Optimal. Leaf size=196 \[ -\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 d (a+b)^2}+\frac {b \cot (c+d x) \left ((4 a+b) \tan ^2(c+d x)+4 a+5 b\right )}{8 a^2 d (a+b)^2 \left ((a+b) \tan ^2(c+d x)+a\right )}-\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} d (a+b)^{5/2}}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a d (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3187, 468, 577, 453, 205} \[ -\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} d (a+b)^{5/2}}-\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 d (a+b)^2}+\frac {b \cot (c+d x) \left ((4 a+b) \tan ^2(c+d x)+4 a+5 b\right )}{8 a^2 d (a+b)^2 \left ((a+b) \tan ^2(c+d x)+a\right )}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a d (a+b) \left ((a+b) \tan ^2(c+d x)+a\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 453
Rule 468
Rule 577
Rule 3187
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^3}{x^2 \left (a+(a+b) x^2\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right ) \left (-4 a-5 b+(-4 a-b) x^2\right )}{x^2 \left (a+(a+b) x^2\right )^2} \, dx,x,\tan (c+d x)\right )}{4 a (a+b) d}\\ &=\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {b \cot (c+d x) \left (4 a+5 b+(4 a+b) \tan ^2(c+d x)\right )}{8 a^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {(2 a+3 b) (4 a+5 b)+(2 a+b) (4 a+b) x^2}{x^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{8 a^2 (a+b)^2 d}\\ &=-\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {b \cot (c+d x) \left (4 a+5 b+(4 a+b) \tan ^2(c+d x)\right )}{8 a^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}-\frac {\left (3 b \left (8 a^2+12 a b+5 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{8 a^3 (a+b)^2 d}\\ &=-\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{8 a^{7/2} (a+b)^{5/2} d}-\frac {(2 a+3 b) (4 a+5 b) \cot (c+d x)}{8 a^3 (a+b)^2 d}+\frac {b \csc (c+d x) \sec ^3(c+d x)}{4 a (a+b) d \left (a+(a+b) \tan ^2(c+d x)\right )^2}+\frac {b \cot (c+d x) \left (4 a+5 b+(4 a+b) \tan ^2(c+d x)\right )}{8 a^2 (a+b)^2 d \left (a+(a+b) \tan ^2(c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.72, size = 214, normalized size = 1.09 \[ \frac {\csc ^6(c+d x) (-2 a+b \cos (2 (c+d x))-b) \left (\frac {4 a^{3/2} b^2 \sin (2 (c+d x))}{a+b}+\frac {3 b \left (8 a^2+12 a b+5 b^2\right ) (2 a-b \cos (2 (c+d x))+b)^2 \tan ^{-1}\left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{5/2}}+\frac {\sqrt {a} b^2 (10 a+7 b) \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(a+b)^2}+8 \sqrt {a} \cot (c+d x) (2 a-b \cos (2 (c+d x))+b)^2\right )}{64 a^{7/2} d \left (a \csc ^2(c+d x)+b\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.53, size = 1003, normalized size = 5.12 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.21, size = 232, normalized size = 1.18 \[ -\frac {\frac {3 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {a^{2} + a b}} + \frac {12 \, a^{2} b^{2} \tan \left (d x + c\right )^{3} + 19 \, a b^{3} \tan \left (d x + c\right )^{3} + 7 \, b^{4} \tan \left (d x + c\right )^{3} + 12 \, a^{2} b^{2} \tan \left (d x + c\right ) + 9 \, a b^{3} \tan \left (d x + c\right )}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{2}} + \frac {8}{a^{3} \tan \left (d x + c\right )}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.58, size = 367, normalized size = 1.87 \[ -\frac {3 b^{2} \left (\tan ^{3}\left (d x +c \right )\right )}{2 d \left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )^{2} a^{2} \left (a +b \right )}-\frac {7 b^{3} \left (\tan ^{3}\left (d x +c \right )\right )}{8 d \,a^{3} \left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )^{2} \left (a +b \right )}-\frac {3 b^{2} \tan \left (d x +c \right )}{2 d \left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )^{2} a \left (a^{2}+2 a b +b^{2}\right )}-\frac {9 b^{3} \tan \left (d x +c \right )}{8 d \,a^{2} \left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )^{2} \left (a^{2}+2 a b +b^{2}\right )}-\frac {3 \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right ) b}{d \left (a^{2}+2 a b +b^{2}\right ) a \sqrt {a \left (a +b \right )}}-\frac {9 \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right ) b^{2}}{2 d \,a^{2} \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}-\frac {15 b^{3} \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{8 d \,a^{3} \left (a^{2}+2 a b +b^{2}\right ) \sqrt {a \left (a +b \right )}}-\frac {1}{d \,a^{3} \tan \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.51, size = 270, normalized size = 1.38 \[ -\frac {\frac {3 \, {\left (8 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {{\left (8 \, a^{4} + 32 \, a^{3} b + 60 \, a^{2} b^{2} + 51 \, a b^{3} + 15 \, b^{4}\right )} \tan \left (d x + c\right )^{4} + 8 \, a^{4} + 16 \, a^{3} b + 8 \, a^{2} b^{2} + {\left (16 \, a^{4} + 48 \, a^{3} b + 60 \, a^{2} b^{2} + 25 \, a b^{3}\right )} \tan \left (d x + c\right )^{2}}{{\left (a^{7} + 4 \, a^{6} b + 6 \, a^{5} b^{2} + 4 \, a^{4} b^{3} + a^{3} b^{4}\right )} \tan \left (d x + c\right )^{5} + 2 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{7} + 2 \, a^{6} b + a^{5} b^{2}\right )} \tan \left (d x + c\right )}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 15.24, size = 251, normalized size = 1.28 \[ -\frac {\frac {1}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^4\,\left (8\,a^3+24\,a^2\,b+36\,a\,b^2+15\,b^3\right )}{8\,a^3\,\left (a+b\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (16\,a^3+48\,a^2\,b+60\,a\,b^2+25\,b^3\right )}{8\,a^2\,\left (a^2+2\,a\,b+b^2\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^5\,\left (a^2+2\,a\,b+b^2\right )+a^2\,\mathrm {tan}\left (c+d\,x\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2\,a^2+2\,b\,a\right )\right )}-\frac {3\,b\,\mathrm {atan}\left (\frac {3\,b\,\mathrm {tan}\left (c+d\,x\right )\,\left (a^5+2\,a^4\,b+a^3\,b^2\right )\,\left (8\,a^2+12\,a\,b+5\,b^2\right )}{a^{7/2}\,{\left (a+b\right )}^{3/2}\,\left (24\,a^2\,b+36\,a\,b^2+15\,b^3\right )}\right )\,\left (8\,a^2+12\,a\,b+5\,b^2\right )}{8\,a^{7/2}\,d\,{\left (a+b\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [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]
________________________________________________________________________________________