Optimal. Leaf size=120 \[ -\frac {\sin ^4(x) (a \cot (x)+b)}{4 \left (a^2+b^2\right )}-\frac {b^5 \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^3}-\frac {\sin ^2(x) \left (a \left (3 a^2+7 b^2\right ) \cot (x)+4 b^3\right )}{8 \left (a^2+b^2\right )^2}+\frac {a x \left (3 a^4+10 a^2 b^2+15 b^4\right )}{8 \left (a^2+b^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3506, 741, 823, 801, 635, 203, 260} \[ \frac {a x \left (10 a^2 b^2+3 a^4+15 b^4\right )}{8 \left (a^2+b^2\right )^3}-\frac {\sin ^4(x) (a \cot (x)+b)}{4 \left (a^2+b^2\right )}-\frac {\sin ^2(x) \left (a \left (3 a^2+7 b^2\right ) \cot (x)+4 b^3\right )}{8 \left (a^2+b^2\right )^2}-\frac {b^5 \log (a \sin (x)+b \cos (x))}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 260
Rule 635
Rule 741
Rule 801
Rule 823
Rule 3506
Rubi steps
\begin {align*} \int \frac {\sin ^4(x)}{a+b \cot (x)} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x) \left (1+\frac {x^2}{b^2}\right )^3} \, dx,x,b \cot (x)\right )}{b}\\ &=-\frac {(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b \operatorname {Subst}\left (\int \frac {-4-\frac {3 a^2}{b^2}-\frac {3 a x}{b^2}}{(a+x) \left (1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \cot (x)\right )}{4 \left (a^2+b^2\right )}\\ &=-\frac {\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac {(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac {b^5 \operatorname {Subst}\left (\int \frac {\frac {3 a^4+7 a^2 b^2+8 b^4}{b^6}+\frac {a \left (3 a^2+7 b^2\right ) x}{b^6}}{(a+x) \left (1+\frac {x^2}{b^2}\right )} \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac {(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac {b^5 \operatorname {Subst}\left (\int \left (\frac {8}{\left (a^2+b^2\right ) (a+x)}+\frac {3 a^5+10 a^3 b^2+15 a b^4-8 b^4 x}{b^4 \left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^2}\\ &=-\frac {b^5 \log (a+b \cot (x))}{\left (a^2+b^2\right )^3}-\frac {\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac {(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}-\frac {b \operatorname {Subst}\left (\int \frac {3 a^5+10 a^3 b^2+15 a b^4-8 b^4 x}{b^2+x^2} \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^3}\\ &=-\frac {b^5 \log (a+b \cot (x))}{\left (a^2+b^2\right )^3}-\frac {\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac {(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}+\frac {b^5 \operatorname {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \cot (x)\right )}{\left (a^2+b^2\right )^3}-\frac {\left (a b \left (3 a^4+10 a^2 b^2+15 b^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \cot (x)\right )}{8 \left (a^2+b^2\right )^3}\\ &=\frac {a \left (3 a^4+10 a^2 b^2+15 b^4\right ) x}{8 \left (a^2+b^2\right )^3}-\frac {b^5 \log (a+b \cot (x))}{\left (a^2+b^2\right )^3}-\frac {b^5 \log (\sin (x))}{\left (a^2+b^2\right )^3}-\frac {\left (4 b^3+a \left (3 a^2+7 b^2\right ) \cot (x)\right ) \sin ^2(x)}{8 \left (a^2+b^2\right )^2}-\frac {(b+a \cot (x)) \sin ^4(x)}{4 \left (a^2+b^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.26, size = 151, normalized size = 1.26 \[ \frac {12 a^5 x-8 a^5 \sin (2 x)+a^5 \sin (4 x)+40 a^3 b^2 x-24 a^3 b^2 \sin (2 x)+2 a^3 b^2 \sin (4 x)-b \left (a^2+b^2\right )^2 \cos (4 x)+4 b \left (a^4+4 a^2 b^2+3 b^4\right ) \cos (2 x)-32 b^5 \log (a \sin (x)+b \cos (x))+60 a b^4 x-16 a b^4 \sin (2 x)+a b^4 \sin (4 x)}{32 \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.91, size = 184, normalized size = 1.53 \[ -\frac {4 \, b^{5} \log \left (2 \, a b \cos \relax (x) \sin \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2}\right ) + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \relax (x)^{4} - 4 \, {\left (a^{4} b + 3 \, a^{2} b^{3} + 2 \, b^{5}\right )} \cos \relax (x)^{2} - {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x - {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \relax (x)^{3} - {\left (5 \, a^{5} + 14 \, a^{3} b^{2} + 9 \, a b^{4}\right )} \cos \relax (x)\right )} \sin \relax (x)}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.96, size = 273, normalized size = 2.28 \[ -\frac {a