
| risch | \(\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (256 a^{4}+256 a^{3} b \right ) \textit {\_Z}^{4}+32 a^{2} \textit {\_Z}^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i x}+\left (-\frac {128 i a^{4}}{b}-128 i a^{3}\right ) \textit {\_R}^{3}+\left (\frac {32 a^{3}}{b}+32 a^{2}\right ) \textit {\_R}^{2}+\left (-\frac {8 i a^{2}}{b}+8 i a \right ) \textit {\_R} +\frac {2 a}{b}-1\right )\) | \(101\) |
| default | \(\frac {\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+b a}-2 a}-a^{\frac {3}{2}} \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}+\sqrt {a +b}\, \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a^{2}\right ) \ln \left (\sqrt {a +b}\, \tan \left (x \right )^{2}+\tan \left (x \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a}\right )}{2 \sqrt {a +b}}+\frac {2 \left (2 a^{2} b -\frac {\left (-a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+b a}-2 a}-a^{\frac {3}{2}} \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}+\sqrt {a +b}\, \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a +\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a^{2}\right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a +b}}\right ) \arctan \left (\frac {2 \sqrt {a +b}\, \tan \left (x \right )+\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 a^{\frac {5}{2}} b}+\frac {\frac {\left (a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+b a}-2 a}+a^{\frac {3}{2}} \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}-\sqrt {a +b}\, \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a^{2}\right ) \ln \left (\sqrt {a +b}\, \tan \left (x \right )^{2}-\tan \left (x \right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}+\sqrt {a}\right )}{2 \sqrt {a +b}}+\frac {2 \left (2 a^{2} b +\frac {\left (a^{\frac {5}{2}} \sqrt {2 \sqrt {a^{2}+b a}-2 a}+a^{\frac {3}{2}} \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}-\sqrt {a +b}\, \sqrt {a^{2}+b a}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a -\sqrt {a +b}\, \sqrt {2 \sqrt {a^{2}+b a}-2 a}\, a^{2}\right ) \sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{2 \sqrt {a +b}}\right ) \arctan \left (\frac {2 \sqrt {a +b}\, \tan \left (x \right )-\sqrt {2 \sqrt {a \left (a +b \right )}-2 a}}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}\right )}{\sqrt {4 \sqrt {a}\, \sqrt {a +b}-2 \sqrt {a \left (a +b \right )}+2 a}}}{4 a^{\frac {5}{2}} b}\) | \(771\) |
Input:
int(1/(a+b*sin(x)^4),x,method=_RETURNVERBOSE)
Output:
sum(_R*ln(exp(2*I*x)+(-128*I/b*a^4-128*I*a^3)*_R^3+(32/b*a^3+32*a^2)*_R^2+ (-8*I/b*a^2+8*I*a)*_R+2/b*a-1),_R=RootOf(1+(256*a^4+256*a^3*b)*_Z^4+32*a^2 *_Z^2))
Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (67) = 134\).
Time = 0.20 (sec) , antiderivative size = 823, normalized size of antiderivative = 9.46 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx =\text {Too large to display} \] Input:
integrate(1/(a+b*sin(x)^4),x, algorithm="fricas")
Output:
-1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b ))*log(1/4*b*cos(x)^2 + 1/2*(a*b*cos(x)*sin(x) + (a^4 + a^3*b)*sqrt(-b/(a^ 5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2 *a^4*b + a^3*b^2)) + 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2*(a^3 + a^2*b)* cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1/4*b) + 1/8*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b))*log(1/4*b*cos(x) ^2 - 1/2*(a*b*cos(x)*sin(x) + (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b ^2))*cos(x)*sin(x))*sqrt(-((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/( a^5 + 2*a^4*b + a^3*b^2)) - 1/4*b) + 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b))*log(-1/4*b*cos(x)^2 + 1/2*(a*b*cos(x )*sin(x) - (a^4 + a^3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x)) *sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2*(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3* b^2)) + 1/4*b) - 1/8*sqrt(((a^2 + a*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b))*log(-1/4*b*cos(x)^2 - 1/2*(a*b*cos(x)*sin(x) - (a^4 + a^ 3*b)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2))*cos(x)*sin(x))*sqrt(((a^2 + a*b)*s qrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) - 1)/(a^2 + a*b)) - 1/4*(a^3 + a^2*b - 2 *(a^3 + a^2*b)*cos(x)^2)*sqrt(-b/(a^5 + 2*a^4*b + a^3*b^2)) + 1/4*b)
Timed out. \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\text {Timed out} \] Input:
integrate(1/(a+b*sin(x)**4),x)
Output:
Timed out
\[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\int { \frac {1}{b \sin \left (x\right )^{4} + a} \,d x } \] Input:
integrate(1/(a+b*sin(x)^4),x, algorithm="maxima")
Output:
integrate(1/(b*sin(x)^4 + a), x)
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (67) = 134\).
