Optimal. Leaf size=227 \[ \frac {2 a^2 (A-i B) \sqrt {1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac {1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{d (m+1) \sqrt {a+i a \tan (c+d x)}}+\frac {2 a (B+(2 m+3) (B+i A)) \sqrt {a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};i \tan (c+d x)+1\right )}{d (2 m+3)}+\frac {2 i a B \sqrt {a+i a \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.70, antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3594, 3601, 3564, 135, 133, 3599, 67, 65} \[ \frac {2 a^2 (A-i B) \sqrt {1+i \tan (c+d x)} \tan ^{m+1}(c+d x) F_1\left (m+1;\frac {1}{2},1;m+2;-i \tan (c+d x),i \tan (c+d x)\right )}{d (m+1) \sqrt {a+i a \tan (c+d x)}}+\frac {2 a (B+(2 m+3) (B+i A)) \sqrt {a+i a \tan (c+d x)} \tan ^m(c+d x) (-i \tan (c+d x))^{-m} \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};i \tan (c+d x)+1\right )}{d (2 m+3)}+\frac {2 i a B \sqrt {a+i a \tan (c+d x)} \tan ^{m+1}(c+d x)}{d (2 m+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 65
Rule 67
Rule 133
Rule 135
Rule 3564
Rule 3594
Rule 3599
Rule 3601
Rubi steps
\begin {align*} \int \tan ^m(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=\frac {2 i a B \tan ^{1+m}(c+d x) \sqrt {a+i a \tan (c+d x)}}{d (3+2 m)}+\frac {2 \int \tan ^m(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {1}{2} a \left (2 i B (1+m)-2 A \left (\frac {3}{2}+m\right )\right )+\frac {1}{2} a (B+(i A+B) (3+2 m)) \tan (c+d x)\right ) \, dx}{3+2 m}\\ &=\frac {2 i a B \tan ^{1+m}(c+d x) \sqrt {a+i a \tan (c+d x)}}{d (3+2 m)}+(2 a (A-i B)) \int \tan ^m(c+d x) \sqrt {a+i a \tan (c+d x)} \, dx+\frac {(i (B+(i A+B) (3+2 m))) \int \tan ^m(c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{3+2 m}\\ &=\frac {2 i a B \tan ^{1+m}(c+d x) \sqrt {a+i a \tan (c+d x)}}{d (3+2 m)}+\frac {\left (2 a^3 (i A+B)\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^m}{\sqrt {a+x} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d}+\frac {\left (i a^2 (B+(i A+B) (3+2 m))\right ) \operatorname {Subst}\left (\int \frac {x^m}{\sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{d (3+2 m)}\\ &=\frac {2 i a B \tan ^{1+m}(c+d x) \sqrt {a+i a \tan (c+d x)}}{d (3+2 m)}+\frac {\left (i a^2 (B+(i A+B) (3+2 m)) (-i \tan (c+d x))^{-m} \tan ^m(c+d x)\right ) \operatorname {Subst}\left (\int \frac {(-i x)^m}{\sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{d (3+2 m)}+\frac {\left (2 a^3 (i A+B) \sqrt {1+i \tan (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {i x}{a}\right )^m}{\sqrt {1+\frac {x}{a}} \left (-a^2+a x\right )} \, dx,x,i a \tan (c+d x)\right )}{d \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {2 a^2 (A-i B) F_1\left (1+m;\frac {1}{2},1;2+m;-i \tan (c+d x),i \tan (c+d x)\right ) \sqrt {1+i \tan (c+d x)} \tan ^{1+m}(c+d x)}{d (1+m) \sqrt {a+i a \tan (c+d x)}}+\frac {2 a (B+(i A+B) (3+2 m)) \, _2F_1\left (\frac {1}{2},-m;\frac {3}{2};1+i \tan (c+d x)\right ) (-i \tan (c+d x))^{-m} \tan ^m(c+d x) \sqrt {a+i a \tan (c+d x)}}{d (3+2 m)}+\frac {2 i a B \tan ^{1+m}(c+d x) \sqrt {a+i a \tan (c+d x)}}{d (3+2 m)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 5.32, size = 0, normalized size = 0.00 \[ \int \tan ^m(c+d x) (a+i a \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, \sqrt {2} {\left ({\left (A - i \, B\right )} a e^{\left (5 i \, d x + 5 i \, c\right )} + {\left (A + i \, B\right )} a e^{\left (3 i \, d x + 3 i \, c\right )}\right )} \left (\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{m} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 3.77, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{m}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (d x +c \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \tan {\left (c + d x \right )}\right ) \tan ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________