Optimal. Leaf size=152 \[ \frac {\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left (a^2+b^2\right )}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {\cos ^2(c+d x) \left (a \left (3 a^2+7 b^2\right ) \tan (c+d x)+4 b^3\right )}{8 d \left (a^2+b^2\right )^2}+\frac {a x \left (3 a^4+10 a^2 b^2+15 b^4\right )}{8 \left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.20, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3506, 741, 823, 801, 635, 203, 260} \[ \frac {\cos ^4(c+d x) (a \tan (c+d x)+b)}{4 d \left (a^2+b^2\right )}+\frac {\cos ^2(c+d x) \left (a \left (3 a^2+7 b^2\right ) \tan (c+d x)+4 b^3\right )}{8 d \left (a^2+b^2\right )^2}+\frac {b^5 \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {a x \left (10 a^2 b^2+3 a^4+15 b^4\right )}{8 \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 741
Rule 801
Rule 823
Rule 3506
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x) \left (1+\frac {x^2}{b^2}\right )^3} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}-\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 \tan (c+d x)\right )}{4 \left (a^2+b^2\right ) d}\\ &=\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}+\frac {\cos ^2(c+d x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\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 \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ &=\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}+\frac {\cos ^2(c+d x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\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 \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ &=\frac {b^5 \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}+\frac {\cos ^2(c+d x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}+\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 \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}\\ &=\frac {b^5 \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}+\frac {\cos ^2(c+d x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}-\frac {b^5 \operatorname {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^3 d}+\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 \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^3 d}\\ &=\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 (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {b^5 \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {\cos ^4(c+d x) (b+a \tan (c+d x))}{4 \left (a^2+b^2\right ) d}+\frac {\cos ^2(c+d x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \tan (c+d x)\right )}{8 \left (a^2+b^2\right )^2 d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 218, normalized size = 1.43 \[ \frac {8 a^5 \sin (2 (c+d x))+a^5 \sin (4 (c+d x))+12 a^5 c+12 a^5 d x+24 a^3 b^2 \sin (2 (c+d x))+2 a^3 b^2 \sin (4 (c+d x))+40 a^3 b^2 c+40 a^3 b^2 d x+b \left (a^2+b^2\right )^2 \cos (4 (c+d x))+4 b \left (a^4+4 a^2 b^2+3 b^4\right ) \cos (2 (c+d x))+32 b^5 \log (a \cos (c+d x)+b \sin (c+d x))+16 a b^4 \sin (2 (c+d x))+a b^4 \sin (4 (c+d x))+60 a b^4 c+60 a b^4 d x}{32 d \left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 208, normalized size = 1.37 \[ \frac {4 \, b^{5} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) + 2 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} d x + 4 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.63, size = 322, normalized size = 2.12 \[ \frac {\frac {8 \, b^{6} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {4 \, b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {6 \, b^{5} \tan \left (d x + c\right )^{4} + 3 \, a^{5} \tan \left (d x + c\right )^{3} + 10 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} + 7 \, a b^{4} \tan \left (d x + c\right )^{3} + 4 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} + 16 \, b^{5} \tan \left (d x + c\right )^{2} + 5 \, a^{5} \tan \left (d x + c\right ) + 14 \, a^{3} b^{2} \tan \left (d x + c\right ) + 9 \, a b^{4} \tan \left (d x + c\right ) + 2 \, a^{4} b + 8 \, a^{2} b^{3} + 12 \, b^{5}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 524, normalized size = 3.45 \[ \frac {b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \left (\tan ^{3}\left (d x +c \right )\right ) a^{5}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {5 \left (\tan ^{3}\left (d x +c \right )\right ) b^{2} a^{3}}{4 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {7 \left (\tan ^{3}\left (d x +c \right )\right ) a \,b^{4}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{3}}{2 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {\left (\tan ^{2}\left (d x +c \right )\right ) b^{5}}{2 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {7 \tan \left (d x +c \right ) b^{2} a^{3}}{4 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {9 \tan \left (d x +c \right ) a \,b^{4}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {5 \tan \left (d x +c \right ) a^{5}}{8 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {a^{4} b}{4 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {a^{2} b^{3}}{d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}+\frac {3 b^{5}}{4 d \left (a^{2}+b^{2}\right )^{3} \left (1+\tan ^{2}\left (d x +c \right )\right )^{2}}-\frac {b^{5} \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {15 \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{4}}{8 d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \arctan \left (\tan \left (d x +c \right )\right ) a^{5}}{8 d \left (a^{2}+b^{2}\right )^{3}}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right ) b^{2} a^{3}}{4 d \left (a^{2}+b^{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 271, normalized size = 1.78 \[ \frac {\frac {8 \, b^{5} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, b^{5} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {4 \, b^{3} \tan \left (d x + c\right )^{2} + {\left (3 \, a^{3} + 7 \, a b^{2}\right )} \tan \left (d x + c\right )^{3} + 2 \, a^{2} b + 6 \, b^{3} + {\left (5 \, a^{3} + 9 \, a b^{2}\right )} \tan \left (d x + c\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \tan \left (d x + c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.20, size = 318, normalized size = 2.09 \[ \frac {\frac {a^2\,b+3\,b^3}{4\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {b^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (3\,a^3+7\,a\,b^2\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (5\,a^3+9\,a\,b^2\right )}{8\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4+2\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-a^2\,3{}\mathrm {i}+9\,a\,b+b^2\,8{}\mathrm {i}\right )}{16\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {b^5\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-3\,a^2+a\,b\,9{}\mathrm {i}+8\,b^2\right )}{16\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cos ^{4}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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