Optimal. Leaf size=202 \[ \frac {a b \left (a^2-11 b^2\right )}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {b \left (a^2-2 b^2\right )}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 b^3 \left (5 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {a x \left (a^4+10 a^2 b^2-15 b^4\right )}{2 \left (a^2+b^2\right )^4} \]
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Rubi [A] time = 0.23, antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3506, 741, 801, 635, 203, 260} \[ \frac {a b \left (a^2-11 b^2\right )}{2 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))}+\frac {b \left (a^2-2 b^2\right )}{2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))^2}+\frac {\cos ^2(c+d x) (a \tan (c+d x)+b)}{2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {2 b^3 \left (5 a^2-b^2\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^4}+\frac {a x \left (10 a^2 b^2+a^4-15 b^4\right )}{2 \left (a^2+b^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 203
Rule 260
Rule 635
Rule 741
Rule 801
Rule 3506
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x)}{(a+b \tan (c+d x))^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \operatorname {Subst}\left (\int \frac {-4-\frac {a^2}{b^2}-\frac {3 a x}{b^2}}{(a+x)^3 \left (1+\frac {x^2}{b^2}\right )} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac {\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {b \operatorname {Subst}\left (\int \left (\frac {2 \left (a^2-2 b^2\right )}{\left (a^2+b^2\right ) (a+x)^3}+\frac {a^3-11 a b^2}{\left (a^2+b^2\right )^2 (a+x)^2}+\frac {4 b^2 \left (-5 a^2+b^2\right )}{\left (a^2+b^2\right )^3 (a+x)}+\frac {-a \left (a^4+10 a^2 b^2-15 b^4\right )+4 b^2 \left (5 a^2-b^2\right ) x}{\left (a^2+b^2\right )^3 \left (b^2+x^2\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac {2 b^3 \left (5 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {b \left (a^2-2 b^2\right )}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (a^2-11 b^2\right )}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {b \operatorname {Subst}\left (\int \frac {-a \left (a^4+10 a^2 b^2-15 b^4\right )+4 b^2 \left (5 a^2-b^2\right ) x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^4 d}\\ &=\frac {2 b^3 \left (5 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {b \left (a^2-2 b^2\right )}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (a^2-11 b^2\right )}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}-\frac {\left (2 b^3 \left (5 a^2-b^2\right )\right ) \operatorname {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{\left (a^2+b^2\right )^4 d}+\frac {\left (a b \left (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 )}{2 \left (a^2+b^2\right )^4 d}\\ &=\frac {a \left (a^4+10 a^2 b^2-15 b^4\right ) x}{2 \left (a^2+b^2\right )^4}+\frac {2 b^3 \left (5 a^2-b^2\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {2 b^3 \left (5 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^4 d}+\frac {b \left (a^2-2 b^2\right )}{2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))^2}+\frac {\cos ^2(c+d x) (b+a \tan (c+d x))}{2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a b \left (a^2-11 b^2\right )}{2 \left (a^2+b^2\right )^3 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [B] time = 6.28, size = 458, normalized size = 2.27 \[ \frac {b^3 \left (\frac {\cos ^2(c+d x) \left (a b \tan (c+d x)+b^2\right )}{2 b^4 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\left (2 a^2-4 b^2\right ) \left (-\frac {2 a}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {1}{2 \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {\left (3 a^2-b^2\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}-\frac {\left (-\frac {a^3-3 a b^2}{\sqrt {-b^2}}+3 a^2-b^2\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}-\frac {\left (\frac {a^3-3 a b^2}{\sqrt {-b^2}}+3 a^2-b^2\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^3}\right )-3 a \left (-\frac {1}{\left (a^2+b^2\right ) (a+b \tan (c+d x))}-\frac {\left (2 a-\frac {a^2-b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}+\frac {2 a \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^2}-\frac {\left (\frac {a^2-b^2}{\sqrt {-b^2}}+2 a\right ) \log \left (\sqrt {-b^2}+b \tan (c+d x)\right )}{2 \left (a^2+b^2\right )^2}\right )}{2 b^2 \left (a^2+b^2\right )}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.12, size = 503, normalized size = 2.49 \[ -\frac {3 \, a^{4} b^{3} - 16 \, a^{2} b^{5} + b^{7} - 2 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{5} b^{2} + 10 \, a^{3} b^{4} - 15 \, a b^{6}\right )} d x - {\left (a^{6} b - a^{4} b^{3} - 45 \, a^{2} b^{5} - 3 \, b^{7} + 2 \, {\left (a^{7} + 9 \, a^{5} b^{2} - 25 \, a^{3} b^{4} + 15 \, a b^{6}\right )} d x\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (5 \, a^{2} b^{5} - b^{7} + {\left (5 \, a^{4} b^{3} - 6 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (5 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 2 \, {\left ({\left (a^{7} + 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{5} b^{2} - 3 \, a^{3} b^{4} + 6 \, a