Optimal. Leaf size=179 \[ \frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac {3}{2},1;m+2;-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 a d (m+1) \sqrt {a+b \tan (c+d x)}}+\frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac {3}{2},1;m+2;-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 a d (m+1) \sqrt {a+b \tan (c+d x)}} \]
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Rubi [A] time = 0.20, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3575, 912, 135, 133} \[ \frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac {3}{2},1;m+2;-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right )}{2 a d (m+1) \sqrt {a+b \tan (c+d x)}}+\frac {\tan ^{m+1}(c+d x) \sqrt {\frac {b \tan (c+d x)}{a}+1} F_1\left (m+1;\frac {3}{2},1;m+2;-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right )}{2 a d (m+1) \sqrt {a+b \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 133
Rule 135
Rule 912
Rule 3575
Rubi steps
\begin {align*} \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^m}{(a+b x)^{3/2} \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {i x^m}{2 (i-x) (a+b x)^{3/2}}+\frac {i x^m}{2 (i+x) (a+b x)^{3/2}}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i \operatorname {Subst}\left (\int \frac {x^m}{(i-x) (a+b x)^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}+\frac {i \operatorname {Subst}\left (\int \frac {x^m}{(i+x) (a+b x)^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 d}\\ &=\frac {\left (i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {x^m}{(i-x) \left (1+\frac {b x}{a}\right )^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 a d \sqrt {a+b \tan (c+d x)}}+\frac {\left (i \sqrt {1+\frac {b \tan (c+d x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {x^m}{(i+x) \left (1+\frac {b x}{a}\right )^{3/2}} \, dx,x,\tan (c+d x)\right )}{2 a d \sqrt {a+b \tan (c+d x)}}\\ &=\frac {F_1\left (1+m;\frac {3}{2},1;2+m;-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 a d (1+m) \sqrt {a+b \tan (c+d x)}}+\frac {F_1\left (1+m;\frac {3}{2},1;2+m;-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \tan ^{1+m}(c+d x) \sqrt {1+\frac {b \tan (c+d x)}{a}}}{2 a d (1+m) \sqrt {a+b \tan (c+d x)}}\\ \end {align*}
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Mathematica [F] time = 19.12, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^m(c+d x)}{(a+b \tan (c+d x))^{3/2}} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 2.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{m}}{b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{m}\left (d x +c \right )}{\left (a +b \tan \left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\tan \left (d x + c\right )^{m}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\mathrm {tan}\left (c+d\,x\right )}^m}{{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\tan ^{m}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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