Optimal. Leaf size=125 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}+\sqrt {b}}} \]
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Rubi [A] time = 0.11, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3217, 1130, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {b} d \sqrt {\sqrt {a}+\sqrt {b}}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 1130
Rule 3217
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (1-\frac {\sqrt {a}}{\sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left (1+\frac {\sqrt {a}}{\sqrt {b}}\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}-\sqrt {b}} \sqrt {b} d}-\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} \sqrt {\sqrt {a}+\sqrt {b}} \sqrt {b} d}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 137, normalized size = 1.10 \[ -\frac {\tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{2 \sqrt {b} d \sqrt {\sqrt {a} \sqrt {b}+a}}-\frac {\tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{2 \sqrt {b} d \sqrt {\sqrt {a} \sqrt {b}-a}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 1087, normalized size = 8.70 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.99, size = 398, normalized size = 3.18 \[ \frac {\frac {{\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (d x + c\right )}{\sqrt {\frac {4 \, a + \sqrt {-16 \, {\left (a - b\right )} a + 16 \, a^{2}}}{a - b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}} - \frac {{\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{2}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \tan \left (d x + c\right )}{\sqrt {\frac {4 \, a - \sqrt {-16 \, {\left (a - b\right )} a + 16 \, a^{2}}}{a - b}}}\right )\right )} {\left | a - b \right |}}{3 \, a^{5} b - 12 \, a^{4} b^{2} + 14 \, a^{3} b^{3} - 4 \, a^{2} b^{4} - a b^{5}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 492, normalized size = 3.94 \[ \frac {a \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {a^{2} \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {a \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {a^{2} \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {b \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {a \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right ) b}{2 d \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {b \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {a \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right ) b}{2 d \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sin \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{4} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 16.19, size = 443, normalized size = 3.54 \[ \frac {\ln \left (a\,b-a^2-\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (a-b\right )\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}\,\left (2\,a\,b^2+a\,\sqrt {a\,b^3}+b\,\sqrt {a\,b^3}\right )}{a\,b+\sqrt {a\,b^3}}\right )\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}}{4\,d}-\frac {\ln \left (a\,b-a^2-\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {1}{a\,b-\sqrt {a\,b^3}}}\,\left (a-b\right )\,\left (a\,\sqrt {a\,b^3}-2\,a\,b^2+b\,\sqrt {a\,b^3}\right )}{a\,b-\sqrt {a\,b^3}}\right )\,\sqrt {\frac {a\,b+\sqrt {a\,b^3}}{16\,\left (a\,b^3-a^2\,b^2\right )}}}{d}+\frac {\ln \left (a\,b-a^2+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\sqrt {-\frac {1}{a\,b-\sqrt {a\,b^3}}}\,\left (a-b\right )\,\left (a\,\sqrt {a\,b^3}-2\,a\,b^2+b\,\sqrt {a\,b^3}\right )}{a\,b-\sqrt {a\,b^3}}\right )\,\sqrt {-\frac {1}{a\,b-\sqrt {a\,b^3}}}}{4\,d}-\frac {\ln \left (a\,b-a^2+\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (a-b\right )\,\sqrt {-\frac {1}{a\,b+\sqrt {a\,b^3}}}\,\left (2\,a\,b^2+a\,\sqrt {a\,b^3}+b\,\sqrt {a\,b^3}\right )}{a\,b+\sqrt {a\,b^3}}\right )\,\sqrt {\frac {a\,b-\sqrt {a\,b^3}}{16\,\left (a\,b^3-a^2\,b^2\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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