Optimal. Leaf size=173 \[ \frac {(A+i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {(A-i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A] time = 0.46, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4241, 3603, 3602, 130, 511, 510} \[ \frac {(A+i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {(A-i B) (a+b \tan (c+d x))^n \left (\frac {b \tan (c+d x)}{a}+1\right )^{-n} F_1\left (\frac {3}{2};1,-n;\frac {5}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 130
Rule 510
Rule 511
Rule 3602
Rule 3603
Rule 4241
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx &=\left (\sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx\\ &=\frac {1}{2} \left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int (1+i \tan (c+d x)) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \, dx+\frac {1}{2} \left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \int (1-i \tan (c+d x)) \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \, dx\\ &=\frac {\left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x} (a+b x)^n}{1-i x} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {\left ((A+i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {x} (a+b x)^n}{1+i x} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {\left ((A-i B) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^2 \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) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {x^2 \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) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {x^2 \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) \sqrt {\cot (c+d x)} \sqrt {\tan (c+d x)} (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}\right ) \operatorname {Subst}\left (\int \frac {x^2 \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 {3}{2};1,-n;\frac {5}{2};-i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {(A-i B) F_1\left (\frac {3}{2};1,-n;\frac {5}{2};i \tan (c+d x),-\frac {b \tan (c+d x)}{a}\right ) (a+b \tan (c+d x))^n \left (1+\frac {b \tan (c+d x)}{a}\right )^{-n}}{3 d \cot ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [F] time = 13.31, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \tan (c+d x))^n (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}}, x\right ) \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.53, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )}{\sqrt {\cot \left (d x +c \right )}}\, dx \]
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 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}}{\sqrt {\cot \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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