Optimal. Leaf size=71 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0673509, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3223, 212, 208, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3223
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a-b x^4} \, dx,x,\sin (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a}-\sqrt{b} x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt{a} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{a}+\sqrt{b} x^2} \, dx,x,\sin (c+d x)\right )}{2 \sqrt{a} d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b} d}\\ \end{align*}
Mathematica [A] time = 0.0235646, size = 54, normalized size = 0.76 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )+\tanh ^{-1}\left (\frac{\sqrt [4]{b} \sin (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b} d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.033, size = 81, normalized size = 1.1 \begin{align*}{\frac{1}{4\,da}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( \sin \left ( dx+c \right ) +\sqrt [4]{{\frac{a}{b}}} \right ) \left ( \sin \left ( dx+c \right ) -\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{1}{2\,da}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sin \left ( dx+c \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 16.0517, size = 165, normalized size = 2.32 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \cos{\left (c \right )}}{\sin ^{4}{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{\sin{\left (c + d x \right )}}{a d} & \text{for}\: b = 0 \\\frac{x \cos{\left (c \right )}}{a - b \sin ^{4}{\left (c \right )}} & \text{for}\: d = 0 \\\frac{1}{3 b d \sin ^{3}{\left (c + d x \right )}} & \text{for}\: a = 0 \\- \frac{\log{\left (- \sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sin{\left (c + d x \right )} \right )}}{4 a^{\frac{3}{4}} b^{2} d \left (\frac{1}{b}\right )^{\frac{7}{4}}} + \frac{\log{\left (\sqrt [4]{a} \sqrt [4]{\frac{1}{b}} + \sin{\left (c + d x \right )} \right )}}{4 a^{\frac{3}{4}} b^{2} d \left (\frac{1}{b}\right )^{\frac{7}{4}}} + \frac{\operatorname{atan}{\left (\frac{\sin{\left (c + d x \right )}}{\sqrt [4]{a} \sqrt [4]{\frac{1}{b}}} \right )}}{2 a^{\frac{3}{4}} b^{2} d \left (\frac{1}{b}\right )^{\frac{7}{4}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 6.58507, size = 302, normalized size = 4.25 \begin{align*} \frac{\frac{2 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} \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 )}{a b} + \frac{2 \, \sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} \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 )}{a b} + \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} \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 b} - \frac{\sqrt{2} \left (-a b^{3}\right )^{\frac{1}{4}} \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 b}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]