Optimal. Leaf size=167 \[ \frac {(A+i B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {1}{2};1,-n;\frac {3}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d}+\frac {(A-i B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {1}{2};1,-n;\frac {3}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d} \]
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Rubi [A] time = 0.33, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3603, 3602, 130, 430, 429} \[ \frac {(A+i B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {1}{2};1,-n;\frac {3}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d}+\frac {(A-i B) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {1}{2};1,-n;\frac {3}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 130
Rule 429
Rule 430
Rule 3602
Rule 3603
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx &=\frac {1}{2} (A-i B) \int \frac {(1+i \tan (c+d x)) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx+\frac {1}{2} (A+i B) \int \frac {(1-i \tan (c+d x)) (a+b \tan (c+d x))^n}{\sqrt {\tan (c+d x)}} \, dx\\ &=\frac {(A-i B) \operatorname {Subst}\left (\int \frac {(a+b x)^n}{(1-i x) \sqrt {x}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {(A+i B) \operatorname {Subst}\left (\int \frac {(a+b x)^n}{(1+i x) \sqrt {x}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {(A-i B) \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^n}{1-i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {(A+i B) \operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^n}{1+i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {\left ((A-i B) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^n}{1-i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}+\frac {\left ((A+i B) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {b x^2}{a}\right )^n}{1+i x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}\\ &=\frac {(A+i B) F_1\left (\frac {1}{2};1,-n;\frac {3}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d}+\frac {(A-i B) F_1\left (\frac {1}{2};1,-n;\frac {3}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{d}\\ \end {align*}
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Mathematica [F] time = 1.21, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\tan \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.38, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )}{\sqrt {\tan \left (d x +c \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{\sqrt {\mathrm {tan}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n}}{\sqrt {\tan {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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