Optimal. Leaf size=185 \[ \frac {b (A b-a B) \tan ^{m+1}(c+d x) \, _2F_1\left (1,m+1;m+2;-\frac {b \tan (c+d x)}{a}\right )}{a d (m+1) \left (a^2+b^2\right )}+\frac {(a A+b B) \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )}{d (m+1) \left (a^2+b^2\right )}-\frac {(A b-a B) \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )}{d (m+2) \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.31, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3613, 3538, 3476, 364, 3634, 64} \[ \frac {(a A+b B) \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )}{d (m+1) \left (a^2+b^2\right )}+\frac {b (A b-a B) \tan ^{m+1}(c+d x) \, _2F_1\left (1,m+1;m+2;-\frac {b \tan (c+d x)}{a}\right )}{a d (m+1) \left (a^2+b^2\right )}-\frac {(A b-a B) \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )}{d (m+2) \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 64
Rule 364
Rule 3476
Rule 3538
Rule 3613
Rule 3634
Rubi steps
\begin {align*} \int \frac {\tan ^m(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=\frac {\int \tan ^m(c+d x) (a A+b B-(A b-a B) \tan (c+d x)) \, dx}{a^2+b^2}+\frac {(b (A b-a B)) \int \frac {\tan ^m(c+d x) \left (1+\tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=-\frac {(A b-a B) \int \tan ^{1+m}(c+d x) \, dx}{a^2+b^2}+\frac {(a A+b B) \int \tan ^m(c+d x) \, dx}{a^2+b^2}+\frac {(b (A b-a B)) \operatorname {Subst}\left (\int \frac {x^m}{a+b x} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {b (A b-a B) \, _2F_1\left (1,1+m;2+m;-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (1+m)}-\frac {(A b-a B) \operatorname {Subst}\left (\int \frac {x^{1+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac {(a A+b B) \operatorname {Subst}\left (\int \frac {x^m}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac {(a A+b B) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{\left (a^2+b^2\right ) d (1+m)}+\frac {b (A b-a B) \, _2F_1\left (1,1+m;2+m;-\frac {b \tan (c+d x)}{a}\right ) \tan ^{1+m}(c+d x)}{a \left (a^2+b^2\right ) d (1+m)}-\frac {(A b-a B) \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{\left (a^2+b^2\right ) d (2+m)}\\ \end {align*}
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Mathematica [A] time = 0.95, size = 144, normalized size = 0.78 \[ \frac {\tan ^{m+1}(c+d x) \left ((a A+b B) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )+\frac {(A b-a B) \left (b (m+2) \, _2F_1\left (1,m+1;m+2;-\frac {b \tan (c+d x)}{a}\right )-a (m+1) \tan (c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )\right )}{a (m+2)}\right )}{d (m+1) \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.80, size = 0, normalized size = 0.00 \[ \int \frac {\left (\tan ^{m}\left (d x +c \right )\right ) \left (A +B \tan \left (d x +c \right )\right )}{a +b \tan \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \tan \left (d x + c\right )^{m}}{b \tan \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{a+b\,\mathrm {tan}\left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}}{a + b \tan {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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