Time = 0.62 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.66 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\frac {{\left (3 \, \sqrt {a^{2} + a b + \sqrt {-a b} {\left (a + b\right )}} a^{2} + 6 \, \sqrt {a^{2} + a b + \sqrt {-a b} {\left (a + b\right )}} a b - \sqrt {a^{2} + a b + \sqrt {-a b} {\left (a + b\right )}} b^{2}\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a + \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a + b}}}\right )\right )} {\left | a + b \right |}}{2 \, {\left (3 \, a^{5} + 12 \, a^{4} b + 14 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - a b^{4}\right )}} + \frac {{\left (3 \, \sqrt {a^{2} + a b - \sqrt {-a b} {\left (a + b\right )}} a^{2} + 6 \, \sqrt {a^{2} + a b - \sqrt {-a b} {\left (a + b\right )}} a b - \sqrt {a^{2} + a b - \sqrt {-a b} {\left (a + b\right )}} b^{2}\right )} {\left (\pi \left \lfloor \frac {x}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (x\right )}{\sqrt {\frac {4 \, a - \sqrt {-16 \, {\left (a + b\right )} a + 16 \, a^{2}}}{a + b}}}\right )\right )} {\left | a + b \right |}}{2 \, {\left (3 \, a^{5} + 12 \, a^{4} b + 14 \, a^{3} b^{2} + 4 \, a^{2} b^{3} - a b^{4}\right )}} \] Input:
integrate(1/(a+b*sin(x)^4),x, algorithm="giac")
Output:
1/2*(3*sqrt(a^2 + a*b + sqrt(-a*b)*(a + b))*a^2 + 6*sqrt(a^2 + a*b + sqrt( -a*b)*(a + b))*a*b - sqrt(a^2 + a*b + sqrt(-a*b)*(a + b))*b^2)*(pi*floor(x /pi + 1/2) + arctan(2*tan(x)/sqrt((4*a + sqrt(-16*(a + b)*a + 16*a^2))/(a + b))))*abs(a + b)/(3*a^5 + 12*a^4*b + 14*a^3*b^2 + 4*a^2*b^3 - a*b^4) + 1 /2*(3*sqrt(a^2 + a*b - sqrt(-a*b)*(a + b))*a^2 + 6*sqrt(a^2 + a*b - sqrt(- a*b)*(a + b))*a*b - sqrt(a^2 + a*b - sqrt(-a*b)*(a + b))*b^2)*(pi*floor(x/ pi + 1/2) + arctan(2*tan(x)/sqrt((4*a - sqrt(-16*(a + b)*a + 16*a^2))/(a + b))))*abs(a + b)/(3*a^5 + 12*a^4*b + 14*a^3*b^2 + 4*a^2*b^3 - a*b^4)
Time = 38.50 (sec) , antiderivative size = 407, normalized size of antiderivative = 4.68 \[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^5\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}-a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^4\,b\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}}{\sqrt {-a^3\,b}}\right )\,\sqrt {-\frac {a^2-\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^5\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}-a^2\,b\,\mathrm {tan}\left (x\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,4{}\mathrm {i}+a^4\,b\,\mathrm {tan}\left (x\right )\,{\left (-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}\right )}^{3/2}\,64{}\mathrm {i}}{\sqrt {-a^3\,b}}\right )\,\sqrt {-\frac {a^2+\sqrt {-a^3\,b}}{16\,a^4+16\,b\,a^3}}\,2{}\mathrm {i} \] Input:
int(1/(a + b*sin(x)^4),x)
Output:
atan((a^3*tan(x)*(-(a^2 - (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(1/2)*4i + a^5*tan(x)*(-(a^2 - (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(3/2)*64i - a^2*b *tan(x)*(-(a^2 - (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(1/2)*4i + a^4*b*tan (x)*(-(a^2 - (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(3/2)*64i)/(-a^3*b)^(1/2 ))*(-(a^2 - (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(1/2)*2i - atan((a^3*tan( x)*(-(a^2 + (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(1/2)*4i + a^5*tan(x)*(-( a^2 + (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(3/2)*64i - a^2*b*tan(x)*(-(a^2 + (-a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(1/2)*4i + a^4*b*tan(x)*(-(a^2 + ( -a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(3/2)*64i)/(-a^3*b)^(1/2))*(-(a^2 + (- a^3*b)^(1/2))/(16*a^3*b + 16*a^4))^(1/2)*2i
\[ \int \frac {1}{a+b \sin ^4(x)} \, dx=\int \frac {1}{\sin \left (x \right )^{4} b +a}d x \] Input:
int(1/(a+b*sin(x)^4),x)
Output:
int(1/(sin(x)**4*b + a),x)