Optimal. Leaf size=213 \[ \frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 (3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{15 a c^2 f \sqrt {c-i c \tan (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.19, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47,
37} \begin {gather*} -\frac {2 (-2 B+3 i A) \sqrt {a+i a \tan (e+f x)}}{15 a c^2 f \sqrt {c-i c \tan (e+f x)}}-\frac {2 (-2 B+3 i A) \sqrt {a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {(-2 B+3 i A) \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac {-B+i A}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 37
Rule 47
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {A+B \tan (e+f x)}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(a+i a x)^{3/2} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}+\frac {((3 A+2 i B) c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}+\frac {(2 (3 A+2 i B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}+\frac {(2 (3 A+2 i B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{15 c f}\\ &=\frac {i A-B}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}-\frac {(3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{5 a f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{15 a c f (c-i c \tan (e+f x))^{3/2}}-\frac {2 (3 i A-2 B) \sqrt {a+i a \tan (e+f x)}}{15 a c^2 f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.01, size = 128, normalized size = 0.60 \begin {gather*} \frac {(\cos (3 (e+f x))+i \sin (3 (e+f x))) (5 (-6 i A+B) \cos (e+f x)+(6 i A-9 B) \cos (3 (e+f x))+(3 A+2 i B) (-5 \sin (e+f x)+3 \sin (3 (e+f x)))) \sqrt {c-i c \tan (e+f x)}}{60 c^3 f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.40, size = 184, normalized size = 0.86
method | result | size |
risch | \(-\frac {3 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+3 B \,{\mathrm e}^{6 i \left (f x +e \right )}+15 i A \,{\mathrm e}^{4 i \left (f x +e \right )}+5 B \,{\mathrm e}^{4 i \left (f x +e \right )}+45 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-15 B \,{\mathrm e}^{2 i \left (f x +e \right )}-15 i A +15 B}{120 c^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) f}\) | \(148\) |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (4 i B \left (\tan ^{5}\left (f x +e \right )\right )+12 i A \left (\tan ^{4}\left (f x +e \right )\right )+6 A \left (\tan ^{5}\left (f x +e \right )\right )+2 i B \left (\tan ^{3}\left (f x +e \right )\right )-8 B \left (\tan ^{4}\left (f x +e \right )\right )+18 i A \left (\tan ^{2}\left (f x +e \right )\right )+3 A \left (\tan ^{3}\left (f x +e \right )\right )-2 i B \tan \left (f x +e \right )-7 B \left (\tan ^{2}\left (f x +e \right )\right )+6 i A -3 A \tan \left (f x +e \right )+B \right )}{15 f a \,c^{3} \left (i+\tan \left (f x +e \right )\right )^{4} \left (i-\tan \left (f x +e \right )\right )^{2}}\) | \(184\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (4 i B \left (\tan ^{5}\left (f x +e \right )\right )+12 i A \left (\tan ^{4}\left (f x +e \right )\right )+6 A \left (\tan ^{5}\left (f x +e \right )\right )+2 i B \left (\tan ^{3}\left (f x +e \right )\right )-8 B \left (\tan ^{4}\left (f x +e \right )\right )+18 i A \left (\tan ^{2}\left (f x +e \right )\right )+3 A \left (\tan ^{3}\left (f x +e \right )\right )-2 i B \tan \left (f x +e \right )-7 B \left (\tan ^{2}\left (f x +e \right )\right )+6 i A -3 A \tan \left (f x +e \right )+B \right )}{15 f a \,c^{3} \left (i+\tan \left (f x +e \right )\right )^{4} \left (i-\tan \left (f x +e \right )\right )^{2}}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.41, size = 170, normalized size = 0.80 \begin {gather*} -\frac {{\left (3 \, {\left (i \, A + B\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + 2 \, {\left (9 i \, A + 4 \, B\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, {\left (6 i \, A - B\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + 8 \, {\left (-6 i \, A - B\right )} e^{\left (3 i \, f x + 3 i \, e\right )} + 30 i \, A e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (-6 i \, A - B\right )} e^{\left (i \, f x + i \, e\right )} - 15 i \, A + 15 \, B\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-i \, f x - i \, e\right )}}{120 \, a c^{3} f} \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*} \int \frac {A + B \tan {\left (e + f x \right )}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {5}{2}}}\, dx \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 = 9.89, size = 186, normalized size = 0.87 \begin {gather*} -\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,45{}\mathrm {i}-15\,B+A\,\cos \left (4\,e+4\,f\,x\right )\,3{}\mathrm {i}+20\,B\,\cos \left (2\,e+2\,f\,x\right )+3\,B\,\cos \left (4\,e+4\,f\,x\right )-30\,A\,\sin \left (2\,e+2\,f\,x\right )-3\,A\,\sin \left (4\,e+4\,f\,x\right )-B\,\sin \left (2\,e+2\,f\,x\right )\,10{}\mathrm {i}+B\,\sin \left (4\,e+4\,f\,x\right )\,3{}\mathrm {i}\right )}{120\,a\,c^2\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________