Optimal. Leaf size=127 \[ -\frac {x}{b}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b d}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3296, 1301,
209, 1180, 211} \begin {gather*} \frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b d \sqrt {\sqrt {a}-\sqrt {b}}}+\frac {\sqrt [4]{a} \text {ArcTan}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b d \sqrt {\sqrt {a}+\sqrt {b}}}-\frac {x}{b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 211
Rule 1180
Rule 1301
Rule 3296
Rubi steps
\begin {align*} \int \frac {\sin ^4(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right ) \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{b \left (1+x^2\right )}+\frac {a \left (1+x^2\right )}{b \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {\text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b d}+\frac {a \text {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b d}\\ &=-\frac {x}{b}+\frac {\left (a \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )\right ) \text {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b d}+\frac {\left (a \left (1+\frac {\sqrt {b}}{\sqrt {a}}\right )\right ) \text {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=-\frac {x}{b}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b d}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.28, size = 143, normalized size = 1.13 \begin {gather*} \frac {-2 (c+d x)+\frac {\sqrt {a} \tan ^{-1}\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}}}-\frac {\sqrt {a} \tanh ^{-1}\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}}}}{2 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.36, size = 163, normalized size = 1.28
method | result | size |
risch | \(-\frac {x}{b}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (a \,b^{4} d^{4}-b^{5} d^{4}\right ) \textit {\_Z}^{4}+32 a \,b^{2} d^{2} \textit {\_Z}^{2}+256 a \right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\left (\frac {1}{32} i a \,b^{2} d^{3}-\frac {1}{32} i b^{3} d^{3}\right ) \textit {\_R}^{3}+\left (-\frac {1}{8} b \,d^{2} a +\frac {1}{8} b^{2} d^{2}\right ) \textit {\_R}^{2}+\left (\frac {1}{2} i a d +\frac {1}{2} i b d \right ) \textit {\_R} -\frac {2 a}{b}-1\right )\right )}{16}\) | \(128\) |
derivativedivides | \(\frac {-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b}}{d}\) | \(163\) |
default | \(\frac {-\frac {\arctan \left (\tan \left (d x +c \right )\right )}{b}+\frac {a \left (a -b \right ) \left (\frac {\left (\sqrt {a b}-b \right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {\left (\sqrt {a b}+b \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{b}}{d}\) | \(163\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1125 vs.
\(2 (91) = 182\).
time = 0.53, size = 1125, normalized size = 8.86 \begin {gather*} \frac {b \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{2} \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - b d \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - \frac {1}{4} \, {\left (2 \, {\left (a b - b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a b - b^{2}\right )} d^{2}\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - \frac {1}{4}\right ) - b \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (\frac {1}{4} \, \cos \left (d x + c\right )^{2} - \frac {1}{2} \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - b d \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {-\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - \frac {1}{4} \, {\left (2 \, {\left (a b - b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a b - b^{2}\right )} d^{2}\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - \frac {1}{4}\right ) + b \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} + \frac {1}{2} \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b d \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - \frac {1}{4} \, {\left (2 \, {\left (a b - b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a b - b^{2}\right )} d^{2}\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \frac {1}{4}\right ) - b \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} \log \left (-\frac {1}{4} \, \cos \left (d x + c\right )^{2} - \frac {1}{2} \, {\left ({\left (a b^{2} - b^{3}\right )} d^{3} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b d \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \sqrt {\frac {{\left (a b^{2} - b^{3}\right )} d^{2} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} - a}{{\left (a b^{2} - b^{3}\right )} d^{2}}} - \frac {1}{4} \, {\left (2 \, {\left (a b - b^{2}\right )} d^{2} \cos \left (d x + c\right )^{2} - {\left (a b - b^{2}\right )} d^{2}\right )} \sqrt {\frac {a}{{\left (a^{2} b^{3} - 2 \, a b^{4} + b^{5}\right )} d^{4}}} + \frac {1}{4}\right ) - 8 \, x}{8 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 912 vs.
\(2 (91) = 182\).
time = 0.91, size = 912, normalized size = 7.18 \begin {gather*} -\frac {\frac {2 \, {\left (d x + c\right )}}{b} + \frac {{\left ({\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 )} b^{2} {\left | -a + b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{3} b - 9 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} + 5 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} a b^{3} + \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left | -a + b \right |} {\left | b \right |} - {\left (3 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b + \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left | -a + b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b + \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} {\left | b \right |}} - \frac {{\left ({\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 )} b^{2} {\left | -a + b \right |} + {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{3} b - 9 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a^{2} b^{2} + 5 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} a b^{3} + \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} b^{4}\right )} {\left | -a + b \right |} {\left | b \right |} - {\left (3 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a^{2} b^{2} - 6 \, \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} a b^{3} - \sqrt {a^{2} - a b - \sqrt {a b} {\left (a - b\right )}} \sqrt {a b} b^{4}\right )} {\left | -a + b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {\tan \left (d x + c\right )}{\sqrt {\frac {a b - \sqrt {a^{2} b^{2} - {\left (a b - b^{2}\right )} a b}}{a b - b^{2}}}}\right )\right )}}{{\left (3 \, a^{5} b^{2} - 15 \, a^{4} b^{3} + 26 \, a^{3} b^{4} - 18 \, a^{2} b^{5} + 3 \, a b^{6} + b^{7}\right )} {\left | b \right |}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 16.27, size = 2991, normalized size = 23.55 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________