Optimal. Leaf size=398 \[ \frac{2 \left (-3 a^2 b^2 \left (15 c^2-13 d^2\right )+20 a^3 b c d+2 a^4 d^2-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^3 \sqrt{a+b \tan (e+f x)}}-\frac{2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.36332, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {3565, 3649, 3616, 3615, 93, 208} \[ \frac{2 \left (-3 a^2 b^2 \left (15 c^2-13 d^2\right )+20 a^3 b c d+2 a^4 d^2-100 a b^3 c d+b^4 \left (15 c^2-23 d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^3 \sqrt{a+b \tan (e+f x)}}-\frac{2 (b c-a d) \left (a^2 d+10 a b c+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))^{3/2}}-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^{5/2}}-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a-i b)^{7/2}}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{f (a+i b)^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3565
Rule 3649
Rule 3616
Rule 3615
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^{5/2}}{(a+b \tan (e+f x))^{7/2}} \, dx &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}+\frac{2 \int \frac{\frac{1}{2} \left (11 b^2 c^2 d+a^2 d^3+a b c \left (5 c^2-7 d^2\right )\right )-\frac{5}{2} b \left (b c^3-3 a c^2 d-3 b c d^2+a d^3\right ) \tan (e+f x)+\frac{1}{2} d \left (\left (a^2+5 b^2\right ) d^2-4 b c (b c-2 a d)\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^{5/2} \sqrt{c+d \tan (e+f x)}} \, dx}{5 b \left (a^2+b^2\right )}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}-\frac{4 \int \frac{-\frac{1}{4} (b c-a d) \left (15 a^2 b c^3-15 b^3 c^3+70 a b^2 c^2 d-27 a^2 b c d^2+23 b^3 c d^2+2 a^3 d^3-8 a b^2 d^3\right )+\frac{15}{4} b (b c-a d) \left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right ) \tan (e+f x)+\frac{1}{2} d (b c-a d)^2 \left (10 a b c+a^2 d+11 b^2 d\right ) \tan ^2(e+f x)}{(a+b \tan (e+f x))^{3/2} \sqrt{c+d \tan (e+f x)}} \, dx}{15 b \left (a^2+b^2\right )^2 (b c-a d)}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}+\frac{8 \int \frac{\frac{15}{8} b (b c-a d)^2 (a c+b d) \left (a^2 c^2-3 b^2 c^2+8 a b c d-3 a^2 d^2+b^2 d^2\right )-\frac{15}{8} b (b c-a d)^3 \left (8 a b c d-b^2 \left (c^2-3 d^2\right )+a^2 \left (3 c^2-d^2\right )\right ) \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{15 b \left (a^2+b^2\right )^3 (b c-a d)^2}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}+\frac{(c-i d)^3 \int \frac{1+i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a-i b)^3}+\frac{(c+i d)^3 \int \frac{1-i \tan (e+f x)}{\sqrt{a+b \tan (e+f x)} \sqrt{c+d \tan (e+f x)}} \, dx}{2 (a+i b)^3}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}+\frac{(c-i d)^3 \operatorname{Subst}\left (\int \frac{1}{(1-i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a-i b)^3 f}+\frac{(c+i d)^3 \operatorname{Subst}\left (\int \frac{1}{(1+i x) \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\tan (e+f x)\right )}{2 (a+i b)^3 f}\\ &=-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}+\frac{(c-i d)^3 \operatorname{Subst}\left (\int \frac{1}{i a+b-(i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^3 f}+\frac{(c+i d)^3 \operatorname{Subst}\left (\int \frac{1}{-i a+b-(-i c+d) x^2} \, dx,x,\frac{\sqrt{a+b \tan (e+f x)}}{\sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^3 f}\\ &=-\frac{i (c-i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a-i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a-i b)^{7/2} f}+\frac{i (c+i d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^{7/2} f}-\frac{2 (b c-a d)^2 \sqrt{c+d \tan (e+f x)}}{5 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^{5/2}}-\frac{2 (b c-a d) \left (10 a b c+a^2 d+11 b^2 d\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^2 f (a+b \tan (e+f x))^{3/2}}+\frac{2 \left (20 a^3 b c d-100 a b^3 c d+2 a^4 d^2+b^4 \left (15 c^2-23 d^2\right )-3 a^2 b^2 \left (15 c^2-13 d^2\right )\right ) \sqrt{c+d \tan (e+f x)}}{15 b \left (a^2+b^2\right )^3 f \sqrt{a+b \tan (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.61735, size = 431, normalized size = 1.08 \[ \frac{\frac{5 (d+i c) \left (\frac{3 (c-i d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{-a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{\sqrt{-a+i b}}+\frac{\sqrt{c+d \tan (e+f x)} ((a d+3 b c-4 i b d) \tan (e+f x)+4 a c-3 i a d-i b c)}{(a+b \tan (e+f x))^{3/2}}\right )}{(a-i b)^3}-\frac{5 (-d+i c) \left (\frac{\sqrt{c+d \tan (e+f x)} ((a d+3 b c+4 i b d) \tan (e+f x)+4 a c+3 i a d+i b c)}{(a+i b)^2 (a+b \tan (e+f x))^{3/2}}-\frac{3 (-c-i d)^{3/2} \tan ^{-1}\left (\frac{\sqrt{-c-i d} \sqrt{a+b \tan (e+f x)}}{\sqrt{a+i b} \sqrt{c+d \tan (e+f x)}}\right )}{(a+i b)^{5/2}}\right )}{a+i b}+\frac{3 (c+d \tan (e+f x))^{5/2}}{(-b+i a) (a+b \tan (e+f x))^{5/2}}-\frac{3 (c+d \tan (e+f x))^{5/2}}{(b+i a) (a+b \tan (e+f x))^{5/2}}}{15 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int{ \left ( c+d\tan \left ( fx+e \right ) \right ) ^{{\frac{5}{2}}} \left ( a+b\tan \left ( fx+e \right ) \right ) ^{-{\frac{7}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]