b^{5} \log \left ({\left | a \tan \relax (x) + b \right |}\right )}{a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}} + \frac {b^{5} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {6 \, b^{5} \tan \relax (x)^{4} + 5 \, a^{5} \tan \relax (x)^{3} + 14 \, a^{3} b^{2} \tan \relax (x)^{3} + 9 \, a b^{4} \tan \relax (x)^{3} - 4 \, a^{4} b \tan \relax (x)^{2} - 12 \, a^{2} b^{3} \tan \relax (x)^{2} + 4 \, b^{5} \tan \relax (x)^{2} + 3 \, a^{5} \tan \relax (x) + 10 \, a^{3} b^{2} \tan \relax (x) + 7 \, a b^{4} \tan \relax (x) - 2 \, a^{4} b - 8 \, a^{2} b^{3}}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \relax (x)^{2} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.24, size = 407, normalized size = 3.39 \[ -\frac {b^{5} \ln \left (a \tan \relax (x )+b \right )}{\left (a^{2}+b^{2}\right )^{3}}-\frac {7 \left (\tan ^{3}\relax (x )\right ) b^{2} a^{3}}{4 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}-\frac {9 \left (\tan ^{3}\relax (x )\right ) a \,b^{4}}{8 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}-\frac {5 \left (\tan ^{3}\relax (x )\right ) a^{5}}{8 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}+\frac {\left (\tan ^{2}\relax (x )\right ) b \,a^{4}}{2 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}+\frac {3 \left (\tan ^{2}\relax (x )\right ) a^{2} b^{3}}{2 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}+\frac {\left (\tan ^{2}\relax (x )\right ) b^{5}}{\left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}-\frac {3 \tan \relax (x ) a^{5}}{8 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}-\frac {5 \tan \relax (x ) b^{2} a^{3}}{4 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}-\frac {7 \tan \relax (x ) a \,b^{4}}{8 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}+\frac {b \,a^{4}}{4 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}+\frac {a^{2} b^{3}}{\left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}+\frac {3 b^{5}}{4 \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\relax (x )\right )^{2}}+\frac {b^{5} \ln \left (1+\tan ^{2}\relax (x )\right )}{2 \left (a^{2}+b^{2}\right )^{3}}+\frac {15 \arctan \left (\tan \relax (x )\right ) a \,b^{4}}{8 \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \arctan \left (\tan \relax (x )\right ) a^{5}}{8 \left (a^{2}+b^{2}\right )^{3}}+\frac {5 \arctan \left (\tan \relax (x )\right ) b^{2} a^{3}}{4 \left (a^{2}+b^{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.68, size = 244, normalized size = 2.03 \[ -\frac {b^{5} \log \left (a \tan \relax (x) + b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {b^{5} \log \left (\tan \relax (x)^{2} + 1\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (5 \, a^{3} + 9 \, a b^{2}\right )} \tan \relax (x)^{3} - 2 \, a^{2} b - 6 \, b^{3} - 4 \, {\left (a^{2} b + 2 \, b^{3}\right )} \tan \relax (x)^{2} + {\left (3 \, a^{3} + 7 \, a b^{2}\right )} \tan \relax (x)}{8 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \relax (x)^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \relax (x)^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.50, size = 263, normalized size = 2.19 \[ \frac {\frac {a^2\,b+3\,b^3}{4\,{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {tan}\relax (x)}^3\,\left (5\,a^3+9\,a\,b^2\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\relax (x)}^2\,\left (a^2\,b+2\,b^3\right )}{2\,{\left (a^2+b^2\right )}^2}-\frac {a\,\mathrm {tan}\relax (x)\,\left (3\,a^2+7\,b^2\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{{\mathrm {tan}\relax (x)}^4+2\,{\mathrm {tan}\relax (x)}^2+1}-\frac {b^5\,\ln \left (b+a\,\mathrm {tan}\relax (x)\right )}{{\left (a^2+b^2\right )}^3}+\frac {\ln \left (\mathrm {tan}\relax (x)-\mathrm {i}\right )\,\left (-3\,a^2+a\,b\,9{}\mathrm {i}+8\,b^2\right )}{16\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}+\frac {\ln \left (\mathrm {tan}\relax (x)+1{}\mathrm {i}\right )\,\left (-a^2\,3{}\mathrm {i}+9\,a\,b+b^2\,8{}\mathrm {i}\right )}{16\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{4}{\relax (x )}}{a + b \cot {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________