3.3.0 ∫ sin2 (c+dx) _____ (a−b sin4(c+dx))2 dx [221]

Integrand size = 24, antiderivative size = 219 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \]

Rubi [A] (veriﬁed)

Time = 0.19 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3296, 1347, 1180, 211} \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \sqrt {b} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \sqrt {b} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\tan (c+d x) \left ((a+b) \tan ^2(c+d x)+a\right )}{4 a d (a-b) \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^2 \left (1+x^2\right )^2}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = -\frac {\tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\text {Subst}\left (\int \frac {-\frac {2 a^2 b}{a-b}-\frac {2 a (3 a-b) b x^2}{a-b}}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{8 a^2 b d} \\ & = -\frac {\tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\left (2 a-\sqrt {a} \sqrt {b}-b\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{8 a \left (\sqrt {a}+\sqrt {b}\right ) \sqrt {b} d}+\frac {\left (2 a+\sqrt {a} \sqrt {b}-b\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{8 a \left (\sqrt {a}-\sqrt {b}\right ) \sqrt {b} d} \\ & = \frac {\left (2 \sqrt {a}-\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}-\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\left (2 \sqrt {a}+\sqrt {b}\right ) \arctan \left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 a^{5/4} \left (\sqrt {a}+\sqrt {b}\right )^{3/2} \sqrt {b} d}-\frac {\tan (c+d x) \left (a+(a+b) \tan ^2(c+d x)\right )}{4 a (a-b) d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )} \\ \end{align*}

Mathematica [A] (veriﬁed)

Time = 1.67 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.16 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\frac {-\frac {\sqrt {a} \left (2 a-\sqrt {a} \sqrt {b}-b\right ) \arctan \left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a+\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {\sqrt {a} \left (2 a+\sqrt {a} \sqrt {b}-b\right ) \text {arctanh}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\sqrt {-a+\sqrt {a} \sqrt {b}} \sqrt {b}}-\frac {4 \sqrt {a} (2 a+b-b \cos (2 (c+d x))) \sin (2 (c+d x))}{8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))}}{8 a^{3/2} (a-b) d} \]

Maple [A] (veriﬁed)

Time = 1.94 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.14


method	result	size

derivativedivides	\(\frac {\frac {-\frac {\left (a +b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{4 \left (a -b \right ) a}-\frac {\tan \left (d x +c \right )}{4 \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b +2 a^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b -2 a^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 a}}{d}\)	\(250\)
default	\(\frac {\frac {-\frac {\left (a +b \right ) \left (\tan ^{3}\left (d x +c \right )\right )}{4 \left (a -b \right ) a}-\frac {\tan \left (d x +c \right )}{4 \left (a -b \right )}}{\left (\tan ^{4}\left (d x +c \right )\right ) a -b \left (\tan ^{4}\left (d x +c \right )\right )+2 a \left (\tan ^{2}\left (d x +c \right )\right )+a}+\frac {\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b +2 a^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (3 a \sqrt {a b}-\sqrt {a b}\, b -2 a^{2}\right ) \operatorname {arctanh}\left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}}{4 a}}{d}\)	\(250\)
risch	\(-\frac {i \left (2 a \,{\mathrm e}^{6 i \left (d x +c \right )}-b \,{\mathrm e}^{6 i \left (d x +c \right )}-8 a \,{\mathrm e}^{4 i \left (d x +c \right )}+3 b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 a \,{\mathrm e}^{2 i \left (d x +c \right )}-3 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}{2 a \left (a -b \right ) d \left ({\mathrm e}^{8 i \left (d x +c \right )} b -4 b \,{\mathrm e}^{6 i \left (d x +c \right )}-16 a \,{\mathrm e}^{4 i \left (d x +c \right )}+6 b \,{\mathrm e}^{4 i \left (d x +c \right )}-4 b \,{\mathrm e}^{2 i \left (d x +c \right )}+b \right )}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (256 a^{8} b^{2} d^{4}-768 a^{7} b^{3} d^{4}+768 a^{6} b^{4} d^{4}-256 a^{5} b^{5} d^{4}\right ) \textit {\_Z}^{4}+\left (128 a^{5} b \,d^{2}+32 a^{4} b^{2} d^{2}-32 a^{3} b^{3} d^{2}\right ) \textit {\_Z}^{2}+16 a^{2}-8 a b +b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {384 i a^{8} b^{2} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {1280 i a^{7} b^{3} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {1536 i a^{6} b^{4} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {768 i a^{5} b^{5} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {128 i a^{4} b^{6} d^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {128 d^{2} a^{7} b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {416 d^{2} b^{2} a^{6}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {480 d^{2} b^{3} a^{5}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {224 d^{2} b^{4} a^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {32 d^{2} b^{5} a^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R}^{2}+\left (\frac {224 i a^{5} d b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {88 i a^{4} b^{2} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {16 i a^{3} b^{3} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {8 i a^{2} b^{4} d}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right ) \textit {\_R} -\frac {32 a^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {32 a^{3} b}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {38 a^{2} b^{2}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}-\frac {11 a \,b^{3}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}+\frac {b^{4}}{32 a^{3} b -28 a^{2} b^{2}+9 a \,b^{3}-b^{4}}\right )\right )}{4}\)	\(997\)

Fricas [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3445 vs. \(2 (169) = 338\).

Time = 0.92 (sec) , antiderivative size = 3445, normalized size of antiderivative = 15.73 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]

Sympy [F(-1)]

Timed out. \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Timed out} \]

Maxima [F]

\[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\int { \frac {\sin \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right )^{4} - a\right )}^{2}} \,d x } \]

Giac [B] (veriﬁcation not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1407 vs. \(2 (169) = 338\).

Time = 0.81 (sec) , antiderivative size = 1407, normalized size of antiderivative = 6.42 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]

Mupad [B] (veriﬁcation not implemented)

Time = 17.17 (sec) , antiderivative size = 3842, normalized size of antiderivative = 17.54 \[ \int \frac {\sin ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx=\text {Too large to display} \]

\(\int \frac {\sin ^2(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\) [221]

Optimal result