Optimal. Leaf size=105 \[ \frac {4 a^2 (i A+B) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a^2 (i A+3 B) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{11/2}}{11 c^2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.047, Rules used = {3669, 78}
\begin {gather*} -\frac {2 a^2 (3 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {4 a^2 (B+i A) (c-i c \tan (e+f x))^{7/2}}{7 f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{11/2}}{11 c^2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 3669
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx &=\frac {(a c) \text {Subst}\left (\int (a+i a x) (A+B x) (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (2 a (A-i B) (c-i c x)^{5/2}-\frac {a (A-3 i B) (c-i c x)^{7/2}}{c}-\frac {i a B (c-i c x)^{9/2}}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {4 a^2 (i A+B) (c-i c \tan (e+f x))^{7/2}}{7 f}-\frac {2 a^2 (i A+3 B) (c-i c \tan (e+f x))^{9/2}}{9 c f}+\frac {2 a^2 B (c-i c \tan (e+f x))^{11/2}}{11 c^2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.21, size = 119, normalized size = 1.13 \begin {gather*} \frac {a^2 c^3 \sec ^5(e+f x) (121 i A-33 B+(121 i A+93 B) \cos (2 (e+f x))+(-77 A+105 i B) \sin (2 (e+f x))) (\cos (3 e+f x)-i \sin (3 e+f x)) \sqrt {c-i c \tan (e+f x)}}{693 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.44, size = 83, normalized size = 0.79
method | result | size |
derivativedivides | \(-\frac {2 i a^{2} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-3 i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {2 \left (-i B c +A c \right ) c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}\right )}{f \,c^{2}}\) | \(83\) |
default | \(-\frac {2 i a^{2} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-3 i B c +A c \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {2 \left (-i B c +A c \right ) c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}\right )}{f \,c^{2}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 81, normalized size = 0.77 \begin {gather*} -\frac {2 i \, {\left (63 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}} B a^{2} + 77 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} {\left (A - 3 i \, B\right )} a^{2} c - 198 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - i \, B\right )} a^{2} c^{2}\right )}}{693 \, c^{2} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 4.42, size = 157, normalized size = 1.50 \begin {gather*} -\frac {32 \, \sqrt {2} {\left (99 \, {\left (-i \, A - B\right )} a^{2} c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} + 11 \, {\left (-11 i \, A + 3 \, B\right )} a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + 2 \, {\left (-11 i \, A + 3 \, B\right )} a^{2} c^{3}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{693 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - a^{2} \left (\int \left (- A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c}\right )\, dx + \int \left (- 2 A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\right )\, dx + \int i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\, dx + \int 2 i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx + \int i A c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\, dx + \int i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int 2 i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx + \int i B c^{3} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{6}{\left (e + f x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 13.89, size = 132, normalized size = 1.26 \begin {gather*} \frac {32\,a^2\,c^3\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,\left (A\,22{}\mathrm {i}-6\,B+A\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,121{}\mathrm {i}+A\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,99{}\mathrm {i}-33\,B\,{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+99\,B\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\right )}{693\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________