Optimal. Leaf size=487 \[ -\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \tan (x)}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{2} (a+b)^{3/4} \tan (x)+\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}} \]
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Rubi [A] time = 1.1374, antiderivative size = 487, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3209, 1169, 634, 618, 204, 628} \[ -\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}-\sqrt{2} (a+b)^{3/4} \tan (x)}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a+b}+\sqrt{a}\right ) \tan ^{-1}\left (\frac{\sqrt{2} (a+b)^{3/4} \tan (x)+\sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}{\sqrt [4]{a} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{\sqrt{a} \sqrt{a+b}+a+b}}+\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)-\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}}-\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left ((a+b)^{3/4} \tan ^2(x)+\sqrt{2} \sqrt [4]{a} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b} \tan (x)+\sqrt{a} \sqrt [4]{a+b}\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{-\sqrt{a} \sqrt{a+b}+a+b}} \]
Antiderivative was successfully verified.
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Rule 3209
Rule 1169
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{a+b \sin ^4(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1+x^2}{a+2 a x^2+(a+b) x^4} \, dx,x,\tan (x)\right )\\ &=\frac{\sqrt [4]{a+b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}-\left (1-\frac{\sqrt{a}}{\sqrt{a+b}}\right ) x}{\frac{\sqrt{a}}{\sqrt{a+b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{2 \sqrt{2} a^{3/4} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}+\frac{\sqrt [4]{a+b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}+\left (1-\frac{\sqrt{a}}{\sqrt{a+b}}\right ) x}{\frac{\sqrt{a}}{\sqrt{a+b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{2 \sqrt{2} a^{3/4} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}\\ &=\frac{\left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{a+b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 (a+b)}+\frac{\left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{a+b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 (a+b)}+\frac{\left (\sqrt [4]{a+b} \left (-1+\frac{\sqrt{a}}{\sqrt{a+b}}\right )\right ) \operatorname{Subst}\left (\int \frac{-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}+2 x}{\frac{\sqrt{a}}{\sqrt{a+b}}-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 \sqrt{2} a^{3/4} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}-\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}+2 x}{\frac{\sqrt{a}}{\sqrt{a+b}}+\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} x}{(a+b)^{3/4}}+x^2} \, dx,x,\tan (x)\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}\\ &=-\frac{\sqrt [4]{a+b} \left (1-\frac{\sqrt{a}}{\sqrt{a+b}}\right ) \log \left (\sqrt{a} \sqrt [4]{a+b}-\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt{2} a^{3/4} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}-\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left (\sqrt{a} \sqrt [4]{a+b}+\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}-\frac{\left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 \sqrt{a} \left (a+b+\sqrt{a} \sqrt{a+b}\right )}{(a+b)^{3/2}}-x^2} \, dx,x,-\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}+2 \tan (x)\right )}{2 (a+b)}-\frac{\left (1+\frac{\sqrt{a+b}}{\sqrt{a}}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{2 \sqrt{a} \left (a+b+\sqrt{a} \sqrt{a+b}\right )}{(a+b)^{3/2}}-x^2} \, dx,x,\frac{\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}+2 \tan (x)\right )}{2 (a+b)}\\ &=-\frac{\left (\sqrt{a}+\sqrt{a+b}\right ) \tan ^{-1}\left (\frac{(a+b)^{3/4} \left (\frac{\sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}-\sqrt{2} \tan (x)\right )}{\sqrt [4]{a} \sqrt{a+b+\sqrt{a} \sqrt{a+b}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{a+b+\sqrt{a} \sqrt{a+b}}}+\frac{\left (\sqrt{a}+\sqrt{a+b}\right ) \tan ^{-1}\left (\frac{(a+b)^{3/4} \left (\frac{\sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}{(a+b)^{3/4}}+\sqrt{2} \tan (x)\right )}{\sqrt [4]{a} \sqrt{a+b+\sqrt{a} \sqrt{a+b}}}\right )}{2 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{a+b+\sqrt{a} \sqrt{a+b}}}-\frac{\sqrt [4]{a+b} \left (1-\frac{\sqrt{a}}{\sqrt{a+b}}\right ) \log \left (\sqrt{a} \sqrt [4]{a+b}-\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt{2} a^{3/4} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}-\frac{\left (\sqrt{a}-\sqrt{a+b}\right ) \log \left (\sqrt{a} \sqrt [4]{a+b}+\sqrt{2} \sqrt [4]{a} \sqrt{a+b-\sqrt{a} \sqrt{a+b}} \tan (x)+(a+b)^{3/4} \tan ^2(x)\right )}{4 \sqrt{2} a^{3/4} \sqrt [4]{a+b} \sqrt{a+b-\sqrt{a} \sqrt{a+b}}}\\ \end{align*}
Mathematica [C] time = 0.299524, size = 148, normalized size = 0.3 \[ \frac{\left (\sqrt{a}-i \sqrt{b}\right ) \sqrt{a+i \sqrt{a} \sqrt{b}} \tan ^{-1}\left (\frac{\sqrt{a+i \sqrt{a} \sqrt{b}} \tan (x)}{\sqrt{a}}\right )-\left (\sqrt{a}+i \sqrt{b}\right ) \sqrt{-a+i \sqrt{a} \sqrt{b}} \tanh ^{-1}\left (\frac{\sqrt{-a+i \sqrt{a} \sqrt{b}} \tan (x)}{\sqrt{a}}\right )}{2 a (a+b)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.169, size = 1677, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \sin \left (x\right )^{4} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.2451, size = 1893, normalized size = 3.89 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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