b^{6} - {\left (a^{6} b + 10 \, a^{4} b^{3} - 15 \, a^{2} b^{5}\right )} d x\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{10} + 3 \, a^{8} b^{2} + 2 \, a^{6} b^{4} - 2 \, a^{4} b^{6} - 3 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{9} b + 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} + 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{8} b^{2} + 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} + 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.14, size = 439, normalized size = 2.17 \[ \frac {\frac {{\left (a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {2 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {4 \, {\left (5 \, a^{2} b^{4} - b^{6}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{8} b + 4 \, a^{6} b^{3} + 6 \, a^{4} b^{5} + 4 \, a^{2} b^{7} + b^{9}} + \frac {10 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} - 2 \, b^{5} \tan \left (d x + c\right )^{2} + a^{5} \tan \left (d x + c\right ) - 2 \, a^{3} b^{2} \tan \left (d x + c\right ) - 3 \, a b^{4} \tan \left (d x + c\right ) + 3 \, a^{4} b + 12 \, a^{2} b^{3} - 3 \, b^{5}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (\tan \left (d x + c\right )^{2} + 1\right )}} - \frac {30 \, a^{2} b^{5} \tan \left (d x + c\right )^{2} - 6 \, b^{7} \tan \left (d x + c\right )^{2} + 68 \, a^{3} b^{4} \tan \left (d x + c\right ) - 4 \, a b^{6} \tan \left (d x + c\right ) + 39 \, a^{4} b^{3} + 4 \, a^{2} b^{5} + b^{7}}{{\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 453, normalized size = 2.24 \[ -\frac {b^{3}}{2 d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {4 b^{3} a}{d \left (a^{2}+b^{2}\right )^{3} \left (a +b \tan \left (d x +c \right )\right )}+\frac {10 b^{3} \ln \left (a +b \tan \left (d x +c \right )\right ) a^{2}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {2 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {\tan \left (d x +c \right ) a^{5}}{2 d \left (a^{2}+b^{2}\right )^{4} \left (1+\tan ^{2}\left (d x +c \right )\right )}-\frac {\tan \left (d x +c \right ) b^{2} a^{3}}{d \left (a^{2}+b^{2}\right )^{4} \left (1+\tan ^{2}\left (d x +c \right )\right )}-\frac {3 \tan \left (d x +c \right ) a \,b^{4}}{2 d \left (a^{2}+b^{2}\right )^{4} \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {3 a^{4} b}{2 d \left (a^{2}+b^{2}\right )^{4} \left (1+\tan ^{2}\left (d x +c \right )\right )}+\frac {a^{2} b^{3}}{d \left (a^{2}+b^{2}\right )^{4} \left (1+\tan ^{2}\left (d x +c \right )\right )}-\frac {b^{5}}{2 d \left (a^{2}+b^{2}\right )^{4} \left (1+\tan ^{2}\left (d x +c \right )\right )}-\frac {5 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b^{3}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{5}}{d \left (a^{2}+b^{2}\right )^{4}}+\frac {5 \arctan \left (\tan \left (d x +c \right )\right ) b^{2} a^{3}}{d \left (a^{2}+b^{2}\right )^{4}}-\frac {15 \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{4}}{2 d \left (a^{2}+b^{2}\right )^{4}}+\frac {\arctan \left (\tan \left (d x +c \right )\right ) a^{5}}{2 d \left (a^{2}+b^{2}\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 458, normalized size = 2.27 \[ \frac {\frac {{\left (a^{5} + 10 \, a^{3} b^{2} - 15 \, a b^{4}\right )} {\left (d x + c\right )}}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {4 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} - \frac {2 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}} + \frac {3 \, a^{4} b - 10 \, a^{2} b^{3} - b^{5} + {\left (a^{3} b^{2} - 11 \, a b^{4}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{4} b - 6 \, a^{2} b^{3} - b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{5} + 3 \, a^{3} b^{2} - 10 \, a b^{4}\right )} \tan \left (d x + c\right )}{a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6} + {\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{8} + 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} + 4 \, a^{2} b^{6} + b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} \tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 419, normalized size = 2.07 \[ \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {10\,b^3}{{\left (a^2+b^2\right )}^3}-\frac {12\,b^5}{{\left (a^2+b^2\right )}^4}\right )}{d}-\frac {\frac {-3\,a^4\,b+10\,a^2\,b^3+b^5}{2\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (11\,a\,b^4-a^3\,b^2\right )}{2\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-a^4\,b+6\,a^2\,b^3+b^5\right )}{\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {a\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^4+3\,a^2\,b^2-10\,b^4\right )}{2\,\left (a^2+b^2\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (a^2+b^2\right )+a^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (b+\frac {a\,1{}\mathrm {i}}{4}\right )}{d\,\left (a^4-a^3\,b\,4{}\mathrm {i}-6\,a^2\,b^2+a\,b^3\,4{}\mathrm {i}+b^4\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (a+b\,4{}\mathrm {i}\right )}{4\,d\,\left (a^4\,1{}\mathrm {i}-4\,a^3\,b-a^2\,b^2\,6{}\mathrm {i}+4\,a\,b^3+b^4\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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